Cross Subspace Alignment and the Asymptotic Capacity of X–Secure T–Private Information Retrieval

Cross Subspace Alignment and the Asymptotic Capacity of X–Secure T–Private Information Retrieval

Zhuqing Jia{}^{\dagger} {}^{\dagger}Center for Pervasive Communications and Computing (CPCC), UC Irvine
{}^{*}Department of Electrical Engineering, University of North Texas
Email: {zhuqingj}@uci.edu, {hua.sun}@unt.edu, {syed}@uci.edu
Hua Sun{}^{*} and Syed A. Jafar{}^{\dagger} {}^{\dagger}Center for Pervasive Communications and Computing (CPCC), UC Irvine
{}^{*}Department of Electrical Engineering, University of North Texas
Email: {zhuqingj}@uci.edu, {hua.sun}@unt.edu, {syed}@uci.edu
Abstract

X–secure and T–private information retrieval (XSTPIR) is a form of private information retrieval where data security is guaranteed against collusion among up to X servers and the user’s privacy is guaranteed against collusion among up to T servers. The capacity of XSTPIR is characterized for arbitrary number of servers N, and arbitrary security and privacy thresholds X and T, in the limit as the number of messages K\rightarrow\infty. Capacity is also characterized for any number of messages if either N=3,X=T=1 or if N\leq X+T. Insights are drawn from these results, about aligning versus decoding noise, dependence of PIR rate on field size, and robustness to symmetric security constraints. In particular, the idea of cross subspace alignment, i.e., introducing a subspace dependence between Reed-Solomon code parameters, emerges as the optimal way to align undesired terms while keeping desired terms resolvable.

1 Introduction

Motivated by the importance of security and privacy in the era of big data and distributed storage, in this work we explore the information theoretic capacity of private information retrieval (PIR) in a secure distributed storage system. Specifically, our focus is on the X-secure and T-private information retrieval problem (XSTPIR). A PIR scheme is said to be T-private if it allows a user to retrieve a desired message from a database of K messages stored at N distributed servers, without revealing any information about the identity of the desired message to any group of up to T colluding servers. Similarly, a distributed storage scheme is said to be X-secure if it guarantees that any group of up to X colluding servers learn nothing about the stored data. The T and X parameters may be chosen arbitrarily depending on the relative importance of security and privacy for any given application.

The rate of a PIR scheme is the ratio of the number of bits retrieved by the user to the total number of bits downloaded from all servers. The supremum of achievable rates is called the capacity of PIR. The capacity of the basic PIR setting was found in [1] to be

\displaystyle C_{\text{PIR}}(N,K) \displaystyle=(1+1/N+1/N^{2}+\dots+1/N^{K-1})^{-1}. (1)

The result was generalized subsequently in [2] to the T-PIR setting, as

C_{\mbox{\tiny TPIR}}(N,K,T)=\left\{\begin{array}[]{ll}\left(1+T/N+T^{2}/N^{2}% +\dots+T^{K-1}/N^{K-1}\right)^{-1},&T<N\\ 1/K,&T\geq N.\end{array}\right. (2)

Further generalizations of T-privacy, e.g., when privacy is required only against certain specified collusion patterns [3, 4] have also been explored. In particular, capacity is known for disjoint colluding sets [4].

The rapidly growing body of literature in this area has produced capacity results for PIR under a rich variety of constraints [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. However, the capacity for the natural setting of secure storage remains unknown, and relatively unexplored. While a number of efforts are motivated by security concerns, such efforts have focused largely on other models, e.g., wiretap models where data security is desired against eavesdroppers listening to the communication between the user and the servers [21, 22], Byzantine models where the servers may respond incorrectly by introducing erasures or errors in their response to the user’s queries [23, 24, 25], and so called symmetric security models [26, 27, 28] that allow the user to learn nothing about the data besides his desired message. An exception in this regard is the recent work in [29] where PIR with distributed storage is explored and the asymptotic (large K) capacity for the X=T=1 setting is bounded as

\displaystyle\left(1-\frac{1}{\sqrt{N}}\right)^{2} \displaystyle\leq\lim_{K\rightarrow\infty}C_{\mbox{\tiny XSTPIR}}(N,K,X=1,T=1)% \leq\left(1-\frac{1}{N}\right). (3)

As the main result of our work, we close this gap and characterize the asymptotic capacity of XSTPIR for all N,X,T as follows.

\displaystyle\lim_{K\rightarrow\infty}C_{\mbox{\tiny XSTPIR}}(N,K,X,T) \displaystyle=\left\{\begin{array}[]{ll}1-\left(\frac{X+T}{N}\right),&N>X+T\\ 0,&N\leq X+T.\end{array}\right. (4)

The asymptotic capacity characterization leads us to supplementary results which include a general upper bound on the capacity of XSTPIR, the exact capacity characterization for any number of messages K if N\leq X+T, and the exact capacity characterization for any K if X=T=1,N=3. The results also lead us to interesting observations about aligning versus decoding noise, dependence of PIR rate on field size, robustness to symmetric security constraints, and a particularly useful idea called cross subspace alignment. When privately retrieving multiple symbols from a desired message in a secure distributed storage system, the structure (say, 1,\beta,\beta^{2},\cdots, for one symbol and 1,\gamma,\gamma^{2},\cdots for another, as in Reed-Solomon (RS) codes) of storage and queries for each symbol determines the number of dimensions occupied by interference and the resolvability of desired symbols. Choosing identical RS codes (\beta=\gamma) for each symbol of the same message would cause desired signals to align among themselves, while making the RS codes insufficiently dependent would cause interference to occupy too many dimensions. Cross subspace alignment is achieved by drawing the code parameters as linear combinations from the same subspace (say, \beta=1+\alpha,\gamma=2+\alpha), which turns out to be the optimal way to align interference while keeping desired symbols resolvable. For a summary of results and a better explanation of the main observations we refer the reader directly to Section 3.

Let us start by defining our notation.

Notation: Let [m:n] denote the set \{m,m+1,\dots,n\} for any two integers m,n such that m\leq n. For sake of simplicity, let X_{[m:n]} denote the set of random variables \{X_{m},X_{m+1},\dots,X_{n}\}. For an index set \mathcal{I}=\{i_{1},i_{2},\dots,i_{n}\}, let X_{\mathcal{I}} denote the set \{X_{i_{1}},X_{i_{2}},\dots,X_{i_{n}}\}. For variables a_{n},n\in[1:N] we denote the N\times 1 vector whose n^{th} term is a_{n}, as

a

. For such vectors

a

and

b

, the notation

ab

denotes their Hadamard product. The notation X\sim Y is used to indicate that X and Y are identically distributed.

2 XSTPIR: Problem Statement

Consider data that is stored at N distributed servers. The data consists of K independent messages, W_{1},W_{2},\cdots,W_{K}, and each message is represented111As usual for an information theoretic formulation, the actual size of each message is allowed to approach infinity. The parameters L and q partition the data into blocks and may be chosen freely by the coding scheme to match the code dimensions. Since the coding scheme for a block can be repeated for each successive block of data with no impact on rate, it suffices to consider one block of data subject to optimization over L and q. by L random symbols from the finite field \mathbb{F}_{q}.

\displaystyle H(W_{1})=H(W_{2})=\dots=H(W_{K})=L, (5)
\displaystyle H(W_{1},W_{2},\dots,W_{K})=KL, (6)

in q-ary units. There are N servers. The information stored at the n^{th} server is denoted by S_{n},n\in[1:N]. An X-secure scheme, 0\leq X<N, guarantees that any X (or fewer) colluding servers learn nothing about the data.

[X-Security] \displaystyle I(S_{\mathcal{X}};W_{1},\dots,W_{K}) \displaystyle=0, \displaystyle\forall\mathcal{X}\subset[1:N],|\mathcal{X}|=X. (7)

Besides X-security, we place no other constraint222The amount of storage at each server is not constrained a priori, however, it is remarkable that none of the XSTPIR schemes in this work end up storing more than KL symbols at each server. Thus the amount of storage used is not worse than a data replication scheme in the absence of security constraints. on the amount of storage or the storage code used at each server. The user generates a desired message index \theta privately and uniformly from [1:K]. In order to retrieve W_{\theta} privately, the user generates N queries, Q_{1}^{[\theta]},Q_{2}^{[\theta]},\dots,Q_{N}^{[\theta]}. The query Q_{n}^{[\theta]} is sent to the n^{th} server. The user has no prior knowledge of the information stored at the servers,

I(S_{[1:N]};Q_{[1:N]}^{[\theta]},\theta)=0. (8)

T-privacy, 1\leq T\leq N, guarantees that any T (or fewer) colluding servers learn nothing about \theta.

[T-Privacy] \displaystyle I(Q_{\mathcal{T}}^{[\theta]},S_{\mathcal{T}};\theta) \displaystyle=0, \displaystyle\forall\mathcal{T}\subset[1:N],|\mathcal{T}|=T. (9)

Upon receiving the query Q_{n}^{[\theta]}, the n^{th} server generates an answering string A_{n}^{[\theta]}, as a function of the query Q_{n}^{[\theta]} and its stored information S_{n}.

H(A_{n}^{[\theta]}|Q_{n}^{[\theta]},S_{n})=0. (10)

From all the answers the user must be able to recover the desired message W_{\theta},

[Correctness] \displaystyle H(W_{\theta}|A_{[1:N]}^{[\theta]},Q_{[1:N]}^{[\theta]},\theta)=0. (11)

The rate of an XSTPIR scheme characterizes how many bits of desired message are retrieved per downloaded bit, (equivalently, how many q-ary symbols of desired message are retrieved per downloaded q-ary symbol),

R=\frac{L}{D}, (12)

where D is the expected value of the number of q-ary symbols downloaded by the user from all servers. The capacity of XSTPIR, denoted C_{\mbox{\tiny XSTPIR}}(N,K,X,T), is the supremum of R.

Finally, note that setting X=0 and T=1 reduces the XSTPIR problem to the basic PIR setting where data storage is not secure and the user’s privacy is only guaranteed if no collusion takes place among servers. Similarly, setting X=0 for arbitrary T, reduces XSTPIR to the T-PIR problem.

3 Capacity of XSTPIR: Results and Observations

The results of this work are presented in this section, followed by some observations.

3.1 Results

Our first result, presented in the following theorem, is an upper bound on the capacity of XSTPIR.

Theorem 1
C_{\mbox{\tiny XSTPIR}}(N,K,X,T)\leq\left(\frac{N-X}{N}\right)C_{\mbox{\tiny TPIR% }}(N-X,K,T). (13)

The proof of Theorem 1 appears in Section 4. While the proof is quite simple, the bound is surprisingly powerful. In fact, we suspect that this bound might be tight in general. An immediate observation is that if we set X=0, i.e., remove the data storage security constraint, then the bound is tight because it gives us the capacity of T-PIR. We further prove the tightness of this bound for the cases identified in our next set of results. The first setting identifies a somewhat degenerate extreme where it is optimal to download everything.

Theorem 2

If N\leq X+T, then333Note that N>X by definition. for arbitrary K,

\displaystyle C_{\mbox{\tiny XSTPIR}}(N,K,X,T) \displaystyle=\left(\frac{N-X}{N}\right)C_{\mbox{\tiny TPIR}}(N-X,K,T) (14)
\displaystyle=\frac{N-X}{NK}. (15)

The proof of Theorem 2 is presented in Section 5. Since the upper bound is already provided by Theorem 1, only a proof of achievability is needed. Furthermore, since retrieving the desired message in this setting amounts to downloading everything stored at all servers regardless of which message is desired, the only thing required for the achievable scheme is a secure storage scheme, which is readily achieved by including X uniformly random noise symbols for every N-X symbols of each message.

Next, the main result of this paper is the asymptotic capacity characterization presented in the following theorem.

Theorem 3

As the number of messages K\rightarrow\infty, for arbitrary N,X,T,

\displaystyle\lim_{K\rightarrow\infty}C_{\mbox{\tiny XSTPIR}}(N,K,X,T) \displaystyle=\lim_{K\rightarrow\infty}\left(\frac{N-X}{N}\right)C_{\mbox{% \tiny TPIR}}(N-X,K,T) (16)
\displaystyle=\left\{\begin{array}[]{ll}1-\left(\frac{X+T}{N}\right),&N>X+T\\ 0,&N\leq X+T.\end{array}\right. (17)

The proof of Theorem 3 appears in Section 6. Theorem 3 is significant for two reasons. First, asymptotic capacity results are particularly relevant for PIR problems because the capacity approaches its asymptotic value extremely quickly — the gap is negligible even for moderate values of K, and K is typically a large value. Second, the asymptotic capacity result showcases a new idea, cross subspace alignment, that is interesting by itself.


Insights from the asymptotically optimal scheme allow us to settle the exact capacity of XSTPIR with X=T=1, N=3 and arbitrary K.

Theorem 4

If the number of servers, N=3, and X=T=1, then for arbitrary number of messages, K,

\displaystyle C_{\mbox{\tiny XSTPIR}}(N=3,K,X=1,T=1) \displaystyle=\left(\frac{N-X}{N}\right)C_{\mbox{\tiny TPIR}}(N-X,K,T) (18)
\displaystyle=\frac{2}{3}\left(1+\frac{1}{2}+\frac{1}{2^{2}}+\cdots+\frac{1}{2% ^{K-1}}\right)^{-1}. (19)

Theorem 4 is proved in Section 7. The capacity achieving scheme introduces a new insight. For almost all PIR settings studied so far, asymptotic capacity achieving schemes have been found that send a uniformly random query vector to each server and download a product of the query vector and information stored at the server. Suppose the query vector is uniform over \mathbb{F}_{q}^{M}. Then with probability 1/q^{M} the query vector is all zero, and the scheme requests nothing from the server. Typically M depends on the number of messages K. As K approaches infinity the probability of requesting nothing approaches zero, so this does not help in the asymptotic sense. However, if the same scheme is used for finite K, then M is also finite, 1/q^{M}>0, and the average download is reduced by the factor (1-1/q^{M}), which improves the achieved rate of the scheme. It is remarkable that the rate achieved in this way depends on the field size. This idea is essential to the capacity achieving scheme for Theorem 4.

Next we present some observations that place our results in perspective.

3.2 Observations

3.2.1 Alignment of Noise and Interference

Figure 1: Suboptimality of the rate achieved by the X=1 secure MDS-PIR alternative that allows the user to decode noise relative to the rate achieved with the asymptotically optimal XSTPIR scheme where the noise is aligned with other interference.

Consider the simplest non-trivial setting for XSTPIR, where X=1,T=1, and the number of servers, N\geq 3. A natural idea for providing X=1 secure storage is to include 1 independent uniformly random noise symbol along with the L symbols of each message, creating a new message with M=L+1 symbols. This new message is stored across N servers according to an (N,M) MDS code, essentially storing a linear combination of the M message symbols at each server, where the coefficients for the noise symbol at each server must be non-zero. Capacity is known for PIR with coded storage (MDS-PIR [5]), and one might wonder if such an MDS-PIR scheme might suffice to achieve capacity with secure storage. It is not difficult to see that the best rate achievable with such an MDS-PIR scheme is

\displaystyle R_{\text{MDS-PIR}}=\frac{M-1}{M}\left(1+\left(\frac{M}{N}\right)% +\dots+\left(\frac{M}{N}\right)^{K-1}\right)^{-1}. (20)

The \frac{M-1}{M} penalty appears because one of the M symbols of the decoded message is the noise symbol. As K\rightarrow\infty, the rate approaches R_{\text{MDS-PIR},\infty}=\frac{M-1}{M}\left(1-\left(\frac{M}{N}\right)\right). This expression takes its maximum value when M=\sqrt{N}, so it can be bounded as,

\displaystyle R_{\text{MDS-PIR},\infty} \displaystyle\leq \displaystyle\frac{\sqrt{N}-1}{\sqrt{N}}\left(1-\left(\frac{\sqrt{N}}{N}\right% )\right)=\left(1-\frac{1}{\sqrt{N}}\right)^{2}. (21)

Note that this expression matches the achievable rate bound of [29]. However, it is strictly smaller than, 1-2/N, the asymptotic capacity of XSTPIR for this setting. Evidently, the natural MDS-PIR solution, and the secret sharing based scheme of [29], are asymptotically suboptimal. In fact, the MDS-PIR solution falls short of the asymptotic (K\rightarrow\infty) capacity of XSTPIR, even if the MDS-PIR scheme is only required to deal with K=2 messages. Denoting the corresponding rate of the MDS-PIR scheme as R_{\text{MDS-PIR},2}, we have,

\displaystyle R_{\text{MDS-PIR},2} \displaystyle\leq\frac{\sqrt{N+1}}{\sqrt{N+1}+1}\left(1+\left(\frac{\sqrt{N+1}% +1}{N}\right)\right)^{-1}\leq 1-\frac{2}{N}. (22)

Figure 1 shows that the gap between the X=1 secure MDS-PIR alternative and the XSTPIR scheme is significant. Intuitively, the reason for this gap is the following. The secure MDS-PIR alternative allows the user to decode the artificial noise symbol which is added to the message to guarantee security. However, in the XSTPIR scheme, the user is able to decode only the desired message, and not the noise protecting it. In fact this noise is aligned with other interfering symbols, e.g., the noise terms protecting other message symbols, thus creating a more efficient solution. Incidentally, the alignment of noise provides another unexpected benefit, in some cases it automatically makes the scheme symmetrically secure, as explained next.

