Secure Degrees of Freedom of the Multiple Access Wiretap Channel with Multiple AntennasThis work was supported by NSF Grants CNS 13-14733, CCF 14-22111, CCF 14-22129, and CNS 15-26608, and presented in part at the Asilomar Conference on Signals, Systems and Computers 2015 and to be presented in part at IEEE ICC 2016.

Secure Degrees of Freedom of the Multiple Access Wiretap Channel with Multiple Antennasthanks: This work was supported by NSF Grants CNS 13-14733, CCF 14-22111, CCF 14-22129, and CNS 15-26608, and presented in part at the Asilomar Conference on Signals, Systems and Computers 2015 and to be presented in part at IEEE ICC 2016.

Pritam Mukherjee   Sennur Ulukus
Department of Electrical and Computer Engineering
University of Maryland
College Park MD 20742
pritamm@umd.edu   ulukus@umd.edu
Abstract

We consider a two-user multiple-input multiple-output (MIMO) multiple access wiretap channel with antennas at each transmitter, antennas at the legitimate receiver, and antennas at the eavesdropper. We determine the optimal sum secure degrees of freedom (s.d.o.f.) for this model for all values of and . We subdivide our problem into several regimes based on the values of and , and provide achievable schemes based on real and vector space alignment techniques for fixed and fading channel gains, respectively. To prove the optimality of the achievable schemes, we provide matching converses for each regime. Our results show how the number of eavesdropper antennas affects the optimal sum s.d.o.f. of the multiple access wiretap channel.

\setstretch

1.2

1 Introduction

We consider the two-user multiple-input multiple-output (MIMO) multiple access wiretap channel where each transmitter has antennas, the legitimate receiver has antennas and the eavesdropper has antennas; see Fig. 1. We consider the case when the channel gains are fixed throughout the duration of the communication, as well as the case when the channel is fast fading and the channel gains vary in an i.i.d. fashion across time. Our goal in this paper is to characterize how the optimal sum secure degrees of freedom (s.d.o.f.) of the MIMO multiple access wiretap channel varies with the number of antennas at the legitimate users and the eavesdropper.

To that end, we partition the range of into various regimes, and propose achievable schemes for each regime. Our schemes are based on a combination of zero-forcing beamforming and vector space interference alignment techniques. When the number of antennas at the eavesdropper is less than the number of antennas at the transmitters, the nullspace of the eavesdropper channel can be exploited to send secure signals to the legitimate transmitter. This strategy is, in fact, optimal when the number of eavesdropper is sufficiently small () and the optimal sum s.d.o.f. is limited by the decoding capability of the legitimate receiver. We note that the optimal scheme requires a single channel use and thus, can be used for both fixed and fading channel gains.

Figure 1: The MIMO multiple access wiretap channel.

However, zero-forcing beamforming does not suffice when . In the regime , the optimal sum s.d.o.f. is of the form , where is an integer. For the case of fading channel gains, we use vector space interference alignment [1] over three time slots to achieve the optimal sum s.d.o.f. The structure of the optimal signaling scheme is inspired by ideas from the optimal real alignment scheme presented in [2] for the single-input single-output (SISO) multiple access wiretap channel. Unlike the previous regime, this scheme for fading channel gains cannot be directly extended to the fixed channel gains case, except for the case , for which the sum s.d.o.f. is an integer and carefully precoded Gaussian signaling suffices. When , the s.d.o.f. has a fractional part, and Gaussian signaling alone is not optimal. This is also observed in the achievable schemes in [3, 4] for the MIMO wiretap channel with one helper, where structured signaling is used when the optimal s.d.o.f. is not an integer. However, references [3, 4] consider complex channel gains, for which an s.d.o.f. of the form can be obtained by using complex symbols (which comprise two real symbols) and one real symbol, where each real symbol belongs to the same PAM constellation and carries s.d.o.f. In our case, the s.d.o.f. is of the form , and such simplification is not possible even with complex channel gains.

In this paper, we consider real channel gains. In order to handle the fractional s.d.o.f., we decompose the channel input at each transmitter into two parts: a Gaussian signaling part carrying (the integer part) d.o.f. of information securely, and a structured signaling part carrying (the fractional part) d.o.f. of information securely. The structure of the Gaussian signals carrying the integer s.d.o.f. resembles that of the schemes for the fading channel gains. When , we design the structured signals carrying sum s.d.o.f. according to the real interference alignment based SISO scheme of [2]. However, when , a new scheme is required to achieve sum s.d.o.f. on the MIMO multiple access wiretap channel with two antennas at every terminal. To that end, we provide a novel optimal scheme for the canonical MIMO multiple access wiretap channel. Interestingly, the scheme relies on asymptotic real interference alignment [5] at each antenna of the legitimate receiver.