3.2.2 Symmetric Security: Capacity of Sym-XSPIR

Let us fix T=1, thereby relaxing the T-privacy constraint to its minimum value for PIR. Now, suppose in addition to X-secure storage, we also include the so called ‘symmetric’ security constraint, that the user should learn nothing about the data besides his desired message, i.e.,

[Sym-Security] \displaystyle I(W_{[1:K]};A_{[1:N]}^{[\theta]}\mid Q_{[1:N]}^{[\theta]},W_{% \theta},\theta) \displaystyle=0. (23)

Capacity of the basic (X=0,T=1,K>1) Sym-PIR setting was shown in [26] to be

\displaystyle C_{\mbox{\tiny Sym-PIR}}(K,N)=1-\frac{1}{N}. (24)

Note that there is a loss of capacity due to the additional symmetric security constraint. Furthermore, the capacity without the symmetric security constraint depends on the number of messages K while the capacity with the symmetric security constraint does not.

XSTPIR with the symmetric security constraint and with T=1, in short the Sym-XSPIR setting (note that we drop the T because T=1 is the degenerate case for T-privacy), reveals a surprising aspect of our XSTPIR schemes, that imposing the symmetric security constraint does not affect444For T>1 our XSTPIR schemes are not symmetrically secure. our capacity results for T=1. This is made explicit in the following corollaries for Sym-XSPIR, that match the corresponding theorems for XSTPIR.

Corollary 1
C_{\mbox{\tiny Sym-XSPIR}}(N,K,X)\leq\left(\frac{N-X}{N}\right)C_{\mbox{\tiny PIR% }}(N-X,K). (25)
Corollary 2

If N=X+1, then555Note that since X<N by definition, and T=1 for XSPIR, the condition N\leq X+T is equivalent to N=X+1. for arbitrary K,

\displaystyle C_{\mbox{\tiny Sym-XSPIR}}(N,K,X) \displaystyle=\left(\frac{N-X}{N}\right)C_{\mbox{\tiny PIR}}(N-X,K) (26)
\displaystyle=\frac{1}{NK}. (27)
Corollary 3

As the number of messages K\rightarrow\infty, for arbitrary N,X,

\displaystyle\lim_{K\rightarrow\infty}C_{\mbox{\tiny Sym-XSPIR}}(N,K,X) \displaystyle=\lim_{K\rightarrow\infty}\left(\frac{N-X}{N}\right)C_{\mbox{% \tiny PIR}}(N-X,K) (28)
\displaystyle=\left\{\begin{array}[]{ll}1-\left(\frac{X+1}{N}\right),&N>X+1\\ 0,&N\leq X+1.\end{array}\right. (29)
Corollary 4

If the number of servers, N=3, and X=1, then for arbitrary number of messages, K,

\displaystyle C_{\mbox{\tiny Sym-XSPIR}}(N=3,K,X=1) \displaystyle=\left(\frac{N-X}{N}\right)C_{\mbox{\tiny PIR}}(N-X,K) (30)
\displaystyle=\frac{2}{3}\left(1+\frac{1}{2}+\frac{1}{2^{2}}+\cdots+\frac{1}{2% ^{K-1}}\right)^{-1}. (31)

The proofs of all 4 corollaries appear in Appendix B. Surprisingly, note that there is no loss of capacity in each case due to the additional symmetric security constraint. Also note that according to Corollary 4, unlike Sym-PIR, the capacity of Sym-XSPIR depends on the number of messages K for all K>1.

3.2.3 Cross Subspace Alignment

Conceptually, the most intriguing aspect of the asymptotically optimal XSTPIR scheme is the extent to which it is able to align interference. Interference alignment is central to PIR [30, 1], and nearly all existing PIR constructions use some form of interference alignment. The strength of XSTPIR lies in the novel idea of cross subspace alignment, that we explain intuitively in this section through an example. Consider the setting of X=2 secure and T=1 PIR with N=5 servers. Let w_{1} be a symbol from a desired message W. For simplicity (and because identical alignments are applied to all messages), it suffices to focus on only this message for the purpose of this explanation. In order to guarantee X=2 security, w_{1} is mixed with 2 random noise symbols z_{11},z_{12}, according to the following RS Code, so that the n^{th} row is stored at the n^{th} server, n\in[1:5].

\displaystyle\left[\begin{matrix}1\\ 1\\ 1\\ 1\\ 1\\ \end{matrix}\right]w_{1}+\left[\begin{matrix}\beta_{1}\\ \beta_{2}\\ \beta_{3}\\ \beta_{4}\\ \beta_{5}\end{matrix}\right]z_{11}+\left[\begin{matrix}\beta_{1}^{2}\\ \beta_{2}^{2}\\ \beta_{3}^{2}\\ \beta_{4}^{2}\\ \beta_{5}^{2}\end{matrix}\right]z_{12}
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{{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{5.182911pt}{8.% 812916pt}\pgfsys@lineto{-5.182911pt}{8.812916pt}\pgfsys@curveto{-7.39205pt}{8.% 812916pt}{-9.182911pt}{7.022055pt}{-9.182911pt}{4.812916pt}\pgfsys@lineto{-9.1% 82911pt}{-4.812916pt}\pgfsys@curveto{-9.182911pt}{-7.022055pt}{-7.39205pt}{-8.% 812916pt}{-5.182911pt}{-8.812916pt}\pgfsys@lineto{5.182911pt}{-8.812916pt}% \pgfsys@curveto{7.39205pt}{-8.812916pt}{9.182911pt}{-7.022055pt}{9.182911pt}{-% 4.812916pt}\pgfsys@lineto{9.182911pt}{4.812916pt}\pgfsys@curveto{9.182911pt}{7% .022055pt}{7.39205pt}{8.812916pt}{5.182911pt}{8.812916pt}\pgfsys@closepath% \pgfsys@moveto{-9.182911pt}{-8.812916pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-5.849911pt}{-3.479947pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\beta^{2}$}}% }{}\pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}z_{12}.

To ensure privacy, the query symbol q_{\theta} (q_{\theta}=1, i.e., this message is desired) is similarly mixed with a noise symbol z_{1}^{\prime}.

\displaystyle\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}q_{\theta}+\leavevmode\hbox to18.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\beta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}z_{1}^{\prime}=\leavevmode\hbox to18.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}+\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\beta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}z_{1}^{\prime}

and the n^{th} row of this query vector is sent to the n^{th} server. Each server returns the product of the noisy query symbol and the noisy stored symbol, so that the user receives the 5 answers (all products of vectors of the form (\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0% .0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$x$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}})(\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$y$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}})=\leavevmode\hbox to25.67pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{6.832885pt}{7.% 832931pt}\pgfsys@lineto{-6.832885pt}{7.832931pt}\pgfsys@curveto{-9.042024pt}{7% .832931pt}{-10.832885pt}{6.04207pt}{-10.832885pt}{3.832931pt}\pgfsys@lineto{-1% 0.832885pt}{-3.832931pt}\pgfsys@curveto{-10.832885pt}{-6.04207pt}{-9.042024pt}% {-7.832931pt}{-6.832885pt}{-7.832931pt}\pgfsys@lineto{6.832885pt}{-7.832931pt}% \pgfsys@curveto{9.042024pt}{-7.832931pt}{10.832885pt}{-6.04207pt}{10.832885pt}% {-3.832931pt}\pgfsys@lineto{10.832885pt}{3.832931pt}\pgfsys@curveto{10.832885% pt}{6.04207pt}{9.042024pt}{7.832931pt}{6.832885pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-10.832885pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.499886pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$xy$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}} are Hadamard products, also known as Schur products).

\displaystyle\left(\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 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\displaystyle=\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 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\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}z_{12}z_{1}^{\prime}.

The desired symbol w_{1} appears along the vector

1

while the remaining 5 undesired symbols align along 3 dimensions. Specifically, w_{1}z_{1}^{\prime} and z_{11} align along

\beta

; z_{11}z_{1}^{\prime} and z_{12} align along

\beta^{2}

and z_{12}z_{1}^{\prime} appears along

\beta^{3}

. This type of alignment, enabled by using the same

\beta

in the storage and query, is indeed very useful and has been used previously by Hollanti et al. for MDS-TPIR [6]. However, note that we have a 5 dimensional space (all vectors are 5\times 1) and we are so far only using 4 dimensions (one desired, three interference), so there is room for improvement.

In order to improve the efficiency of the retrieval scheme, suppose we try to retrieve another symbol, w_{2}, from the same desired message W=(w_{1},w_{2}). The challenge is that because of the X=2 security requirement w_{2} is mixed with new (independent) noise symbols z_{21},z_{22}, so any attempt to retrieve w_{2} will add new interference terms. Since we already have 3 dimensions of interference, the new interference added due to the noise protecting w_{2} must align completely within the existing interference. This will be accomplished by cross-alignment, i.e., introducing additional structure across the storage and query codes for the different symbols to be retrieved. The answers from the 5 servers are now expressed as follows.

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\pgfsys@closepath\pgfsys@moveto{-12.932854pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.599854pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\beta^{3}\gamma$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}z_{12}z_{1}^{\prime}+\leavevmode\hbox to29.87pt{\vbox to21.63% pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{8.932854pt}{8.% 812916pt}\pgfsys@lineto{-8.932854pt}{8.812916pt}\pgfsys@curveto{-11.141993pt}{% 8.812916pt}{-12.932854pt}{7.022055pt}{-12.932854pt}{4.812916pt}\pgfsys@lineto{% -12.932854pt}{-4.812916pt}\pgfsys@curveto{-12.932854pt}{-7.022055pt}{-11.14199% 3pt}{-8.812916pt}{-8.932854pt}{-8.812916pt}\pgfsys@lineto{8.932854pt}{-8.81291% 6pt}\pgfsys@curveto{11.141993pt}{-8.812916pt}{12.932854pt}{-7.022055pt}{12.932% 854pt}{-4.812916pt}\pgfsys@lineto{12.932854pt}{4.812916pt}\pgfsys@curveto{12.9% 32854pt}{7.022055pt}{11.141993pt}{8.812916pt}{8.932854pt}{8.812916pt}% \pgfsys@closepath\pgfsys@moveto{-12.932854pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.599854pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\beta\gamma^{3}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}z_{22}z_{2}^{\prime}. (33)

Note that we cannot choose \leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\beta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}=\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\gamma$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}, because the two desired symbols (w_{1},w_{2}) must not align in the same dimension. Also note that by cross-multiplying the first set of answers with

\gamma

and the second with

\beta

we have achieved cross alignment of 4 terms along

\beta\gamma

. 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{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{8.932854pt}{8.% 812916pt}\pgfsys@lineto{-8.932854pt}{8.812916pt}\pgfsys@curveto{-11.141993pt}{% 8.812916pt}{-12.932854pt}{7.022055pt}{-12.932854pt}{4.812916pt}\pgfsys@lineto{% -12.932854pt}{-4.812916pt}\pgfsys@curveto{-12.932854pt}{-7.022055pt}{-11.14199% 3pt}{-8.812916pt}{-8.932854pt}{-8.812916pt}\pgfsys@lineto{8.932854pt}{-8.81291% 6pt}\pgfsys@curveto{11.141993pt}{-8.812916pt}{12.932854pt}{-7.022055pt}{12.932% 854pt}{-4.812916pt}\pgfsys@lineto{12.932854pt}{4.812916pt}\pgfsys@curveto{12.9% 32854pt}{7.022055pt}{11.141993pt}{8.812916pt}{8.932854pt}{8.812916pt}% \pgfsys@closepath\pgfsys@moveto{-12.932854pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.599854pt}{-3.479947pt}{}\hbox{{% 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\beta\gamma^{3}

. Since the overall space is only 5 dimensional and we need two dimensions for desired symbols, we need to restrict interference to no more than 3 dimensions. Surprisingly, it is possible to do this by cross subspace alignment as we show next. Let us introduce a structural relationship between \beta and \gamma. In particular, let us set,

\beta

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\gamma

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so that the answers from the 5 servers are now expressed as,

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\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\right)I (36)

where the interference I is

\displaystyle I \displaystyle=\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(w_{1}z_{1}^{\prime}+z_{11}+w_{2}z_{2}^{\prime}+z_{21})+\left(% \leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% 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\displaystyle+\left(\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 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{{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% 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\pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\alpha$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}^{2}\right)z_{22}z_{2}^{\prime}. (37)

Note that there are still 5 interference vectors, no two of which align directly with each other. However, the 5 interference vectors align into a 3 dimensional subspace of the 5 dimensional vector space. This is what we mean by cross subspace alignment and it is essential to this work. To see explicitly how the interference aligns into a 3 dimensional subspace, we can rewrite I as,

\displaystyle I \displaystyle=\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(w_{1}z_{1}^{\prime}+z_{11}+w_{2}z_{2}^{\prime}+z_{21}+z_{11}z% _{1}^{\prime}+z_{12}+2z_{21}z_{1}^{\prime}+2z_{22}+z_{12}z_{1}^{\prime}+4z_{22% }z_{2}^{\prime})
\displaystyle+\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\alpha$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(z_{11}z_{1}^{\prime}+z_{12}+z_{21}z_{1}^{\prime}+z_{22}+2z_{1% 2}z_{1}^{\prime}+4z_{22}z_{2}^{\prime})
\displaystyle+\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\alpha$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}^{2}(z_{12}z_{1}^{\prime}+z_{22}z_{2}^{\prime}). (38)

Thus, due to cross subspace alignment, all of I aligns within a 3 dimensional space, leaving the remaining 2 dimensions interference-free for the desired symbols. Exactly the same alignments apply to all messages as explained in the formal descriptions of the schemes provided in this paper.

4 Proof of Theorem 1

Without loss of generality, suppose the expected number of bits downloaded from each server is the same. Now, relax the constraints so that S_{[1:X]}, i.e., the stored information at the first X servers is made available globally (to all servers and to the user) for free, the messages W_{1},W_{2},\cdots,W_{K} are made available to all servers, and the data-security constraint is eliminated. None of this can hurt capacity because any XSTPIR scheme from before can still be used with the relaxed constraints. So any upper bound on capacity of this relaxed setting is still an upper bound on the capacity of the original XSTPIR setting. The relaxed setting is simply the T-PIR problem with K messages and N-X servers, for which we already know the optimal download per server from the existing capacity results for T-PIR. Therefore, a general upper bound for XSTPIR is obtained as

\displaystyle C_{\mbox{\tiny XSTPIR}}(N,K,X,T) \displaystyle\leq \displaystyle\left(\frac{N-X}{N}\right)C_{\mbox{\tiny TPIR}}(N-X,K,T). (39)

This completes the proof of Theorem 1. \hfill\square

5 Proof of Theorem 2

Let each message consist of L=N-X symbols in \mathbb{F}_{q}, q\geq N, and append X instances of 0 symbols, to create artificial messages of length N,

\displaystyle\bar{W}_{k} \displaystyle=(W_{k1},W_{k_{2}},\cdots,W_{k(N-X)},\underbrace{0,0,\cdots,0}_{X% }), \displaystyle\forall k\in[1:K]. (40)

Corresponding to each message W_{k}, let Z_{k}=(Z_{k1},Z_{k2},\cdots,Z_{kX})\in\mathbb{F}_{q}^{X} be X independent uniform noise symbols, to be used for X-security. Let Z_{k} be encoded with an (N,X) MDS code to produce \bar{Z}_{k}\in\mathbb{F}_{q}^{N}. For each k\in[1:K] and n\in[1:N], the n^{th} server stores the n^{th} symbol of \bar{W}_{k}+\bar{Z}_{k}. Thus, each server stores a total of K symbols. The MDS property of \bar{Z}_{k} ensures that the data storage is X-secure. Retrieval is trivial — in order to retrieve the desired message W_{\theta}, the user simply downloads everything from all servers. Since the queries do not depend on the desired message, the scheme is N-private, so it is also T-private. The rate achieved is \frac{N-X}{NK} which matches the capacity for this setting.\hfill\square

6 Proof of Theorem 3

Let us present an XSTPIR scheme for arbitrary X, T, N, K, that is asymptotically optimal (as K\rightarrow\infty). The asymptotic capacity is zero for N\leq X+T, so we only need to consider N>X+T. Throughout this scheme we will set

\displaystyle L \displaystyle=N-X-T (41)

and we will use the compact notation,

\displaystyle\Delta \displaystyle=\prod_{i=1}^{L}(i+\alpha). (42)

\Delta_{n} will represent the value of \Delta when \alpha is replaced with \alpha_{n}.

Each message W_{k},k\in[1:K], consists of L=N-X-T symbols, W_{k}=(W_{k1},W_{k2},\cdots,W_{kL}) from a finite field \mathbb{F}_{q}. The field \mathbb{F}_{q} is assumed to have size q\geq L+N, and characteristic greater than L-1. For the design of this scheme, we will need constants \alpha_{n},n\in[1:N] that are distinct elements of \mathbb{G},

\displaystyle\mathbb{G} \displaystyle=\{\alpha\in\mathbb{F}_{q}:\alpha+i\neq 0,~{}\forall i\in[1:L]\}. (43)

Such \alpha_{n},n\in[1:N] must exist because q\geq L+N. These constants will be globally known. In the following description of the scheme, we will explain explicitly how the values of these constants are chosen. For now, let us note that because the characteristic of the field is assumed to be greater than L-1, the values \alpha+1,\alpha+2,\cdots,\alpha+L are distinct for any \alpha\in\mathbb{F}_{q}.

Let us split the messages into L vectors, so that {\bf W}_{l}=(W_{1l},W_{2l},\cdots,W_{Kl}), l\in[1:L], contains the l^{th} symbol of every message. Let {\bf Z}_{lx},l\in[1:L],x\in[1:X], be independent uniformly random noise vectors from \mathbb{F}_{q}^{1\times K}, that are used to guarantee security. Similarly, let {\bf Z}^{\prime}_{lt},l\in[1:L],t\in[1:T], be independent uniformly random noise vectors from \mathbb{F}_{q}^{K\times 1}, that are used to guarantee privacy. The independence between noise vectors, messages, and the user’s desired message index \theta is specified as follows.

\displaystyle H\Big{(}\left({\bf W}_{l}\right)_{l\in[1:L]},\left({\bf Z}_{lx}% \right)_{l\in[1:L],x\in[1:X]},\left({\bf Z}^{\prime}_{lt}\right)_{l\in[1:L],t% \in[1:T]},\theta\Big{)} \displaystyle=H(\left({\bf W}_{l}\right)_{l\in[1:L]})+H(\theta)+L(X+T) (44)

in q-ary units. Let {\bf Q}_{\theta} represent666Note that the XSTPIR scheme described in this section works even if {\bf Q}_{\theta} is an arbitrary vector, i.e., if instead of retrieving one of the K messages, the user wishes to compute an arbitrary linear function of the K messages over \mathbb{F}_{q}. Thus, the scheme automatically settles the asymptotic capacity of the natural X-secure and T-private generalization of the private computation problem introduced in [31] (also known as private function retrieval [32]). the \theta^{th} column of the K\times K identity matrix, so it contains a 1 in the \theta^{th} position and zeros everywhere else. Note that

\displaystyle({\bf W}_{1}{\bf Q}_{\theta},{\bf W}_{2}{\bf Q}_{\theta},\cdots,{% \bf W}_{L}{\bf Q}_{\theta}) \displaystyle=(W_{\theta 1},W_{\theta 2},\cdots,W_{\theta L})=W_{\theta} (45)

is the message desired by the user. A succinct summary of the storage at each server, the queries, and a partitioning of signal and interference dimensions contained in the answers from each server, is provided below.

\displaystyle\begin{array}[]{cc}\hline&\mbox{Server `$n$' (Replace $\alpha,% \Delta$ with $\alpha_{n},\Delta_{n}$)}\\ \hline\mbox{Storage}&\mathbf{W}_{1}+(1+\alpha)\mathbf{Z}_{11}+\cdots+(1+\alpha% )^{X}\mathbf{Z}_{1X},\\ (S_{n})&\mathbf{W}_{2}+(2+\alpha)\mathbf{Z}_{21}+\cdots+(2+\alpha)^{X}\mathbf{% Z}_{2X},\\ &\vdots\\ &\mathbf{W}_{L}+(L+\alpha)\mathbf{Z}_{L1}+\cdots+(L+\alpha)^{X}\mathbf{Z}_{LX}% \\ \hline\mbox{Query}&\frac{\Delta}{1+\alpha}\Big{(}\mathbf{Q_{\theta}}+(1+\alpha% )\mathbf{Z}_{11}^{\prime}+\cdots+(1+\alpha)^{T}\mathbf{Z}_{1T}^{\prime}\Big{)}% ,\\ (Q_{n}^{[\theta]})&\frac{\Delta}{2+\alpha}\Big{(}\mathbf{Q_{\theta}}+(2+\alpha% )\mathbf{Z}_{21}^{\prime}+\cdots+(2+\alpha)^{T}\mathbf{Z}_{2T}^{\prime}\Big{)}% ,\\ &\vdots\\ &\frac{\Delta}{L+\alpha}\Big{(}\mathbf{Q_{\theta}}+(L+\alpha)\mathbf{Z}_{L1}^{% \prime}+\cdots+(L+\alpha)^{T}\mathbf{Z}_{LT}^{\prime}\Big{)}\\ \hline\omit\span\omit\mbox{Desired symbols appear along vectors}\\ \omit\span\omit\leavevmode\hbox 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Initially, the user knows only his desired message index \theta and the noise terms {\bf Z}^{\prime}_{lt},l\in[1:L],t\in[1:T], all of which are privately generated by the user. Each server n\in[1:N] knows only its stored information S_{n}. The storage S_{n} at Server n may be viewed as a 1\times LK row vector formed by concatenating the L row vectors, {\bf W}_{l}+\sum_{x=1}^{X}(l+\alpha_{n})^{x}{\bf Z}_{lx}, l\in[1:L]. Similarly, the query Q_{n}^{[\theta]} may be viewed as an LK\times 1 column vector formed by concatenating the L column vectors, \frac{\Delta_{n}}{l+\alpha_{n}}\left({\bf Q}_{\theta}+\sum_{t=1}^{T}(l+\alpha_% {n})^{t}{\bf Z}_{lt}^{\prime}\right), l\in[1:L].

Upon receiving the query Q^{[\theta]}_{n} from the user, Server n responds with the answer A_{n}^{[\theta]} that is exactly one symbol in \mathbb{F}_{q}, found by multiplying S_{n} with Q^{[\theta]}_{n}.

\displaystyle A_{n}^{[\theta]} \displaystyle=S_{n}Q^{[\theta]}_{n}. (46)

This produces a single equation in a total of L(X+1)(T+1) terms. Out of these, L terms are desired message symbols {\bf W}_{l}{\bf Q}_{\theta}, l\in[1:L], and the remaining L(X+1)(T+1)-L terms are undesired, or interference terms. The interference terms include LT terms of the type {\bf W}_{l}{\bf Z}_{lt}^{\prime}, LX terms of the type {\bf Z}_{lx}{\bf Q}_{\theta}, and LXT terms of the type {\bf Z}_{lx}{\bf Z}_{lt}^{\prime}. The user obtains one such equation from each server, for a total of N equations, from which he must be able to retrieve his L desired symbols. The key to this is the alignment of L(X+1)(T+1)-L interference terms into N-L dimensions, leaving L dimensions free from interference from which the L desired symbols can be decoded.

First let us identify the desired signal dimensions, i.e., the vectors along which desired symbols are seen by the user. Each answer A_{n}^{[\theta]} contains the desired symbols \frac{\Delta_{n}}{l+\alpha_{n}}{\bf W}_{l}{\bf Q}_{\theta}=\frac{\Delta_{n}}{l% +\alpha_{n}}W_{\theta l}, l\in[1:L]. These L desired symbols appear along the following L vectors.

\displaystyle\left[\begin{matrix}\frac{\Delta_{1}}{1+\alpha_{1}}\\ \frac{\Delta_{2}}{1+\alpha_{2}}\\ \vdots\\ \frac{\Delta_{N}}{1+\alpha_{N}}\\ \end{matrix}\right],\left[\begin{matrix}\frac{\Delta_{1}}{2+\alpha_{1}}\\ \frac{\Delta_{2}}{2+\alpha_{2}}\\ \vdots\\ \frac{\Delta_{N}}{2+\alpha_{N}}\\ \end{matrix}\right],\cdots,\left[\begin{matrix}\frac{\Delta_{1}}{L+\alpha_{1}}% \\ \frac{\Delta_{2}}{L+\alpha_{2}}\\ \vdots\\ \frac{\Delta_{N}}{L+\alpha_{N}}\\ \end{matrix}\right]\triangleq\leavevmode\hbox to18.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} 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{-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{1}{2+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\cdots,\leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{1}{L+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\Big{)}. (47)

Recall that the products among vectors in the \leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$a$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$b$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}} notation are Hadamard products. Similarly, the vectors along which interference symbols appear are identified as follows.

\displaystyle{\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\Big{(}\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to48.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{18.082714pt}{7% .832931pt}\pgfsys@lineto{-18.082714pt}{7.832931pt}\pgfsys@curveto{-20.291853pt% }{7.832931pt}{-22.082714pt}{6.04207pt}{-22.082714pt}{3.832931pt}\pgfsys@lineto% {-22.082714pt}{-3.832931pt}\pgfsys@curveto{-22.082714pt}{-6.04207pt}{-20.29185% 3pt}{-7.832931pt}{-18.082714pt}{-7.832931pt}\pgfsys@lineto{18.082714pt}{-7.832% 931pt}\pgfsys@curveto{20.291853pt}{-7.832931pt}{22.082714pt}{-6.04207pt}{22.08% 2714pt}{-3.832931pt}\pgfsys@lineto{22.082714pt}{3.832931pt}\pgfsys@curveto{22.% 082714pt}{6.04207pt}{20.291853pt}{7.832931pt}{18.082714pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-22.082714pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.749714pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\cdots,\leavevmode\hbox to69.17pt{\vbox to21.63pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{28.582554pt}{8% .812916pt}\pgfsys@lineto{-28.582554pt}{8.812916pt}\pgfsys@curveto{-30.791693pt% }{8.812916pt}{-32.582554pt}{7.022055pt}{-32.582554pt}{4.812916pt}% \pgfsys@lineto{-32.582554pt}{-4.812916pt}\pgfsys@curveto{-32.582554pt}{-7.0220% 55pt}{-30.791693pt}{-8.812916pt}{-28.582554pt}{-8.812916pt}\pgfsys@lineto{28.5% 82554pt}{-8.812916pt}\pgfsys@curveto{30.791693pt}{-8.812916pt}{32.582554pt}{-7% .022055pt}{32.582554pt}{-4.812916pt}\pgfsys@lineto{32.582554pt}{4.812916pt}% \pgfsys@curveto{32.582554pt}{7.022055pt}{30.791693pt}{8.812916pt}{28.582554pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-32.582554pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.249554pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha)^{X+T-1}$}}}{}\pgfsys@endscope}}}}{% }\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to48.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{18.082714pt}{7% .832931pt}\pgfsys@lineto{-18.082714pt}{7.832931pt}\pgfsys@curveto{-20.291853pt% }{7.832931pt}{-22.082714pt}{6.04207pt}{-22.082714pt}{3.832931pt}\pgfsys@lineto% {-22.082714pt}{-3.832931pt}\pgfsys@curveto{-22.082714pt}{-6.04207pt}{-20.29185% 3pt}{-7.832931pt}{-18.082714pt}{-7.832931pt}\pgfsys@lineto{18.082714pt}{-7.832% 931pt}\pgfsys@curveto{20.291853pt}{-7.832931pt}{22.082714pt}{-6.04207pt}{22.08% 2714pt}{-3.832931pt}\pgfsys@lineto{22.082714pt}{3.832931pt}\pgfsys@curveto{22.% 082714pt}{6.04207pt}{20.291853pt}{7.832931pt}{18.082714pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-22.082714pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.749714pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(2+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\cdots,\leavevmode\hbox to69.17pt{\vbox to21.63pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{28.582554pt}{8% .812916pt}\pgfsys@lineto{-28.582554pt}{8.812916pt}\pgfsys@curveto{-30.791693pt% }{8.812916pt}{-32.582554pt}{7.022055pt}{-32.582554pt}{4.812916pt}% \pgfsys@lineto{-32.582554pt}{-4.812916pt}\pgfsys@curveto{-32.582554pt}{-7.0220% 55pt}{-30.791693pt}{-8.812916pt}{-28.582554pt}{-8.812916pt}\pgfsys@lineto{28.5% 82554pt}{-8.812916pt}\pgfsys@curveto{30.791693pt}{-8.812916pt}{32.582554pt}{-7% .022055pt}{32.582554pt}{-4.812916pt}\pgfsys@lineto{32.582554pt}{4.812916pt}% \pgfsys@curveto{32.582554pt}{7.022055pt}{30.791693pt}{8.812916pt}{28.582554pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-32.582554pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.249554pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(2+\alpha)^{X+T-1}$}}}{}\pgfsys@endscope}}}}{% }\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\cdots,}
\displaystyle{         \cdots,\leavevmode\hbox to48.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{18.082714pt}{7% .832931pt}\pgfsys@lineto{-18.082714pt}{7.832931pt}\pgfsys@curveto{-20.291853pt% }{7.832931pt}{-22.082714pt}{6.04207pt}{-22.082714pt}{3.832931pt}\pgfsys@lineto% {-22.082714pt}{-3.832931pt}\pgfsys@curveto{-22.082714pt}{-6.04207pt}{-20.29185% 3pt}{-7.832931pt}{-18.082714pt}{-7.832931pt}\pgfsys@lineto{18.082714pt}{-7.832% 931pt}\pgfsys@curveto{20.291853pt}{-7.832931pt}{22.082714pt}{-6.04207pt}{22.08% 2714pt}{-3.832931pt}\pgfsys@lineto{22.082714pt}{3.832931pt}\pgfsys@curveto{22.% 082714pt}{6.04207pt}{20.291853pt}{7.832931pt}{18.082714pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-22.082714pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.749714pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(L+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\cdots,\leavevmode\hbox to69.17pt{\vbox to21.63pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{28.582554pt}{8% .812916pt}\pgfsys@lineto{-28.582554pt}{8.812916pt}\pgfsys@curveto{-30.791693pt% }{8.812916pt}{-32.582554pt}{7.022055pt}{-32.582554pt}{4.812916pt}% \pgfsys@lineto{-32.582554pt}{-4.812916pt}\pgfsys@curveto{-32.582554pt}{-7.0220% 55pt}{-30.791693pt}{-8.812916pt}{-28.582554pt}{-8.812916pt}\pgfsys@lineto{28.5% 82554pt}{-8.812916pt}\pgfsys@curveto{30.791693pt}{-8.812916pt}{32.582554pt}{-7% .022055pt}{32.582554pt}{-4.812916pt}\pgfsys@lineto{32.582554pt}{4.812916pt}% \pgfsys@curveto{32.582554pt}{7.022055pt}{30.791693pt}{8.812916pt}{28.582554pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-32.582554pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.249554pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(L+\alpha)^{X+T-1}$}}}{}\pgfsys@endscope}}}}{% }\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\Big{)}}. (48)

Thus, the vector of answers from all N servers can be expressed as

A^{[\theta]}

\displaystyle=\sum_{l=1}^{L}W_{\theta l}\leavevmode\hbox to26.42pt{\vbox to% 23.27pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{l+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}+\sum_{l=1}^{L}\sum_{i=0}^{X+T-1}\leavevmode\hbox to59.87pt{% \vbox to21.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{23.932624pt}{8% .812916pt}\pgfsys@lineto{-23.932624pt}{8.812916pt}\pgfsys@curveto{-26.141763pt% }{8.812916pt}{-27.932624pt}{7.022055pt}{-27.932624pt}{4.812916pt}% \pgfsys@lineto{-27.932624pt}{-4.812916pt}\pgfsys@curveto{-27.932624pt}{-7.0220% 55pt}{-26.141763pt}{-8.812916pt}{-23.932624pt}{-8.812916pt}\pgfsys@lineto{23.9% 32624pt}{-8.812916pt}\pgfsys@curveto{26.141763pt}{-8.812916pt}{27.932624pt}{-7% .022055pt}{27.932624pt}{-4.812916pt}\pgfsys@lineto{27.932624pt}{4.812916pt}% \pgfsys@curveto{27.932624pt}{7.022055pt}{26.141763pt}{8.812916pt}{23.932624pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-27.932624pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-24.599625pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(l+\alpha)^{i}$}}}{}\pgfsys@endscope}}}% }{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}I_{li} (49)

for some interference terms I_{li} that are sums of various {\bf W}_{l}{\bf Z}_{lt}^{\prime}, {\bf Z}_{lx}{\bf Q}_{\theta}, and {\bf Z}_{lx}{\bf Z}_{lt}^{\prime} terms. The exact form of I_{li} terms is not important for our analysis. Using binomial expansion to write each

(l+\alpha)^{i}

vector as \sum_{j=0}^{i}\binom{i}{j}l^{j}\leavevmode\hbox to30.77pt{\vbox to21.63pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{9.382847pt}{8.% 812916pt}\pgfsys@lineto{-9.382846pt}{8.812916pt}\pgfsys@curveto{-11.591985pt}{% 8.812916pt}{-13.382846pt}{7.022055pt}{-13.382846pt}{4.812916pt}\pgfsys@lineto{% -13.382846pt}{-4.812916pt}\pgfsys@curveto{-13.382846pt}{-7.022055pt}{-11.59198% 5pt}{-8.812916pt}{-9.382846pt}{-8.812916pt}\pgfsys@lineto{9.382847pt}{-8.81291% 6pt}\pgfsys@curveto{11.591986pt}{-8.812916pt}{13.382847pt}{-7.022055pt}{13.382% 847pt}{-4.812916pt}\pgfsys@lineto{13.382847pt}{4.812916pt}\pgfsys@curveto{13.3% 82847pt}{7.022055pt}{11.591986pt}{8.812916pt}{9.382847pt}{8.812916pt}% \pgfsys@closepath\pgfsys@moveto{-13.382846pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.049846pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\alpha^{i-j}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}, and grouping terms by the vectors

\alpha^{i}

, we can write,

A^{[\theta]}

\displaystyle=\sum_{l=1}^{L}W_{\theta l}\leavevmode\hbox to26.42pt{\vbox to% 23.27pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{l+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}+\sum_{i=0}^{X+T-1}\leavevmode\hbox to29.87pt{\vbox to21.63pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{8.932854pt}{8.% 812916pt}\pgfsys@lineto{-8.932854pt}{8.812916pt}\pgfsys@curveto{-11.141993pt}{% 8.812916pt}{-12.932854pt}{7.022055pt}{-12.932854pt}{4.812916pt}\pgfsys@lineto{% -12.932854pt}{-4.812916pt}\pgfsys@curveto{-12.932854pt}{-7.022055pt}{-11.14199% 3pt}{-8.812916pt}{-8.932854pt}{-8.812916pt}\pgfsys@lineto{8.932854pt}{-8.81291% 6pt}\pgfsys@curveto{11.141993pt}{-8.812916pt}{12.932854pt}{-7.022055pt}{12.932% 854pt}{-4.812916pt}\pgfsys@lineto{12.932854pt}{4.812916pt}\pgfsys@curveto{12.9% 32854pt}{7.022055pt}{11.141993pt}{8.812916pt}{8.932854pt}{8.812916pt}% \pgfsys@closepath\pgfsys@moveto{-12.932854pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.599854pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha^{i}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}I_{i}^{\prime}. (50)

Thus, all interference is aligned within the subspace spanned by vectors

\Delta

,

\Delta\alpha

, \dots,

\Delta\alpha^{X+T-1}

. As explained in Section 3.2.3, this is because of cross subspace alignment.