When the number of eavesdropper antennas is large enough , the optimal sum s.d.o.f. is given by , which is always an integer. In this regime Gaussian signaling along with vector space alignment techniques suffices. In fact, the scheme uses only one time slot and can be used with both fixed and fading channel gains. When the number of antennas at the eavesdropper is very large (), the two-user multiple access wiretap channel reduces to a wiretap channel with one helper, and, thus, the scheme for the MIMO wiretap channel with one helper in [4] is optimal.

To establish the optimality of our achievable schemes, we present matching converses in each regime. A simple upper bound is obtained by allowing cooperation between the two transmitters. This reduces the two-user multiple access wiretap channel to a MIMO wiretap channel with antennas at the transmitter, antennas at the legitimate receiver and antennas at the eavesdropper. The optimal s.d.o.f. of this MIMO wiretap channel is well known to be [6, 7], and this serves as an upper bound for the sum s.d.o.f. of the two-user multiple access wiretap channel. This bound is optimal when the number of eavesdropper antennas is either quite small (), or quite large (). When is small, the sum s.d.o.f. is limited by the decoding capability of the legitimate receiver, and the optimal sum s.d.o.f. is which is optimal even without any secrecy constraints. When is large, the s.d.o.f. is limited by the requirement of secrecy from a very strong eavesdropper. For intermediate values of , the distributed nature of the transmitters dominates, and we employ a generalization of the SISO converse techniques of [2] for the converse proof in the MIMO case, similar to [4].

Related Work: The multiple access wiretap channel is introduced by [8, 9], where the technique of cooperative jamming is introduced to improve the rates achievable with Gaussian signaling. Reference [10] provides outer bounds and identifies cases where these outer bounds are within 0.5 bits per channel use of the rates achievable by Gaussian signaling. While the exact secrecy capacity remains unknown, the achievable rates in [8, 9, 10] all yield zero s.d.o.f. Reference [11] proposes scaling-based and ergodic alignment techniques to achieve a sum s.d.o.f. of for the -user MAC-WT; thus, showing that an alignment based scheme strictly outperforms i.i.d. Gaussian signaling with or without cooperative jamming at high SNR. Finally, references [2, 12] establish the optimal sum s.d.o.f. to be and the full s.d.o.f. region, respectively, for the SISO multiple access wiretap channel. Other related channel models are the wiretap channel with helpers and the interference channel with confidential messages, for which the optimal sum s.d.o.f. is known for the SISO and MIMO cases in [2] and [3, 4], and in [13] and [14], respectively.

2 System Model

The two-user multiple access wiretap channel, see Fig. 1, is described by,

(1)
(2)

where is an dimensional column vector denoting the th user’s channel input, is an dimensional vector denoting the legitimate receiver’s channel output, and is a dimensional vector denoting the eavesdropper’s channel output, at time . In addition, and are and dimensional white Gaussian noise vectors, respectively, with and , where denotes the identity matrix. Here, and are the and channel matrices from transmitter to the legitimate receiver and the eavesdropper, respectively, at time . When the channel gains are fixed, the entries of and are drawn from an arbitrary but fixed continuous distribution with bounded support in an i.i.d. fashion prior to the start of the communication, and remain fixed throughout the duration of the communication, i.e., for . When the channel gains are fading, the entries of and are drawn from the fixed continuous distribution with bounded support in an i.i.d. fashion at every time slot . We assume that the channel matrices and are known with full precision at all terminals, at time . All channel inputs satisfy the average power constraint , where denotes the Euclidean (or the spectral norm) of the vector (or matrix) .

Transmitter wishes to send a message , uniformly distributed in , securely to the legitimate receiver in the presence of the eavesdropper. A secure rate pair , with is achievable if there exists a sequence of codes which satisfy the reliability constraints at the legitimate receiver, namely, , for , and the secrecy constraint, namely,

(3)

where as . An s.d.o.f. pair is said to be achievable if a rate pair is achievable with

(4)

The sum s.d.o.f.  is the largest achievable .

3 Main Result

The main result of this paper is the determination of the optimal sum s.d.o.f. of the MIMO multiple access wiretap channel. We have the following theorem.

Theorem 1

The optimal sum s.d.o.f. of the MIMO multiple access wiretap channel with antennas at the transmitters, antennas at the legitimate receiver and antennas at the eavesdropper is given by

(5)

for almost all channel gains.

We present the converse proof for this theorem in Section 4. The achievable schemes for the case of fading channel gains are presented in Section 5, while the achievable schemes for the case of fixed channel gains are presented in Section 6.

Figure 2: versus .

Fig. 2 shows the variation of the optimal sum s.d.o.f. with the number of eavesdropper antennas . Note that as in the SISO case, the optimal sum s.d.o.f. is higher for the multiple access wiretap channel than for the wiretap channel with one helper [4], when . However, when the number of eavesdropper antennas is large enough, i.e., when , the optimal sum s.d.o.f. of the multiple access wiretap channel is the same as the optimal s.d.o.f. of the wiretap channel with a helper.