In matrix notation, we have,

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where the N\times N square matrix (note that L+X+T=N)

\displaystyle M_{N} \displaystyle=\left[\begin{matrix}\frac{\Delta_{1}}{1+\alpha_{1}}&\cdots&\frac% {\Delta_{1}}{L+\alpha_{1}}&\Delta_{1}&\Delta_{1}\alpha_{1}&\cdots&\Delta_{1}% \alpha_{1}^{X+T-1}\\ \frac{\Delta_{2}}{1+\alpha_{2}}&\cdots&\frac{\Delta_{2}}{L+\alpha_{2}}&\Delta_% {2}&\Delta_{2}\alpha_{2}&\cdots&\Delta_{2}\alpha_{2}^{X+T-1}\\ \vdots\\ \frac{\Delta_{N}}{1+\alpha_{N}}&\cdots&\frac{\Delta_{N}}{L+\alpha_{N}}&\Delta_% {N}&\Delta_{N}\alpha_{N}&\cdots&\Delta_{N}\alpha_{N}^{X+T-1}\end{matrix}\right] (52)
\displaystyle=\left[\begin{matrix}\leavevmode\hbox to26.42pt{\vbox to23.27pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{1+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}&\cdots&\leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{L+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}&\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% 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\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}&\cdots&\leavevmode\hbox to46.67pt{\vbox to21.63pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{17.332725pt}{8% .812916pt}\pgfsys@lineto{-17.332725pt}{8.812916pt}\pgfsys@curveto{-19.541864pt% }{8.812916pt}{-21.332725pt}{7.022055pt}{-21.332725pt}{4.812916pt}% \pgfsys@lineto{-21.332725pt}{-4.812916pt}\pgfsys@curveto{-21.332725pt}{-7.0220% 55pt}{-19.541864pt}{-8.812916pt}{-17.332725pt}{-8.812916pt}\pgfsys@lineto{17.3% 32725pt}{-8.812916pt}\pgfsys@curveto{19.541864pt}{-8.812916pt}{21.332725pt}{-7% .022055pt}{21.332725pt}{-4.812916pt}\pgfsys@lineto{21.332725pt}{4.812916pt}% \pgfsys@curveto{21.332725pt}{7.022055pt}{19.541864pt}{8.812916pt}{17.332725pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-21.332725pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.999726pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha^{X+T-1}$}}}{}\pgfsys@endscope}}}% }{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\end{matrix}\right] (53)

is called the decoding matrix. Evidently, if the decoding matrix is invertible, then the user can recover his L desired message symbols.

Next we invoke Theorem 5 from Appendix C, which guarantees the existence of \alpha_{n}\in\mathbb{G},n\in[1:N] for which M_{N} is invertible,777Note that such \alpha_{n} must be distinct, otherwise M_{N} cannot be invertible provided the following conditions are satisfied.

  1. The maximum degree of the N polynomials in \mathbb{F}[\alpha], namely \frac{\Delta}{1+\alpha}, \frac{\Delta}{2+\alpha}, \cdots, \frac{\Delta}{L+\alpha}, \Delta, \Delta\alpha, \cdots, \Delta\alpha^{X+T-1}, is strictly smaller than |\mathbb{G}|.

  2. These N polynomials are linearly independent.

The first condition is easily verified, because the maximum degree is L+X+T-1=N-1, and N-1 is strictly smaller than |\mathbb{G}|=q-L because q\geq N+L by assumption.

The second condition is proved as follows.

Suppose these N polynomials are not linearly independent. Then there must exist c_{1},c_{2},\cdots,c_{N}\in\mathbb{F}_{q}, at least one of which is not zero, such that

\displaystyle g(\alpha) \displaystyle=\sum_{i=1}^{L}c_{i}\left(\frac{\Delta}{i+\alpha}\right)+\sum_{i=% L+1}^{N}c_{i}\Delta\alpha^{i-(L+1)} (54)

is the zero polynomial, i.e., the coefficients of all monomials in g(\alpha) must be zero. The coefficient of \alpha^{N-1} is c_{N} so we must have c_{N}=0. Then, the remaining coefficient of \alpha^{N-2} is c_{N-1}, so we must have c_{N-1}=0. Similarly, we find c_{L+1}=c_{L+2}=\cdots=c_{N}=0, leaving us with

\displaystyle g(\alpha) \displaystyle=\sum_{i=1}^{L}c_{i}\left(\frac{\Delta}{i+\alpha}\right). (55)

Now, if this g(\alpha) is the zero polynomial, then it must be zero for every \alpha. Choosing \alpha such888Note that \frac{\Delta}{i+\alpha} is simply a compact notation for \prod_{l\in[1:L],l\neq i}(l+\alpha), i.e., it only means that the (i+\alpha) factor is eliminated from \Delta, so there is no ‘division by 0’ when we set i+\alpha=0 in \frac{\Delta}{i+\alpha}. that (i+\alpha)=0, gives us c_{i}=0 for every i\in[1:L]. Thus, we have c_{1}=c_{2}=\cdots=c_{N}=0, which implies that the N polynomials are linearly independent. Since both conditions are satisfied, Theorem 5 guarantees the existence of \alpha_{n}\in\mathbb{G},n\in[1:N] such that the decoding matrix M_{N} is invertible. Fixing the values of \alpha_{1},\cdots,\alpha_{N} according to this non-singular realization completes the design of the scheme.

Now consider the security guarantee. For any X colluding servers, i_{1},i_{2},\cdots,i_{X}, the X observations, U_{kl1},\cdots,U_{klX}, of each message symbol W_{kl},k\in[1:K],l\in[1:L], are protected by noise terms as follows.

\displaystyle\left[\begin{matrix}U_{kl1}\\ \vdots\\ U_{klX}\end{matrix}\right] \displaystyle=\left[\begin{matrix}W_{kl}\\ \vdots\\ W_{kl}\end{matrix}\right]+\underbrace{\left[\begin{matrix}l+\alpha_{i_{1}}&(l+% \alpha_{i_{1}})^{2}&\cdots&(l+\alpha_{i_{1}})^{X}\\ l+\alpha_{i_{2}}&(l+\alpha_{i_{2}})^{2}&\cdots&(l+\alpha_{i_{2}})^{X}\\ \vdots&\vdots&&\vdots\\ l+\alpha_{i_{X}}&(l+\alpha_{i_{X}})^{2}&\cdots&(l+\alpha_{i_{X}})^{X}\end{% matrix}\right]}_{P}\underbrace{\left[\begin{matrix}{\bf Z}_{l1}(k)\\ \vdots\\ {\bf Z}_{lX}(k)\end{matrix}\right]}_{Z} (56)
\displaystyle=W_{kl}\underbrace{\left[\begin{matrix}1\\ \vdots\\ 1\end{matrix}\right]}_{\bf 1}+\left[\begin{matrix}l+\alpha_{i_{1}}&0&\cdots&0% \\ 0&l+\alpha_{i_{2}}&\cdots&0\\ 0&0&\ddots&0\\ 0&0&\cdots&l+\alpha_{i_{X}}\end{matrix}\right]\left[\begin{matrix}1&l+\alpha_{% i_{1}}&\cdots&(l+\alpha_{i_{1}})^{X-1}\\ 1&l+\alpha_{i_{2}}&\cdots&(l+\alpha_{i_{2}})^{X-1}\\ \vdots&\vdots&&\vdots\\ 1&l+\alpha_{i_{X}}&\cdots&(l+\alpha_{i_{X}})^{X-1}\end{matrix}\right]\left[% \begin{matrix}{\bf Z}_{l1}(k)\\ \vdots\\ {\bf Z}_{lX}(k)\end{matrix}\right]. (57)

where {\bf Z}_{lx}(k) is the k^{th} element of the vector {\bf Z}_{lx}. Note that P is a product of a diagonal matrix which is invertible because (l+\alpha_{i_{j}}) are non-zero, and a Vandermonde matrix which is invertible because (l+\alpha_{i_{j}}) are distinct. Therefore, P is invertible, and the observations are independent of the message symbols as shown below.

\displaystyle I(W_{kl};\left(U_{klx}\right)_{x\in[1:X]}) \displaystyle=I(W_{kl};W_{kl}{\bf 1}+PZ)=I(W_{kl};W_{kl}P^{-1}{\bf 1}+Z)=I(W_{% kl};Z)=0. (58)

Furthermore, since the noise terms protecting each message symbol W_{kl},k\in[1:K],l\in[1:L], i.e., {\bf Z}_{lx}(k),x\in[1:X] are independent across (k,l,x), security is preserved for all data.

The noise terms protecting each query also have the same structure and independence properties by design. Therefore, it follows from the same reasoning that user’s privacy is protected from any T colluding servers.

Finally, note that the user is able to retrieve L=N-X-T desired q-ary symbols by downloading N q-ary symbols, one from each server. The rate achieved is L/N=1-(X+T)/N, which is the asymptotic capacity for this general setting. This completes the proof of Theorem 3. \hfill\square

6.1 Example: (X=1) Secure, (T=1) Private Scheme with N=5 Servers

Each message consists of L=3 symbols from a finite field \mathbb{F}_{q}, q\geq N+L=8, and characteristic greater than 2. For this setting, \Delta=(1+\alpha)(2+\alpha)(3+\alpha).

\displaystyle\begin{array}[]{cc}\hline&\mbox{Server `$n$' (Replace $\alpha,% \Delta$ with $\alpha_{n},\Delta_{n}$)}\\ \hline\mbox{Storage}&\mathbf{W}_{1}+(1+\alpha)\mathbf{Z}_{1},\\ (S_{n})&\mathbf{W}_{2}+(2+\alpha)\mathbf{Z}_{2},\\ &\mathbf{W}_{3}+(3+\alpha)\mathbf{Z}_{3}\\ \hline\mbox{Query}&\frac{\Delta}{1+\alpha}\Big{(}\mathbf{Q_{\theta}}+(1+\alpha% )\mathbf{Z}_{1}^{\prime}\Big{)},\\ (Q_{n}^{[\theta]})&\frac{\Delta}{2+\alpha}\Big{(}\mathbf{Q_{\theta}}+(2+\alpha% )\mathbf{Z}_{2}^{\prime}\Big{)},\\ &\frac{\Delta}{3+\alpha}\Big{(}\mathbf{Q_{\theta}}+(3+\alpha)\mathbf{Z}_{3}^{% \prime}\Big{)}\\ \hline\omit\span\omit\mbox{Desired symbols appear along vectors}\\ \omit\span\omit\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% 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The answers from all N=5 servers may be written explicitly as,

A^{[\theta]}

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7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{1+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}{\bf W}_{1}{\bf Q}_{\theta}+\leavevmode\hbox to26.42pt{\vbox to% 23.27pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{2+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}{\bf W}_{2}{\bf Q}_{\theta}+\leavevmode\hbox to26.42pt{\vbox to% 23.27pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{3+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}{\bf W}_{3}{\bf Q}_{\theta}
\displaystyle         +\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf W}_{1}{\bf Z}_{1}^{\prime}+{\bf W}_{2}{% \bf Z}_{2}^{\prime}+{\bf W}_{3}{\bf Z}_{3}^{\prime}+{\bf Z}_{1}{\bf Q}_{\theta% }+{\bf Z}_{2}{\bf Q}_{\theta}+{\bf Z}_{3}{\bf Q}_{\theta}\right)}_{I_{10}+I_{2% 0}+I_{30}}
\displaystyle         +\leavevmode\hbox to55.67pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{21.832656pt}{7% .832931pt}\pgfsys@lineto{-21.832656pt}{7.832931pt}\pgfsys@curveto{-24.041795pt% }{7.832931pt}{-25.832656pt}{6.04207pt}{-25.832656pt}{3.832931pt}\pgfsys@lineto% {-25.832656pt}{-3.832931pt}\pgfsys@curveto{-25.832656pt}{-6.04207pt}{-24.04179% 5pt}{-7.832931pt}{-21.832656pt}{-7.832931pt}\pgfsys@lineto{21.832656pt}{-7.832% 931pt}\pgfsys@curveto{24.041795pt}{-7.832931pt}{25.832656pt}{-6.04207pt}{25.83% 2656pt}{-3.832931pt}\pgfsys@lineto{25.832656pt}{3.832931pt}\pgfsys@curveto{25.% 832656pt}{6.04207pt}{24.041795pt}{7.832931pt}{21.832656pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-25.832656pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.499657pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(1+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf Z}_{1}{\bf Z}_{1}^{\prime}\right)}_{I_{% 11}}+\leavevmode\hbox to55.67pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox t% o 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{21.832656pt}{7% .832931pt}\pgfsys@lineto{-21.832656pt}{7.832931pt}\pgfsys@curveto{-24.041795pt% }{7.832931pt}{-25.832656pt}{6.04207pt}{-25.832656pt}{3.832931pt}\pgfsys@lineto% {-25.832656pt}{-3.832931pt}\pgfsys@curveto{-25.832656pt}{-6.04207pt}{-24.04179% 5pt}{-7.832931pt}{-21.832656pt}{-7.832931pt}\pgfsys@lineto{21.832656pt}{-7.832% 931pt}\pgfsys@curveto{24.041795pt}{-7.832931pt}{25.832656pt}{-6.04207pt}{25.83% 2656pt}{-3.832931pt}\pgfsys@lineto{25.832656pt}{3.832931pt}\pgfsys@curveto{25.% 832656pt}{6.04207pt}{24.041795pt}{7.832931pt}{21.832656pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-25.832656pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.499657pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(2+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf Z}_{2}{\bf Z}_{2}^{\prime}\right)}_{I_{% 21}}+\leavevmode\hbox to55.67pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox t% o 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{21.832656pt}{7% .832931pt}\pgfsys@lineto{-21.832656pt}{7.832931pt}\pgfsys@curveto{-24.041795pt% }{7.832931pt}{-25.832656pt}{6.04207pt}{-25.832656pt}{3.832931pt}\pgfsys@lineto% {-25.832656pt}{-3.832931pt}\pgfsys@curveto{-25.832656pt}{-6.04207pt}{-24.04179% 5pt}{-7.832931pt}{-21.832656pt}{-7.832931pt}\pgfsys@lineto{21.832656pt}{-7.832% 931pt}\pgfsys@curveto{24.041795pt}{-7.832931pt}{25.832656pt}{-6.04207pt}{25.83% 2656pt}{-3.832931pt}\pgfsys@lineto{25.832656pt}{3.832931pt}\pgfsys@curveto{25.% 832656pt}{6.04207pt}{24.041795pt}{7.832931pt}{21.832656pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-25.832656pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.499657pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(3+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf Z}_{3}{\bf Z}_{3}^{\prime}\right)}_{I_{% 31}} (59)
\displaystyle=\leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{1+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}W_{\theta 1}+\leavevmode\hbox to26.42pt{\vbox to23.27pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{2+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}W_{\theta 2}+\leavevmode\hbox to26.42pt{\vbox to23.27pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{3+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}W_{\theta 3}
\displaystyle         +\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(I_{10}+I_{20}+I_{30}+I_{11}+2I_{21}+3I_{31})+\leavevmode\hbox to% 25.67pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0.0pt{% \pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{6.832885pt}{7.% 832931pt}\pgfsys@lineto{-6.832885pt}{7.832931pt}\pgfsys@curveto{-9.042024pt}{7% .832931pt}{-10.832885pt}{6.04207pt}{-10.832885pt}{3.832931pt}\pgfsys@lineto{-1% 0.832885pt}{-3.832931pt}\pgfsys@curveto{-10.832885pt}{-6.04207pt}{-9.042024pt}% {-7.832931pt}{-6.832885pt}{-7.832931pt}\pgfsys@lineto{6.832885pt}{-7.832931pt}% \pgfsys@curveto{9.042024pt}{-7.832931pt}{10.832885pt}{-6.04207pt}{10.832885pt}% {-3.832931pt}\pgfsys@lineto{10.832885pt}{3.832931pt}\pgfsys@curveto{10.832885% pt}{6.04207pt}{9.042024pt}{7.832931pt}{6.832885pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-10.832885pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.499886pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(I_{11}+I_{21}+I_{31}). (60)

Privacy and security are guaranteed since 1+\alpha_{n}\neq 0,\forall n\in[1:5], the messages and queries are hidden behind the noise.

Interference terms align into the space spanned by the two vectors, \leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to25.67pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{6.832885pt}{7.% 832931pt}\pgfsys@lineto{-6.832885pt}{7.832931pt}\pgfsys@curveto{-9.042024pt}{7% .832931pt}{-10.832885pt}{6.04207pt}{-10.832885pt}{3.832931pt}\pgfsys@lineto{-1% 0.832885pt}{-3.832931pt}\pgfsys@curveto{-10.832885pt}{-6.04207pt}{-9.042024pt}% {-7.832931pt}{-6.832885pt}{-7.832931pt}\pgfsys@lineto{6.832885pt}{-7.832931pt}% \pgfsys@curveto{9.042024pt}{-7.832931pt}{10.832885pt}{-6.04207pt}{10.832885pt}% {-3.832931pt}\pgfsys@lineto{10.832885pt}{3.832931pt}\pgfsys@curveto{10.832885% pt}{6.04207pt}{9.042024pt}{7.832931pt}{6.832885pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-10.832885pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.499886pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}, while the 3 symbols of the desired message appear along \leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 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\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{1+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% 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{-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{3+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}. Independence of the 3 desired signal dimensions from the two interference dimensions is trivially verified, because the highest exponent of \alpha along desired signal dimensions is 2, but each interference dimension has an \alpha^{3} term (contributed by \Delta). Independence of the 3 desired signal dimensions among themselves is also easily verified, because for

\displaystyle c_{1}(2+\alpha)(3+\alpha)+c_{2}(1+\alpha)(3+\alpha)+c_{3}(1+% \alpha)(2+\alpha) (61)

to be the zero polynomial it must be zero everywhere, but in that case, setting \alpha+i=0 for i=1,2,3, leads us to c_{1}=c_{2}=c_{3}=0, thus proving their independence. The rate achieved is 3/5, which matches the asymptotic capacity for this setting.

6.2 Example: (X=2) Secure, (T=1) Private Scheme with N=4 Servers

Each message consists of L=1 symbol from a finite field \mathbb{F}_{q}, q\geq N+L=5. \Delta=(1+\alpha).

\displaystyle\begin{array}[]{cc}\hline&\mbox{Server `$n$' (Replace $\alpha,% \Delta$ with $\alpha_{n},\Delta_{n}$)}\\ \hline\mbox{Storage $(S_{n})$}&\mathbf{W}_{1}+(1+\alpha)\mathbf{Z}_{11}+(1+% \alpha)^{2}\mathbf{Z}_{12}\\ \hline\mbox{Query $(Q_{n}^{[\theta]})$}&\mathbf{Q_{\theta}}+(1+\alpha)\mathbf{% Z}_{1}^{\prime}\\ \hline\omit\span\omit\mbox{Desired symbols appear along vector}\\ \omit\span\omit\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\\ \omit\span\omit\mbox{Interference symbols appear along vectors}\\ \omit\span\omit\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\Big{(}\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 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0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{20.182682pt}{8% .812916pt}\pgfsys@lineto{-20.182682pt}{8.812916pt}\pgfsys@curveto{-22.391821pt% }{8.812916pt}{-24.182682pt}{7.022055pt}{-24.182682pt}{4.812916pt}% \pgfsys@lineto{-24.182682pt}{-4.812916pt}\pgfsys@curveto{-24.182682pt}{-7.0220% 55pt}{-22.391821pt}{-8.812916pt}{-20.182682pt}{-8.812916pt}\pgfsys@lineto{20.1% 82682pt}{-8.812916pt}\pgfsys@curveto{22.391821pt}{-8.812916pt}{24.182682pt}{-7% .022055pt}{24.182682pt}{-4.812916pt}\pgfsys@lineto{24.182682pt}{4.812916pt}% \pgfsys@curveto{24.182682pt}{7.022055pt}{22.391821pt}{8.812916pt}{20.182682pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-24.182682pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.849682pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha)^{2}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\Big{)}\\ \hline\end{array}

The answers from all N=4 servers may be written explicitly as,

A^{[\theta]}

\displaystyle=\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}{\bf W}_{1}{\bf Q}_{\theta}+\leavevmode\hbox to48.17pt{\vbox to% 19.67pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{18.082714pt}{7% .832931pt}\pgfsys@lineto{-18.082714pt}{7.832931pt}\pgfsys@curveto{-20.291853pt% }{7.832931pt}{-22.082714pt}{6.04207pt}{-22.082714pt}{3.832931pt}\pgfsys@lineto% {-22.082714pt}{-3.832931pt}\pgfsys@curveto{-22.082714pt}{-6.04207pt}{-20.29185% 3pt}{-7.832931pt}{-18.082714pt}{-7.832931pt}\pgfsys@lineto{18.082714pt}{-7.832% 931pt}\pgfsys@curveto{20.291853pt}{-7.832931pt}{22.082714pt}{-6.04207pt}{22.08% 2714pt}{-3.832931pt}\pgfsys@lineto{22.082714pt}{3.832931pt}\pgfsys@curveto{22.% 082714pt}{6.04207pt}{20.291853pt}{7.832931pt}{18.082714pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-22.082714pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.749714pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\left({\bf W}_{1}{\bf Z}_{1}^{\prime}+{\bf Z}_{11}{\bf Q}_{% \theta}\right)+\leavevmode\hbox to52.37pt{\vbox to21.63pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{20.182682pt}{8% .812916pt}\pgfsys@lineto{-20.182682pt}{8.812916pt}\pgfsys@curveto{-22.391821pt% }{8.812916pt}{-24.182682pt}{7.022055pt}{-24.182682pt}{4.812916pt}% \pgfsys@lineto{-24.182682pt}{-4.812916pt}\pgfsys@curveto{-24.182682pt}{-7.0220% 55pt}{-22.391821pt}{-8.812916pt}{-20.182682pt}{-8.812916pt}\pgfsys@lineto{20.1% 82682pt}{-8.812916pt}\pgfsys@curveto{22.391821pt}{-8.812916pt}{24.182682pt}{-7% .022055pt}{24.182682pt}{-4.812916pt}\pgfsys@lineto{24.182682pt}{4.812916pt}% \pgfsys@curveto{24.182682pt}{7.022055pt}{22.391821pt}{8.812916pt}{20.182682pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-24.182682pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.849682pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha)^{2}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\left({\bf Z}_{11}{\bf Z}_{1}^{\prime}+{\bf Z}_{12}{\bf Q}_{% \theta}\right)+\leavevmode\hbox to56.57pt{\vbox to22.71pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{22.28265pt}{9.% 352908pt}\pgfsys@lineto{-22.28265pt}{9.352908pt}\pgfsys@curveto{-24.491789pt}{% 9.352908pt}{-26.28265pt}{7.562047pt}{-26.28265pt}{5.352908pt}\pgfsys@lineto{-2% 6.28265pt}{-5.352908pt}\pgfsys@curveto{-26.28265pt}{-7.562047pt}{-24.491789pt}% {-9.352908pt}{-22.28265pt}{-9.352908pt}\pgfsys@lineto{22.28265pt}{-9.352908pt}% \pgfsys@curveto{24.491789pt}{-9.352908pt}{26.28265pt}{-7.562047pt}{26.28265pt}% {-5.352908pt}\pgfsys@lineto{26.28265pt}{5.352908pt}\pgfsys@curveto{26.28265pt}% {7.562047pt}{24.491789pt}{9.352908pt}{22.28265pt}{9.352908pt}\pgfsys@closepath% \pgfsys@moveto{-26.28265pt}{-9.352908pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-22.94965pt}{-2.939955pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha_{n% })^{3}$}}}{}\pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}{\bf Z}_{12}{\bf Z}_{1}^{\prime} (62)
\displaystyle=\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}W_{\theta 1}+\leavevmode\hbox to18.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\left(\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf W}_{1}{\bf Z}_{1}^{\prime}+{\bf Z}_{11}% {\bf Q}_{\theta}\right)}_{I_{10}}+\leavevmode\hbox to48.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{18.082714pt}{7% .832931pt}\pgfsys@lineto{-18.082714pt}{7.832931pt}\pgfsys@curveto{-20.291853pt% }{7.832931pt}{-22.082714pt}{6.04207pt}{-22.082714pt}{3.832931pt}\pgfsys@lineto% {-22.082714pt}{-3.832931pt}\pgfsys@curveto{-22.082714pt}{-6.04207pt}{-20.29185% 3pt}{-7.832931pt}{-18.082714pt}{-7.832931pt}\pgfsys@lineto{18.082714pt}{-7.832% 931pt}\pgfsys@curveto{20.291853pt}{-7.832931pt}{22.082714pt}{-6.04207pt}{22.08% 2714pt}{-3.832931pt}\pgfsys@lineto{22.082714pt}{3.832931pt}\pgfsys@curveto{22.% 082714pt}{6.04207pt}{20.291853pt}{7.832931pt}{18.082714pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-22.082714pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.749714pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf Z}_{11}{\bf Z}_{1}^{\prime}+{\bf Z}_{12% }{\bf Q}_{\theta}\right)}_{I_{11}}+\leavevmode\hbox to52.37pt{\vbox to21.63pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{20.182682pt}{8% .812916pt}\pgfsys@lineto{-20.182682pt}{8.812916pt}\pgfsys@curveto{-22.391821pt% }{8.812916pt}{-24.182682pt}{7.022055pt}{-24.182682pt}{4.812916pt}% \pgfsys@lineto{-24.182682pt}{-4.812916pt}\pgfsys@curveto{-24.182682pt}{-7.0220% 55pt}{-22.391821pt}{-8.812916pt}{-20.182682pt}{-8.812916pt}\pgfsys@lineto{20.1% 82682pt}{-8.812916pt}\pgfsys@curveto{22.391821pt}{-8.812916pt}{24.182682pt}{-7% .022055pt}{24.182682pt}{-4.812916pt}\pgfsys@lineto{24.182682pt}{4.812916pt}% \pgfsys@curveto{24.182682pt}{7.022055pt}{22.391821pt}{8.812916pt}{20.182682pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-24.182682pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.849682pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$(1+\alpha)^{2}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{{\bf Z}_{12}{\bf Z}_{1}^{\prime}}_{I_{12}}\right) (63)
\displaystyle=\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$1$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}W_{\theta 1}+\leavevmode\hbox to18.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% 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.04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left(I_{10}+I_{11}+I_{12}\right)}_{I_{0}^{\prime}% }+\leavevmode\hbox to25.67pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to % 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 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\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha^{2}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{(I_{12})}_{I_{2}^{\prime}}. (64)

Interference aligns in the space spanned by the three vectors, \leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to25.67pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% 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pt}{6.04207pt}{9.042024pt}{7.832931pt}{6.832885pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-10.832885pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.499886pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to29.87pt{\vbox to21.63pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{8.932854pt}{8.% 812916pt}\pgfsys@lineto{-8.932854pt}{8.812916pt}\pgfsys@curveto{-11.141993pt}{% 8.812916pt}{-12.932854pt}{7.022055pt}{-12.932854pt}{4.812916pt}\pgfsys@lineto{% -12.932854pt}{-4.812916pt}\pgfsys@curveto{-12.932854pt}{-7.022055pt}{-11.14199% 3pt}{-8.812916pt}{-8.932854pt}{-8.812916pt}\pgfsys@lineto{8.932854pt}{-8.81291% 6pt}\pgfsys@curveto{11.141993pt}{-8.812916pt}{12.932854pt}{-7.022055pt}{12.932% 854pt}{-4.812916pt}\pgfsys@lineto{12.932854pt}{4.812916pt}\pgfsys@curveto{12.9% 32854pt}{7.022055pt}{11.141993pt}{8.812916pt}{8.932854pt}{8.812916pt}% \pgfsys@closepath\pgfsys@moveto{-12.932854pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.599854pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha^{2}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}, while the desired symbol appears along the vector of all ones. The independence of these directions is easily established. Privacy is guaranteed because 1+\alpha_{n}\neq 0, \forall n\in[1:4], so the queries are hidden behind random noise. Security is guaranteed because for any X=2 colluding servers, i and j, the independent noise protecting each message W_{k}, k\in[1:K],

\displaystyle\underbrace{\left[\begin{matrix}1+\alpha_{i}&(1+\alpha_{i})^{2}\\ 1+\alpha_{j}&(1+\alpha_{j})^{2}\end{matrix}\right]}_{P_{ij}}\left[\begin{% matrix}{\bf Z}_{11}(k)\\ {\bf Z}_{12}(k)\end{matrix}\right] \displaystyle=\left[\begin{matrix}1+\alpha_{i}&0\\ 0&1+\alpha_{j}\end{matrix}\right]\left[\begin{matrix}1&(1+\alpha_{i})\\ 1&(1+\alpha_{j})\end{matrix}\right]\left[\begin{matrix}{\bf Z}_{11}(k)\\ {\bf Z}_{12}(k)\end{matrix}\right] (65)

spans X=2 dimensions, because P_{ij} is invertible for distinct and non-zero values of (1+\alpha_{i}),(1+\alpha_{j}). The rate achieved is 1/4 which matches the asymptotic capacity for this setting.

6.3 Example: (X=1) Secure, (T=2) Private Scheme with N=5 Servers

Each message consists of L=N-X-T=2 symbols from \mathbb{F}_{q}, q\geq 7.

\displaystyle\Delta \displaystyle=(1+\alpha)(2+\alpha). (66)
\displaystyle\begin{array}[]{cc}\hline&\mbox{Server `$n$' (Replace $\alpha,% \Delta$ with $\alpha_{n},\Delta_{n}$)}\\ \hline\mbox{Storage}&\mathbf{W}_{1}+(1+\alpha)\mathbf{Z}_{1},\\ (S_{n})&\mathbf{W}_{2}+(2+\alpha)\mathbf{Z}_{2}\\ \hline\mbox{Query}&\frac{\Delta}{1+\alpha}\Big{(}\mathbf{Q_{\theta}}+(1+\alpha% )\mathbf{Z}_{11}^{\prime}+(1+\alpha)^{2}\mathbf{Z}_{12}^{\prime}\Big{)},\\ (Q_{n}^{[\theta]})&\frac{\Delta}{2+\alpha}\Big{(}\mathbf{Q_{\theta}}+(2+\alpha% )\mathbf{Z}_{21}^{\prime}+(2+\alpha)^{2}\mathbf{Z}_{22}^{\prime}\Big{)}\\ \hline\omit\span\omit\mbox{Desired symbols appear along vectors}\\ \omit\span\omit\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 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\pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\Big{(}\leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% 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The answers from all N=5 servers may be written explicitly as,

A^{[\theta]}

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7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{1+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}{\bf W}_{1}{\bf Q}_{\theta}+\leavevmode\hbox to26.42pt{\vbox to% 23.27pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% 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\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(2+\alpha)^{2}$}}}{}\pgfsys@endscope}}}% }{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{{\bf Z}_{2}{\bf Z}_{22}^{\prime}}_{I_{22}}
\displaystyle         +\leavevmode\hbox to55.67pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{21.832656pt}{7% .832931pt}\pgfsys@lineto{-21.832656pt}{7.832931pt}\pgfsys@curveto{-24.041795pt% }{7.832931pt}{-25.832656pt}{6.04207pt}{-25.832656pt}{3.832931pt}\pgfsys@lineto% {-25.832656pt}{-3.832931pt}\pgfsys@curveto{-25.832656pt}{-6.04207pt}{-24.04179% 5pt}{-7.832931pt}{-21.832656pt}{-7.832931pt}\pgfsys@lineto{21.832656pt}{-7.832% 931pt}\pgfsys@curveto{24.041795pt}{-7.832931pt}{25.832656pt}{-6.04207pt}{25.83% 2656pt}{-3.832931pt}\pgfsys@lineto{25.832656pt}{3.832931pt}\pgfsys@curveto{25.% 832656pt}{6.04207pt}{24.041795pt}{7.832931pt}{21.832656pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-25.832656pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.499657pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(1+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf Z}_{1}{\bf Z}_{11}^{\prime}+{\bf W}_{1}% {\bf Z}_{12}^{\prime}\right)}_{I_{11}}+\leavevmode\hbox to55.67pt{\vbox to% 19.67pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{21.832656pt}{7% .832931pt}\pgfsys@lineto{-21.832656pt}{7.832931pt}\pgfsys@curveto{-24.041795pt% }{7.832931pt}{-25.832656pt}{6.04207pt}{-25.832656pt}{3.832931pt}\pgfsys@lineto% {-25.832656pt}{-3.832931pt}\pgfsys@curveto{-25.832656pt}{-6.04207pt}{-24.04179% 5pt}{-7.832931pt}{-21.832656pt}{-7.832931pt}\pgfsys@lineto{21.832656pt}{-7.832% 931pt}\pgfsys@curveto{24.041795pt}{-7.832931pt}{25.832656pt}{-6.04207pt}{25.83% 2656pt}{-3.832931pt}\pgfsys@lineto{25.832656pt}{3.832931pt}\pgfsys@curveto{25.% 832656pt}{6.04207pt}{24.041795pt}{7.832931pt}{21.832656pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-25.832656pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.499657pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(2+\alpha)$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{\left({\bf Z}_{2}{\bf Z}_{21}^{\prime}+{\bf W}_{2}% {\bf Z}_{22}^{\prime}\right)}_{I_{21}}+\leavevmode\hbox to59.87pt{\vbox to% 21.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{23.932624pt}{8% .812916pt}\pgfsys@lineto{-23.932624pt}{8.812916pt}\pgfsys@curveto{-26.141763pt% }{8.812916pt}{-27.932624pt}{7.022055pt}{-27.932624pt}{4.812916pt}% \pgfsys@lineto{-27.932624pt}{-4.812916pt}\pgfsys@curveto{-27.932624pt}{-7.0220% 55pt}{-26.141763pt}{-8.812916pt}{-23.932624pt}{-8.812916pt}\pgfsys@lineto{23.9% 32624pt}{-8.812916pt}\pgfsys@curveto{26.141763pt}{-8.812916pt}{27.932624pt}{-7% .022055pt}{27.932624pt}{-4.812916pt}\pgfsys@lineto{27.932624pt}{4.812916pt}% \pgfsys@curveto{27.932624pt}{7.022055pt}{26.141763pt}{8.812916pt}{23.932624pt}% {8.812916pt}\pgfsys@closepath\pgfsys@moveto{-27.932624pt}{-8.812916pt}% \pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-24.599625pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta(1+\alpha)^{2}$}}}{}\pgfsys@endscope}}}% }{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}\underbrace{{\bf Z}_{1}{\bf Z}_{12}^{\prime}}_{I_{12}} (67)
\displaystyle=\leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{1+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}W_{\theta 1}+\leavevmode\hbox to26.42pt{\vbox to23.27pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{2+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}W_{\theta 2}+\leavevmode\hbox to18.17pt{\vbox to19.67pt{% \pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(I_{10}+I_{20}+I_{11}+2I_{21}+I_{12}+4I_{22})
\displaystyle         +\leavevmode\hbox to25.67pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{6.832885pt}{7.% 832931pt}\pgfsys@lineto{-6.832885pt}{7.832931pt}\pgfsys@curveto{-9.042024pt}{7% .832931pt}{-10.832885pt}{6.04207pt}{-10.832885pt}{3.832931pt}\pgfsys@lineto{-1% 0.832885pt}{-3.832931pt}\pgfsys@curveto{-10.832885pt}{-6.04207pt}{-9.042024pt}% {-7.832931pt}{-6.832885pt}{-7.832931pt}\pgfsys@lineto{6.832885pt}{-7.832931pt}% \pgfsys@curveto{9.042024pt}{-7.832931pt}{10.832885pt}{-6.04207pt}{10.832885pt}% {-3.832931pt}\pgfsys@lineto{10.832885pt}{3.832931pt}\pgfsys@curveto{10.832885% pt}{6.04207pt}{9.042024pt}{7.832931pt}{6.832885pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-10.832885pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.499886pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(I_{11}+I_{21}+2I_{12}+4I_{22})+\leavevmode\hbox to29.87pt{% \vbox to21.63pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{8.932854pt}{8.% 812916pt}\pgfsys@lineto{-8.932854pt}{8.812916pt}\pgfsys@curveto{-11.141993pt}{% 8.812916pt}{-12.932854pt}{7.022055pt}{-12.932854pt}{4.812916pt}\pgfsys@lineto{% -12.932854pt}{-4.812916pt}\pgfsys@curveto{-12.932854pt}{-7.022055pt}{-11.14199% 3pt}{-8.812916pt}{-8.932854pt}{-8.812916pt}\pgfsys@lineto{8.932854pt}{-8.81291% 6pt}\pgfsys@curveto{11.141993pt}{-8.812916pt}{12.932854pt}{-7.022055pt}{12.932% 854pt}{-4.812916pt}\pgfsys@lineto{12.932854pt}{4.812916pt}\pgfsys@curveto{12.9% 32854pt}{7.022055pt}{11.141993pt}{8.812916pt}{8.932854pt}{8.812916pt}% \pgfsys@closepath\pgfsys@moveto{-12.932854pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.599854pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha^{2}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}(I_{12}+I_{22}). (68)

Thus, interference aligns into the space spanned by the 3 vectors: \leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% \pgfsys@curveto{5.292082pt}{-7.832931pt}{7.082943pt}{-6.04207pt}{7.082943pt}{-% 3.832931pt}\pgfsys@lineto{7.082943pt}{3.832931pt}\pgfsys@curveto{7.082943pt}{6% .04207pt}{5.292082pt}{7.832931pt}{3.082943pt}{7.832931pt}\pgfsys@closepath% \pgfsys@moveto{-7.082943pt}{-7.832931pt}\pgfsys@fill\pgfsys@invoke{ }{}% \pgfsys@endscope}{{{{}}{\pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{-3.749943pt}{-2.499962pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\Delta$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to25.67pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{6.832885pt}{7.% 832931pt}\pgfsys@lineto{-6.832885pt}{7.832931pt}\pgfsys@curveto{-9.042024pt}{7% .832931pt}{-10.832885pt}{6.04207pt}{-10.832885pt}{3.832931pt}\pgfsys@lineto{-1% 0.832885pt}{-3.832931pt}\pgfsys@curveto{-10.832885pt}{-6.04207pt}{-9.042024pt}% {-7.832931pt}{-6.832885pt}{-7.832931pt}\pgfsys@lineto{6.832885pt}{-7.832931pt}% \pgfsys@curveto{9.042024pt}{-7.832931pt}{10.832885pt}{-6.04207pt}{10.832885pt}% {-3.832931pt}\pgfsys@lineto{10.832885pt}{3.832931pt}\pgfsys@curveto{10.832885% pt}{6.04207pt}{9.042024pt}{7.832931pt}{6.832885pt}{7.832931pt}% \pgfsys@closepath\pgfsys@moveto{-10.832885pt}{-7.832931pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.499886pt}{-2.499962pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to29.87pt{\vbox to21.63pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{8.932854pt}{8.% 812916pt}\pgfsys@lineto{-8.932854pt}{8.812916pt}\pgfsys@curveto{-11.141993pt}{% 8.812916pt}{-12.932854pt}{7.022055pt}{-12.932854pt}{4.812916pt}\pgfsys@lineto{% -12.932854pt}{-4.812916pt}\pgfsys@curveto{-12.932854pt}{-7.022055pt}{-11.14199% 3pt}{-8.812916pt}{-8.932854pt}{-8.812916pt}\pgfsys@lineto{8.932854pt}{-8.81291% 6pt}\pgfsys@curveto{11.141993pt}{-8.812916pt}{12.932854pt}{-7.022055pt}{12.932% 854pt}{-4.812916pt}\pgfsys@lineto{12.932854pt}{4.812916pt}\pgfsys@curveto{12.9% 32854pt}{7.022055pt}{11.141993pt}{8.812916pt}{8.932854pt}{8.812916pt}% \pgfsys@closepath\pgfsys@moveto{-12.932854pt}{-8.812916pt}\pgfsys@fill% \pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{\pgfsys@beginscope% \pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.599854pt}{-3.479947pt}{}\hbox{{% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@color@rgb@fill{0}{0}{0}{$\Delta\alpha^{2}$}}}{}\pgfsys@endscope}}}}{}% \pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}, while the 2 desired symbols appear along \leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture\makeatletter\hbox to 0.% 0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb}{0,0,0}% \pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{1+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}},\leavevmode\hbox to26.42pt{\vbox to23.27pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{7.20788pt}{9.6% 329pt}\pgfsys@lineto{-7.20788pt}{9.6329pt}\pgfsys@curveto{-9.417019pt}{9.6329% pt}{-11.20788pt}{7.842039pt}{-11.20788pt}{5.6329pt}\pgfsys@lineto{-11.20788pt}% {-5.6329pt}\pgfsys@curveto{-11.20788pt}{-7.842039pt}{-9.417019pt}{-9.6329pt}{-% 7.20788pt}{-9.6329pt}\pgfsys@lineto{7.20788pt}{-9.6329pt}\pgfsys@curveto{9.417% 019pt}{-9.6329pt}{11.20788pt}{-7.842039pt}{11.20788pt}{-5.6329pt}% \pgfsys@lineto{11.20788pt}{5.6329pt}\pgfsys@curveto{11.20788pt}{7.842039pt}{9.% 417019pt}{9.6329pt}{7.20788pt}{9.6329pt}\pgfsys@closepath\pgfsys@moveto{-11.20% 788pt}{-9.6329pt}\pgfsys@fill\pgfsys@invoke{ }{}\pgfsys@endscope}{{{{}}{% \pgfsys@beginscope\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.87488pt}{-3.14995% 2pt}{}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{% 0}{0}{0}\pgfsys@color@rgb@fill{0}{0}{0}{$\frac{\Delta}{2+\alpha}$}}}{}% \pgfsys@endscope}}}}{}\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}. Note that the highest exponent of \alpha along a desired signal dimension is 1, but every interference dimension contains \alpha^{2} (contributed by \Delta), so the desired signals are independent of the interference. The independence of desired signals among themselves is also easily verified because if c_{1}\frac{\Delta}{1+\alpha}+c_{2}\frac{\Delta}{2+\alpha}=c_{1}(2+\alpha)+c_{2% }(1+\alpha) is the zero polynomial, then by substituting i+\alpha=0 for i=1,2 we find that we must have c_{1}=c_{2}=0. Privacy and security are guaranteed by the MDS coded independent noise terms mixed with the message and query symbols. The rate achieved is 2/5, which matches the asymptotic capacity for this setting.

6.4 Example: (X=2) Secure, (T=2) Private Scheme with N=7 Servers

Each message consists of L=3 symbols from a finite field \mathbb{F}_{q}, of size q\geq 10 and characteristic greater than 2.

\Delta=(1+\alpha)(2+\alpha)(3+\alpha).
\displaystyle\begin{array}[]{cc}\hline&\mbox{Server `$n$' (Replace $\alpha,% \Delta$ with $\alpha_{n},\Delta_{n})$}\\ \hline\mbox{Storage}&\mathbf{W}_{1}+(1+\alpha)\mathbf{Z}_{11}+(1+\alpha)^{2}% \mathbf{Z}_{12},\\ (S_{n})&\mathbf{W}_{2}+(2+\alpha)\mathbf{Z}_{21}+(2+\alpha)^{2}\mathbf{Z}_{22}% ,\\ &\mathbf{W}_{3}+(3+\alpha)\mathbf{Z}_{31}+(3+\alpha)^{2}\mathbf{Z}_{32}\\ \hline\mbox{Query}&\frac{\Delta}{1+\alpha}\Big{(}\mathbf{Q_{\theta}}+(1+\alpha% )\mathbf{Z}_{11}^{\prime}+(1+\alpha)^{2}\mathbf{Z}_{12}^{\prime}\Big{)},\\ (Q_{n}^{[\theta]})&\frac{\Delta}{2+\alpha}\Big{(}\mathbf{Q_{\theta}}+(2+\alpha% )\mathbf{Z}_{21}^{\prime}+(2+\alpha)^{2}\mathbf{Z}_{22}^{\prime}\Big{)},\\ &\frac{\Delta}{3+\alpha}\Big{(}\mathbf{Q_{\theta}}+(3+\alpha)\mathbf{Z}_{31}^{% \prime}+(3+\alpha)^{2}\mathbf{Z}_{32}^{\prime}\Big{)}\\ \hline\omit\span\omit\mbox{Desired symbols appear along vectors}\\ \omit\span\omit\leavevmode\hbox to18.17pt{\vbox to19.67pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope\definecolor{pgfstrokecolor}{rgb% }{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{}% \pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{ \pgfsys@beginscope {{}}\hbox{\hbox{{\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{% 0.93,0.93,0.93}\pgfsys@color@gray@fill{0.93}{}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {\pgfsys@beginscope\definecolor{pgffillcolor}{rgb}{0.93,0.93,0.93}% \pgfsys@color@gray@fill{0.93}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{% }{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{3.082943pt}{7.% 832931pt}\pgfsys@lineto{-3.082943pt}{7.832931pt}\pgfsys@curveto{-5.292082pt}{7% .832931pt}{-7.082943pt}{6.04207pt}{-7.082943pt}{3.832931pt}\pgfsys@lineto{-7.0% 82943pt}{-3.832931pt}\pgfsys@curveto{-7.082943pt}{-6.04207pt}{-5.292082pt}{-7.% 832931pt}{-3.082943pt}{-7.832931pt}\pgfsys@lineto{3.082943pt}{-7.832931pt}% 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\pgfsys@endscope}}} \pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath{}{}{}\pgfsys@endscope\hss}% \endpgfpicture}}. Independence of desired signals from interference is trivially verified – highest exponent of \alpha along any desired signal dimension is 2, but each interference dimension has an \alpha^{3} term (contributed by \Delta). The desired signal dimensions are easily verified to be linearly independent among themselves because in order for

\displaystyle c_{1}(2+\alpha)(3+\alpha)+c_{2}(1+\alpha)(3+\alpha)+c_{3}(1+% \alpha)(2+\alpha) (69)

to be the zero polynomial it must be zero everywhere, but in that case, setting \alpha+i=0 for i=1,2,3 leads us to c_{1}=c_{2}=c_{3}=0. Privacy and security are guaranteed by the MDS coded independent noise terms mixed with the message and query symbols. The rate achieved is 3/7, which matches the asymptotic capacity for this setting.

7 Proof of Theorem 4

In Section 6 we presented an XSTPIR scheme for arbitrary X,T,N,K that achieves capacity as K\rightarrow\infty. Since the scheme also works for any K, a natural starting point for finite K settings is to apply the same scheme. A key insight here is that the rate achieved by the scheme improves as K decreases. Let us elaborate. Note that the query Q_{n}^{[\theta]} that is sent to each server is uniformly distributed in \mathbb{F}_{q}^{LK}. Therefore, with probability \frac{1}{q^{LK}}, the query vector is the all zero vector. Whenever this happens, no download is needed from the server. Thus, the average download is reduced by the factor (1-1/q^{LK}) and the rate achieved is expressed as follows.

Lemma 1

The asymptotically capacity achieving XSTPIR scheme of Section 6 achieves the rate

\displaystyle R \displaystyle=\left(1-\frac{1}{q^{KL}}\right)^{-1}\left(1-\left(\frac{X+T}{N}% \right)\right) (70)

for arbitrary X,L,K,N values, where N>X+T.

Note that N\leq X+T is excluded as the degenerate setting where we already know the capacity for all parameters, according to Theorem 2. Remarkably, the rate in Lemma 1 depends on the message size L and the field size q used by the scheme. As presented, the scheme uses q\geq L+N and L=N-X-T. So the achieved rate for finite K becomes

\displaystyle R \displaystyle=\left(1-\frac{1}{(2N-X-T)^{K(N-X-T)}}\right)^{-1}\left(1-\left(% \frac{X+T}{N}\right)\right). (71)

Consider the simplest non-trivial setting of interest, i.e., the setting for Theorem 4, where T=X=1, N=3 and K is arbitrary. The scheme of Section 6 uses L=1,q\geq 4, so the rate achieved for arbitrary K is

\displaystyle R \displaystyle=\frac{1}{3}\left(1-\frac{1}{4^{K}}\right)^{-1}. (72)

However, note that if the field size could be reduced to q=2, then the rate achieved by the scheme would become

\frac{1}{3}\left(1-\frac{1}{2^{K}}\right)^{-1}=\frac{2}{3}\left(\frac{1-\frac{% 1}{2}}{1-\frac{1}{2^{K}}}\right)=\frac{2}{3}\left(1+\frac{1}{2}+\frac{1}{2^{2}% }+\cdots+\frac{1}{2^{K-1}}\right)^{-1}

which matches the capacity upper bound from Theorem 1. Surprisingly, this can be done with some modification to the structure of the scheme, as explained below.

Suppose each message W_{k},k\in[1:K] consists of L=1 symbol (bit) from \mathbb{F}_{2}. Let {\bf W}=(W_{1},W_{2},\cdots,W_{K}) be a random row vector in \mathbb{F}_{2}^{1\times K}, containing all messages. Let {\bf Z} and {\bf Z}^{\prime} be uniformly random noise vectors from \mathbb{F}_{2}^{1\times K} and \mathbb{F}_{2}^{K\times 1}, that are used to guarantee data security and user privacy, respectively. The noise vectors are independent of each other and of the message vector and \theta, i.e., H({\bf W},{\bf Z},{\bf Z}^{\prime},\theta)=H({\bf W})+H({\bf Z})+H({\bf Z}^{% \prime})+H(\theta). Let {\bf Q}_{\theta} represent the \theta^{th} column of {\bf I}_{K} (the K\times K identity matrix). Note that {\bf W}{\bf Q}_{\theta}=W_{\theta} is the message desired by the user. The storage at the servers, the queries and the answers are listed below.

\displaystyle\begin{array}[]{|c|c|c|c|}\hline&\mbox{Server $1$}&\mbox{Server $% 2$}&\mbox{Server $3$}\\ \hline\mbox{Storage $S_{n}$}&\mathbf{W}+\mathbf{Z}&\mathbf{W}+\mathbf{Z}% \mathbf{B}&\mathbf{Z}\\ \hline\mbox{Query $Q_{n}^{[\theta]}$}&\mathbf{Z^{\prime}}&\mathbf{Q_{\theta}}+% \mathbf{Z^{\prime}}&(\mathbf{I}_{K}+\mathbf{B})\mathbf{Z^{\prime}}+\mathbf{B}{% \bf Q}_{\theta}\\ \hline\mbox{Answer $A_{n}^{[\theta]}$}&\mathbf{W}\mathbf{Z^{\prime}}+\mathbf{Z% }\mathbf{Z}^{\prime}&\mathbf{W}\mathbf{Q_{\theta}}+\mathbf{W}\mathbf{Z^{\prime% }}+\mathbf{Z}\mathbf{B}\mathbf{Z}^{\prime}+\mathbf{Z}\mathbf{B}\mathbf{Q_{% \theta}}&\mathbf{Z}\mathbf{Z}^{\prime}+\mathbf{Z}\mathbf{B}\mathbf{Z}^{\prime}% +\mathbf{Z}\mathbf{B}\mathbf{Q_{\theta}}\\ \hline\end{array}

where {\bf B} is a K\times K deterministic binary matrix such that {\bf B} and {\bf I}_{K}+{\bf B} are both full rank. Any such choice of {\bf B} will work for our scheme. The existence of such {\bf B} is established in the following lemma whose proof appears in Appendix A.

Lemma 2

For all K\geq 2, there exists a matrix {\bf B}\in\mathbb{F}_{2}^{K\times K} such that {\bf B} and {\bf I}_{K}+{\bf B} are both invertible.

Now, let us check the correctness, security and privacy of this scheme. The scheme is obviously correct because by adding the three answers shown in the table above, the user recovers W_{\theta}. It is obviously secure because {\bf B} is invertible, so {\bf Z}{\bf B}\sim{\bf Z}, is still uniform noise independent of {\bf W}. And similarly, it is also obviously private, because {\bf I}_{K}+{\bf B} is also invertible, so ({\bf I}_{K}+{\bf B}){\bf Z}^{\prime}\sim{\bf Z}^{\prime} is still uniform noise independent of {\bf B}{\bf Q}_{\theta}. Thus, surprisingly, we have achieved the capacity of XSTPIR for arbitrary K, when X=T=1 and N=3, completing the proof of Theorem 4. \hfill\square

8 Conclusion

The XSTPIR problem is timely due to the growing importance of privacy and security concerns in modern information storage and retrieval systems. It is a conceptually rich topic that reveals new insights into alignment of noise terms, dependence of coding and query structures, cost of symmetric security, significance of field size for the rate of information retrieval, etc. As indicated by various open problems identified here, XSTPIR is a fertile research avenue for future work. In particular, the capacity characterization for arbitrary K could reveal fundamentally new schemes for PIR. Especially intriguing would be the role that field size might play in such a result. Capacity of Sym-XSTPIR is another promising open problem. XSTPIR with constraints on the amount of storage per server, coded storage, multi-message retrieval are other open problems that merit investigation.

Appendix A Proof of Lemma 2

Let {\bf J}_{k} denote the k\times k anti-diagonal identity matrix, and let {\bf 0}_{k_{1}\times k_{2}} denote the k_{1}\times k_{2} matrix where all elements are equal to 0 (when k_{1}=k_{2}, this notation is further simplified to {\bf 0}_{k_{1}}). Define

\displaystyle{\bf I}^{\prime}_{k}=\left[\begin{array}[]{cc}{\bf I}_{k}&{\bf 0}% _{k\times 1}\\ {\bf 0}_{1\times k}&0\end{array}\right]. (73)

Choose {\bf B} as follows.

\displaystyle{\bf B} \displaystyle=\left\{\begin{array}[]{ll}\left[\begin{array}[]{cc}{\bf I}_{% \frac{K}{2}}&{\bf J}_{\frac{K}{2}}\\ {\bf J}_{\frac{K}{2}}&{\bf 0}_{\frac{K}{2}}\end{array}\right],&\mbox{if }K% \mbox{ is even,}\\ \\ \left[\begin{array}[]{c|c}{\bf J}_{\frac{K+1}{2}}+{\bf I}^{\prime}_{\frac{K-1}% {2}}+{\bf I}_{\frac{K+1}{2}}&\begin{array}[]{cc}{\bf J}_{\frac{K-1}{2}}\\ {\bf 0}_{1\times\frac{K-1}{2}}\end{array}\\ \hdashline{\bf J}_{\frac{K-1}{2}}~{}~{}\begin{array}[]{c}{\bf 0}_{\frac{K-1}{2% }\times 1}\end{array}&{\bf 0}_{\frac{K-1}{2}}\end{array}\right],&\mbox{if }K% \mbox{ is odd.}\\ \end{array}\right. (74)

For example,

when K=4: \displaystyle{\bf B}=\left[\begin{array}[]{cccc}1&0&0&1\\ 0&1&1&0\\ 0&1&0&0\\ 1&0&0&0\end{array}\right], and when K=5: \displaystyle{\bf B}=\left[\begin{array}[]{ccccc}0&0&1&0&1\\ 0&1&0&1&0\\ 1&0&1&0&0\\ 0&1&0&0&0\\ 1&0&0&0&0\end{array}\right]. (75)

Let us show that {\bf B} and {\bf I}_{K}+{\bf B} are both invertible.

First, consider {\bf B}. Regardless of whether K is even or odd, {\bf B} is an upper anti-triangular matrix where all anti-diagonal elements are 1 so that \det({\bf B})=1 and {\bf B} has full rank.

Next, consider {\bf I}_{K}+{\bf B}.

When K is even: \displaystyle~{}~{}{\bf I}_{K}+{\bf B}=\left[\begin{array}[]{cc}{\bf I}_{\frac% {K}{2}}&{\bf 0}_{\frac{K}{2}}\\ {\bf 0}_{\frac{K}{2}}&{\bf I}_{\frac{K}{2}}\end{array}\right]+\left[\begin{% array}[]{cc}{\bf I}_{\frac{K}{2}}&{\bf J}_{\frac{K}{2}}\\ {\bf J}_{\frac{K}{2}}&{\bf 0}_{\frac{K}{2}}\end{array}\right]=\left[\begin{% array}[]{cc}{\bf 0}_{\frac{K}{2}}&{\bf J}_{\frac{K}{2}}\\ {\bf J}_{\frac{K}{2}}&{\bf I}_{\frac{K}{2}}\end{array}\right] (76)
\displaystyle~{}~{}\Rightarrow~{}~{}\det({\bf I}_{K}+{\bf B})=1. (77)
When K is odd: \displaystyle~{}~{}{\bf I}_{K}+{\bf B}=\left[\begin{array}[]{c|c}{\bf I}_{% \frac{K+1}{2}}&{\bf 0}_{\frac{K+1}{2}\times\frac{K-1}{2}}\\ \hdashline{\bf 0}_{\frac{K-1}{2}\times\frac{K+1}{2}}&{\bf I}_{\frac{K-1}{2}}% \end{array}\right]+\left[\begin{array}[]{c|c}{\bf J}_{\frac{K+1}{2}}+{\bf I}^{% \prime}_{\frac{K-1}{2}}+{\bf I}_{\frac{K+1}{2}}&\begin{array}[]{cc}{\bf J}_{% \frac{K-1}{2}}\\ {\bf 0}_{1\times\frac{K-1}{2}}\end{array}\\ \hdashline{\bf J}_{\frac{K-1}{2}}~{}~{}\begin{array}[]{c}{\bf 0}_{\frac{K-1}{2% }\times 1}\end{array}&{\bf 0}_{\frac{K-1}{2}}\end{array}\right] (78)
\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\left[\begin{array}[]{c|c}{\bf J% }_{\frac{K+1}{2}}+{\bf I}^{\prime}_{\frac{K-1}{2}}&\begin{array}[]{cc}{\bf J}_% {\frac{K-1}{2}}\\ {\bf 0}_{1\times\frac{K-1}{2}}\end{array}\\ \hdashline{\bf J}_{\frac{K-1}{2}}~{}~{}\begin{array}[]{c}{\bf 0}_{\frac{K-1}{2% }\times 1}\end{array}&{\bf I}_{\frac{K-1}{2}}\end{array}\right] (79)
\displaystyle~{}~{}\Rightarrow~{}~{}\det({\bf I}_{K}+{\bf B})=\det\left({\bf J% }_{\frac{K+1}{2}}+{\bf I}^{\prime}_{\frac{K-1}{2}}+\left[\begin{array}[]{cc}{% \bf J}_{\frac{K-1}{2}}\\ {\bf 0}_{1\times\frac{K-1}{2}}\end{array}\right]{\bf I}_{\frac{K-1}{2}}^{-1}% \left[{\bf J}_{\frac{K-1}{2}}~{}~{}\begin{array}[]{c}{\bf 0}_{\frac{K-1}{2}% \times 1}\end{array}\right]\right) (80)
\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}=\det\left({\bf J}_{\frac{K+1}{2}}+{\bf I}^{\prime}_{\frac{K-1}{2}}+% {\bf I}^{\prime}_{\frac{K-1}{2}}\right)=\det\left({\bf J}_{\frac{K+1}{2}}% \right)=1. (81)

The proof is thus complete.\hfill\square

Appendix B Proof of Corollaries 1, 2, 3, 4

The proof of Corollary 1 is trivial because imposing the symmetric security constraint cannot increase capacity.

B.1 Proof of Corollary 2

To prove Corollary 2 we provide a scheme as follows. Each message W_{k},k\in[1:K], consists of L=1 symbol from some finite field \mathbb{F}_{q}. Let Z_{x,k,m},x\in[1:X],k\in[1:K],m\in[1:K] be independent uniform noise symbols from \mathbb{F}_{q}. The subscript, m, in Z_{x,k,m} is interpreted modulo K, i.e., Z_{x,k,m}=Z_{x,k,m+K}. The storage at each server is specified as,

\displaystyle S_{n} \displaystyle=\{Z_{n,k,m},k\in[1:K],m\in[1:K]\}, \displaystyle n\in[1:X], (82)
\displaystyle S_{n} \displaystyle=\left\{W_{k}+\sum_{x=1}^{X}Z_{x,k,m},k\in[1:K],m\in[1:K]\right\}, \displaystyle n=N. (83)

The queries from each server are specified as,

\displaystyle Q^{[\theta]}_{n} \displaystyle:\mbox{Ask for }\{Z_{n,k,m_{o}},k\in[1:K]\}, \displaystyle n\in[1:X], (84)
\displaystyle Q^{[\theta]}_{n} \displaystyle:\mbox{Ask for }\{W_{k}+\sum_{x=1}^{X}Z_{x,k,m_{o}-\theta+k},k\in% [1:K]\}, \displaystyle n=N, (85)

where m_{o} is chosen privately and uniformly randomly by the user from [1:K]. Thus, in order to retrieve 1 desired message symbol, the user downloads a total of KN symbols from all servers. The scheme is X-secure because each message symbol is protected by independent uniform noise terms. It is correct because for k=\theta the download from Server N, contains the symbol W_{\theta}+\sum_{x=1}^{X}Z_{x,\theta,m_{o}} and the downloads from the first X servers include all the noise terms Z_{x,\theta,m_{o}}. The scheme is private because m_{o} is chosen uniformly and privately by the user. It satisfies symmetric security because all the undesired message symbols W_{k},k\neq\theta, contained in the answers are protected by noise terms Z_{x,k,m_{o}-\theta+k} and these noise terms are independent of the noise terms downloaded from servers n\in[1:X] because m_{o}-\theta+k\neq m_{o} when k\neq\theta. The rate achieved is \frac{1}{KN}, which is the capacity for this setting.\hfill\square

Note that in the Sym-XSPIR scheme described above, each server stores K^{2} symbols, when the total data is only KL=K symbols. Thus, this Sym-XSPIR scheme takes advantage of unconstrained storage when K is large, more so than the XSTPIR schemes which store no more than KL symbols at each server.

B.2 Proof of Corollary 3

Define

\displaystyle{\bf W}_{i}^{c} \displaystyle=\{{\bf W}_{l},l\in[1:L],l\neq i\} (86)
\displaystyle{\bf Z}_{ij}^{c} \displaystyle=\{{\bf Z}_{lx},l\in[1:L],x\in[1:X],(l,x)\neq(i,j)\}. (87)
\displaystyle I\left({\bf W}_{[1:L]};A_{[1:N]}^{[\theta]}\mid W_{\theta},Q^{[% \theta]}_{[1:N]},\theta\right)\leq\sum_{l\in[1:L]}I\left({\bf W}_{l};A_{[1:N]}% ^{[\theta]}\mid{\bf W}_{l}^{c},W_{\theta},Q^{[\theta]}_{[1:N]},\theta\right) (88)
\displaystyle\leq\sum_{l\in[1:L]}I\left({\bf W}_{l};A_{[1:N]}^{[\theta]}\mid{% \bf Z}_{l1}^{c},{\bf W}_{l}^{c},W_{\theta},Q^{[\theta]}_{[1:N]},\theta\right) (89)

where we used the fact that I(A;B\mid C)\leq I(A;B\mid C,D) if I(A;D\mid C)=0. To prove Corollary 3 it suffices to show that each of the terms in the summation is zero. Without loss of generality, let us consider l=1, and define

\displaystyle{A^{\prime}}^{[\theta]}_{n} \displaystyle=\left({\bf W}_{1}+(1+\alpha_{n}){\bf Z}_{11}\right)\left(\frac{% \Delta_{n}}{1+\alpha_{n}}\right)\left({\bf Q}_{\theta}+(1+\alpha_{n}){\bf Z}^{% \prime}_{1}\right) (90)
\displaystyle=\left(\frac{\Delta_{n}}{1+\alpha_{n}}\right){\bf W}_{1}{\bf Q}_{% \theta}+\Delta_{n}({\bf W}_{1}{\bf Z}^{\prime}_{1}+{\bf Z}_{11}{\bf Q}_{\theta% })+\Delta_{n}(1+\alpha_{n}){\bf Z}_{11}{\bf Z}_{1}^{\prime} (91)
\displaystyle=\left(\frac{\Delta_{n}}{1+\alpha_{n}}\right)W_{\theta 1}+\Delta_% {n}({\bf W}_{1}{\bf Z}^{\prime}_{1}+{\bf Z}_{11}(\theta))+\Delta_{n}(1+\alpha_% {n}){\bf Z}_{11}{\bf Z}_{1}^{\prime} (92)

where {\bf Z}_{11}(i) is the i^{th} element of the vector {\bf Z}_{11}. Note that after removing the terms containing only those random variables whose values are fixed by the conditioning on {\bf Z}_{11}^{c},{\bf W}_{1}^{c},W_{\theta},Q^{[\theta]}_{[1:N]},\theta, all that is left of A_{[1:N]}^{[\theta]} is {A^{\prime}}^{[\theta]}_{[1:N]}.

\displaystyle I\left({\bf W}_{1};A_{[1:N]}^{[\theta]}\mid{\bf Z}_{11}^{c},{\bf W% }_{1}^{c},W_{\theta},Q^{[\theta]}_{[1:N]},\theta\right) (93)
\displaystyle=I\left({\bf W}_{1};{A^{\prime}}_{[1:N]}^{[\theta]}\mid{\bf Z}_{1% 1}^{c},{\bf W}_{1}^{c},W_{\theta},Q^{[\theta]}_{[1:N]},\theta\right) (94)
\displaystyle=I\left({\bf W}_{1};{A^{\prime}}_{[1:N]}^{[\theta]}\mid W_{\theta 1% },Q^{[\theta]}_{[1:N]},\theta\right) (95)
\displaystyle\leq I({\bf W}_{1};{\bf W}_{1}{\bf Z}_{1}^{\prime}+{\bf Z}_{11}(% \theta),{\bf Z}_{11}{\bf Z}_{1}^{\prime}\mid W_{\theta 1},{\bf Q}_{\theta},{% \bf Z}_{1}^{\prime},\theta) (96)
\displaystyle=0. (97)

The last step is justified as follows. There are only three possibilities: {\bf Z}_{1}^{\prime} is either the zero vector, or it is equal to \mu{\bf Q}_{\theta} for some non-zero \mu\in\mathbb{F}_{q}, or it is neither zero nor equal to \mu{\bf Q}_{\theta}. If {\bf Z}_{1}^{\prime} is the zero vector, then the mutual information is automatically zero because {\bf W}_{1} is eliminated entirely. If {\bf Z}_{1}^{\prime}=\mu{\bf Q}_{\theta} for some non-zero \mu, then {\bf W}_{1}{\bf Z}_{1}^{\prime}=\mu W_{\theta 1} and the mutual information is again zero because of the conditioning on W_{\theta 1}. Finally, if {\bf Z}_{1}^{\prime} is neither zero nor a scaled version of {\bf Q}_{\theta}, then {\bf Z}_{11}{\bf Z}_{1}^{\prime} is a uniformly random noise term in \mathbb{F}_{q} that is independent of {\bf Z}_{11}(\theta) and {\bf Z}_{1}^{\prime}. So in that case also the mutual information is zero. This completes the proof of Corollary 3. \hfill\square

B.3 Proof of Corollary 4

The proof of Corollary 4 is presented next. Recall that in the scheme of the proof of Theorem 4, the user obtains the following three symbols from the answers,

\displaystyle\mathbf{W}\mathbf{Q_{\theta}}=W_{\theta} (98)
\displaystyle\mathbf{W}\mathbf{Z}^{\prime}+\mathbf{Z}\mathbf{Z}^{\prime} (99)
\displaystyle\mathbf{W}\mathbf{Z}^{\prime}+\mathbf{Z}\mathbf{B}\mathbf{Z}^{% \prime}+\mathbf{Z}\mathbf{B}\mathbf{Q}_{\theta}. (100)

We show that symmetric security holds, i.e., conditioned on {\bf Z}^{\prime}, from these three symbols the user learns nothing about the undesired messages W_{1},\cdots,W_{\theta-1},W_{\theta+1},\cdots,W_{K}. When \mathbf{Z}^{\prime} is the zero vector, the symbol \mathbf{W}\mathbf{Z}^{\prime} is zero as well, leaking nothing about the undesired messages. Now consider (99). If \mathbf{Z}^{\prime} is not the zero vector, then the symbol \mathbf{W}\mathbf{Z}^{\prime} is protected by an independent noise term. Similarly, consider (100) and consider three possibilities: {\bf B(Z^{\prime}+Q_{\theta})} is either zero, or equal to {\bf Z}^{\prime}, or not zero and not equal to {\bf Z}^{\prime}. If \mathbf{B}(\mathbf{Z}^{\prime}+{\bf Q}_{\theta}) is the zero vector, then because {\bf B} is invertible, we must have {\bf Z}^{\prime}={\bf Q}_{\theta}, so the symbol \mathbf{W}\mathbf{Z}^{\prime}=\mathbf{W}\mathbf{Q}_{\theta} is the desired message, again leaking nothing about undesired messages. If \mathbf{B}(\mathbf{Z}^{\prime}+{\bf Q}_{\theta})={\bf Z}^{\prime} then (100) is redundant, i.e., same as (99), so it leaks no new information. Finally, if \mathbf{B}(\mathbf{Z}^{\prime}+{\bf Q}_{\theta}) is not zero and not equal to {\bf Z}^{\prime}, then {\bf ZB(Z^{\prime}+Q_{\theta})} is independent of {\bf ZZ}^{\prime}, so that (100) is protected by an independent noise term. Therefore, in all cases, the user learns nothing about undesired messages, and this completes the proof of symmetric security.\hfill\square

Appendix C Properties of Polynomials

(Zero Polynomial) A polynomial f\in\mathbb{F}[x_{1},x_{2},\cdots,x_{n}] is said to be a zero polynomial if the coefficients of all monomials in f are zero. If f is not a zero polynomial, then it is called a non-zero polynomial.

(Zero on \mathbb{G}) Let f\in\mathbb{F}[x_{1},x_{2},\cdots,x_{n}] and let \mathbb{G}\subset\mathbb{F}^{n}. We say that f is zero on \mathbb{G} if f(x_{1},x_{2},\cdots,x_{n}) evaluates to zero for all (x_{1},x_{2},\cdots,x_{n})\in\mathbb{G}.

Lemma 3

Let f\in\mathbb{F}_{q}[x_{1},x_{2},\cdots,x_{n}] be a polynomial of degree d, let \mathbb{G}\subset\mathbb{F}_{q}^{n} and let d<|\mathbb{G}|/q^{n-1}. Then f is zero on \mathbb{G} if and only if it is a zero polynomial.

Proof: If f is a zero polynomial then it is trivially zero on \mathbb{G}. For the other direction, we need to show that if f is zero on \mathbb{G} then it must be a zero polynomial. Equivalently, we need to show that if f is a non-zero polynomial then it cannot be zero on \mathbb{G}. This follows from a direct application of the Schwartz-Zippel lemma, which implies that for a non-zero polynomial f\in\mathbb{F}_{q}[x_{1},x_{2},\cdots,x_{n}] of degree d, the probability that it evaluates to zero for (x_{1},x_{2},\cdots,x_{n}) chosen uniformly over \mathbb{F}_{q}^{n}, is not more than d/q. Therefore, if d/q<|\mathbb{G}|/q^{n}, then there must be a realization of (x_{1},x_{2},\cdots,x_{n})\in\mathbb{G} for which f evaluates to a non-zero value. In other words, f cannot be zero on \mathbb{G}. \hfill\square


(Linear Independence) Polynomials f_{1},f_{2},\cdots,f_{K}\in\mathbb{F}[x_{1},x_{2},\cdots,x_{n}] are said to be linearly independent if \sum_{i=1}^{K}c_{i}f_{i}(x_{1},\cdots,x_{n}) is a non-zero polynomial for every {\bf c}=(c_{1},c_{2},\cdots,c_{K})\in\mathbb{F}^{K} except for {\bf c}=(0,0,\cdots,0). It follows from the definition that any subset of linearly independent polynomials is also linearly independent. Also note that the zero polynomial cannot be included in any set of linearly independent polynomials.

Theorem 5

Let f_{1},f_{2},\cdots,f_{m}\in\mathbb{F}_{q}[x_{1},x_{2},\cdots,x_{n}] be linearly independent polynomials of maximum degree d, let \mathbb{G}\subset\mathbb{F}_{q}^{n}, and let d<|\mathbb{G}|/q^{n-1}. Then there exist {\bf x}_{i}=(x_{1i},\cdots,x_{ni})\in\mathbb{G},i\in[1:m], such that the m\times m matrix

\displaystyle M_{m} \displaystyle=\left(\begin{matrix}f_{1}({\bf x}_{1})&f_{2}({\bf x}_{1})&\cdots% &f_{m}({\bf x}_{1})\\ f_{1}({\bf x}_{2})&f_{2}({\bf x}_{2})&\cdots&f_{m}({\bf x}_{2})\\ \vdots&\vdots&\vdots&\\ f_{1}({\bf x}_{m})&f_{2}({\bf x}_{m})&\cdots&f_{m}({\bf x}_{m})\\ \end{matrix}\right) (101)

has determinant not equal to zero.

Remark: The proof of Theorem 5 follows along the lines of similar arguments over complex numbers in [33], with minor modifications for our finite field setting.

Proof: Let M_{\mu} represent the \mu\times\mu sub-matrix obtained by including only the first \mu rows and the first \mu columns of M_{m}. The proof is by induction on \mu. If \mu=1 then the claim is trivially true.999The determinant |M_{1}|=f_{1}({\bf x}_{1}). Since f_{1} belongs to a set of linearly independent polynomials, it cannot be a zero polynomial. Since its degree is less than |\mathbb{G}|/q^{n-1}, according to Lemma 3 it cannot be zero on \mathbb{G}. Now, consider 1\leq\mu<m, and assume that there exist {\bf x}_{i}\in\mathbb{G},i\in[1:\mu], such that |M_{\mu}|\neq 0, and let us prove the claim for \mu+1. Fix the values of these {\bf x}_{i}\in\mathbb{G},i\in[1:\mu] such that |M_{\mu}|\neq 0. Now, to set up a proof by contradiction, suppose |M_{\mu+1}|=0 for all {\bf x}_{\mu+1}\in\mathbb{G}. Since the first \mu rows of M_{\mu+1} are linearly independent and fixed, there is a fixed (up to scaling) non-zero101010At least one of the elements of {\bf c} must be non-zero. column vector {\bf c}=(c_{1},c_{2},\cdots,c_{\mu+1})^{T}\in\mathbb{F}_{q}^{(\mu+1)\times 1}, such that M_{\mu+1}{\bf c}=0 for all {\bf x}_{\mu+1}\in\mathbb{G}. But this implies that

\displaystyle c_{1}f_{1}({\bf x}_{\mu+1})+c_{2}f_{2}({\bf x}_{\mu+1})+\cdots+c% _{\mu+1}f_{\mu+1}({\bf x}_{\mu+1}) \displaystyle= \displaystyle 0 (102)

for all (a_{\mu+1},b_{\mu+1})\in\mathbb{G}. Equivalently, the polynomial g\in\mathbb{F}_{q}[{\bf x}] defined as

\displaystyle g(a,b) \displaystyle= \displaystyle c_{1}f_{1}({\bf x})+c_{2}f_{2}({\bf x})+\cdots+c_{\mu+1}f_{\mu+1% }({\bf x}) (103)

is zero on \mathbb{G}. Since the degree of this polynomial is strictly less than |\mathbb{G}|/q^{n-1}, according to Lemma 3 it must be a zero polynomial. However, since f_{1},f_{2},\cdots,f_{\nu+1} are linearly independent, their linear combination can produce a zero polynomial only if c_{1}=c_{2}=\cdots=c_{\mu+1}=0. This is a contradiction because {\bf c} is a non-zero vector. The contradiction completes the proof.\hfill\square

References

  • [1] H. Sun and S. A. Jafar, “The Capacity of Private Information Retrieval,” IEEE Transactions on Information Theory, vol. 63, no. 7, pp. 4075–4088, July 2017.
  • [2] ——, “The Capacity of Robust Private Information Retrieval with Colluding Databases,” IEEE Transactions on Information Theory, vol. 64, no. 4, pp. 2361–2370, April 2018.
  • [3] R. Tajeddine, O. W. Gnilke, D. Karpuk, R. Freij-Hollanti, C. Hollanti, and S. El Rouayheb, “Private information retrieval schemes for codec data with arbitrary collusion patterns,” IEEE International Symposium on Information Theory (ISIT), pp. 1908–1912, 2017.
  • [4] Z. Jia, H. Sun, and S. Jafar, “The capacity of private information retrieval with disjoint colluding sets,” in IEEE GLOBECOM, 2017.
  • [5] K. Banawan and S. Ulukus, “The Capacity of Private Information Retrieval from Coded Databases,” IEEE Transactions on Information Theory, vol. 64, no. 3, pp. 1945–1956, 2018.
  • [6] R. Freij-Hollanti, O. Gnilke, C. Hollanti, and D. Karpuk, “Private Information Retrieval from Coded Databases with Colluding Servers,” SIAM Journal on Applied Algebra and Geometry, vol. 1, no. 1, pp. 647–664, 2017.
  • [7] H. Sun and S. A. Jafar, “Private Information Retrieval from MDS Coded Data with Colluding Servers: Settling a Conjecture by Freij-Hollanti et al.” IEEE Transactions on Information Theory, vol. 64, no. 2, pp. 1000–1022, February 2018.
  • [8] H.-Y. Lin, S. Kumar, E. Rosnes, and A. G. i Amat, “A capacity-achieving PIR protocol for distributed storage using an arbitrary linear code,” arXiv preprint arXiv:1801.04923, 2018.
  • [9] R. T. Mohamed Adel Attia, Deepak Kumar, “The capacity of private information retrieval from uncoded storage constrained databases,” arXiv preprint arXiv:1805.04104, 2018.
  • [10] K. Banawan and S. Ulukus, “Multi-message private information retrieval: Capacity results and near-optimal schemes,” arXiv preprint arXiv:1702.01739, 2017.
  • [11] H. Sun and S. A. Jafar, “Optimal download cost of private information retrieval for arbitrary message length,” IEEE Transactions on Information Forensics and Security, vol. 12, no. 12, pp. 2920–2932, 2017.
  • [12] ——, “Multiround Private Information Retrieval: Capacity and Storage Overhead,” IEEE Transactions on Information Theory, vol. 64, no. 8, pp. 5743–5754, August 2018.
  • [13] C. Tian, H. Sun, and J. Chen, “A Shannon-theoretic approach to the storage-retrieval tradeoff in pir systems,” IEEE International Symposium on Information Theory, ISIT 2018, pp. 1904–1908, 2018.
  • [14] S. Kadhe, B. Garcia, A. Heidarzadeh, S. E. Rouayheb, and A. Sprintson, “Private information retrieval with side information,” arXiv preprint arXiv:1709.00112, 2017.
  • [15] Z. Chen, Z. Wang, and S. Jafar, “The capacity of private information retrieval with private side information,” arXiv preprint arXiv:1709.03022, 2017.
  • [16] R. Tandon, “The capacity of cache aided private information retrieval,” arXiv preprint arXiv:1706.07035, 2017.
  • [17] Y.-P. Wei, K. Banawan, and S. Ulukus, “Fundamental limits of cache-aided private information retrieval with unknown and uncoded prefetching,” arXiv preprint arXiv:1709.01056, 2017.
  • [18] ——, “The capacity of private information retrieval with partially known private side information,” arXiv preprint arXiv:1710.00809, 2017.
  • [19] S. P. Shariatpanahi, M. J. Siavoshani, and M. A. Maddah-Ali, “Multi-message private information retrieval with private side information,” arXiv preprint arXiv:1805.11892, 2018.
  • [20] S. Li and M. Gastpar, “Single-server multi-message private information retrieval with side information,” arXiv preprint arXiv:1808.05797, 2018.
  • [21] K. Banawan and S. Ulukus, “Private information retrieval through wiretap channel ii: Privacy meets security,” arXiv preprint arXiv:1801.06171, 2018.
  • [22] Q. Wang, H. Sun, and M. Skoglund, “The capacity of private information retrieval with eavesdroppers,” arXiv preprint arXiv:1804.10189, 2018.
  • [23] K. Banawan and S. Ulukus, “The capacity of private information retrieval from byzantine and colluding databases,” arXiv preprint arXiv:1706.01442, 2017.
  • [24] Y. Zhang and G. Ge, “Private information retrieval from MDS coded databases with colluding servers under several variant models,” arXiv preprint arXiv:1705.03186, 2017.
  • [25] R. Tajeddine, O. W. Gnilke, D. Karpuk, R. Freij-Hollanti, and C. Hollanti, “Private information retrieval from coded storage systems with colluding, byzantine, and unresponsive servers,” arXiv preprint arXiv:1806.08006, 2018.
  • [26] H. Sun and S. A. Jafar, “The capacity of symmetric private information retrieval,” IEEE Transactions on Information Theory, 2018.
  • [27] Q. Wang and M. Skoglund, “Linear symmetric private information retrieval for MDS coded distributed storage with colluding servers,” arXiv preprint arXiv:1708.05673, 2017.
  • [28] ——, “Secure symmetric private information retrieval from colluding databases with adversaries,” arXiv preprint arXiv:1707.02152, 2017.
  • [29] H. Yang, W. Shin, and J. Lee, “Private information retrieval for secure distributed storage systems,” IEEE Transactions on Information Forensics and Security, vol. 13, no. 12, pp. 2953–2964, December 2018.
  • [30] H. Sun and S. A. Jafar, “Blind interference alignment for private information retrieval,” 2016 IEEE International Symposium on Information Theory (ISIT), pp. 560–564, 2016.
  • [31] ——, “The capacity of private computation,” arXiv preprint arXiv:1710.11098, 2017.
  • [32] M. Mirmohseni and M. A. Maddah-Ali, “Private function retrieval,” arXiv preprint arXiv:1711.04677, 2017.
  • [33] V. Annapureddy and A. El Gamal and V. Veeravalli, “Degrees of Freedom of Interference Channels With CoMP Transmission and Reception,” IEEE Trans. on Information Theory, vol. 58, no. 9, pp. 5740–5760, Sep. 2012.
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