Further, note that when the number of eavesdropper antennas is small enough (), the optimal sum s.d.o.f. is , which is the optimal d.o.f. of the multiple access channel without any secrecy constraints. Thus, there is no penalty for imposing the secrecy constraints in this regime. Also note that allowing cooperation beteen the transmitters does not increase the sum s.d.o.f. in this regime. Heuristically, the eavesdropper is quite weak in this regime, and the optimal sum s.d.o.f. is limited by the decoding capabilities of the legitimate receiver.

On the other hand, when the number of antennas is quite large (), the optimal sum s.d.o.f. is , which is the optimal s.d.o.f. obtained by allowing cooperation between the transmitters. Intuitively, the eavesdropper is very strong in this regime and the sum s.d.o.f. is limited by the requirement of secrecy from this strong eavesdropper. In the intermediate regime, when , the distributed nature of the transmitters becomes a key factor and the upper bound obtained by allowing cooperation between the transmitters is no longer achievable; see Fig. 3.

4 Proof of the Converse

We prove the following upper bounds which are combined to give the converse for the full range of and ,

(6)
(7)

where denotes .

It can be verified from Fig. 3 that the minimum of the two bounds in (6)-(7) gives the converse to the sum s.d.o.f. stated in (5) for all ranges of and . Thus, we next provide proofs of each of the bounds in (6) and (7).

4.1 Proof of

This bound follows by allowing cooperation between the transmitters, which reduces the two-user multiple access wiretap channel to a single-user MIMO wiretap channel with antennas at the transmitter, antennas at the legitimate receiver and antennas at the eavesdropper. The optimal s.d.o.f. for this MIMO wiretap channel is known to be [6, 7].

4.2 Proof of

We only show that , when , and note that the bound for follows from the fact that increasing the number of eavesdropper antennas cannot increase the sum s.d.o.f.; thus, the sum s.d.o.f. when is upper-bounded by the sum s.d.o.f. for the case of , which is .

To prove when , we follow [2, 4]. We define noisy versions of as where with . The secrecy penalty lemma [2] can then be derived as

(8)
(9)
(10)
(11)
(12)

Now consider a stochastically equivalent version of given by , where is an independent Gaussian noise vector, distributed as . Further, let and , where is the matrix with the first columns of , has the last columns of , is a vector with the top elements of , while has the remaining elements of . Then, we have

Figure 3: The two upper bounds.
(13)
(14)
(15)
(16)
(17)

Using (17) in (12), we have

(18)

The role of a helper lemma [2] also generalizes to the MIMO case as

(19)
(20)
(21)

Adding (18) and (21), we have

(22)
(23)
(24)

First dividing by and letting , and then dividing by and letting , we have

(25)

By reversing the roles of the transmitters, we have

(26)

Combining (25) and (26), we have the required bound

(27)

This completes the proof of the converse of Theorem 1.

5 Achievable Schemes for Fading Channel Gains

We provide separate achievable schemes for each of the following regimes:

Each scheme described in the following sections can be outlined as follows. We neglect the impact of noise at high SNR. Then, to achieve a certain sum s.d.o.f., , we achieve the s.d.o.f. pair with . We send symbols and symbols from the first and second transmitters, respectively, in slots, such that and . Finally, we show that the leakage of information symbols at the eavesdropper is . We however want a stronger guarantee of security, namely,

(28)

as . To achieve this, we view the slots described in the scheme as a block and treat the equivalent channel from and to and as a memoryless multiple access wiretap channel with being the output at the legitimate receiver and being the output at the eavesdropper. The following sum secure rate is achievable [15]:

(29)

where . Using the proposed scheme, and can be reconstructed from to within noise distortion. Thus,

(30)

Also, for each scheme, by design

(31)

Thus, from (29), the achievable sum secure rate in each block is . Since our block contains channel uses, the effective sum secure rate is

(32)

Thus, the achievable sum s.d.o.f. is , with the stringent security requirement as well.

In the following subsections, we present the achievable scheme for each regime.

5.1

In this regime, the optimal sum s.d.o.f. is . In our scheme, transmitter sends independent Gaussian symbols while transmitter sends independent Gaussian symbols , in one time slot. This can be done by beamforming the information streams at both transmitters to directions that are orthogonal to the eavesdropper’s channel. To this end, the transmitted signals are:

(33)
(34)

where is a matrix whose columns span the dimensional nullspace of , and is a matrix with linearly independent vectors drawn from the dimensional nullspace of . This can be done since . The channel outputs are:

(35)
(36)

Note that is an matrix with full rank almost surely, and thus, both and can be decoded at the legitimate receiver to within noise variance. On the other hand, they do not appear in the eavesdropper’s observation and thus their security is guaranteed.

5.2

The optimal sum s.d.o.f. in this regime is . Thus, transmitter sends Gaussian symbols , each drawn independently from , in time slots for , where and is chosen to satisfy the power constraint. Intuitively, transmitter sends the symbols by beamforming orthogonal to the eavesdropper in each time slot . The remaining symbols are sent over time slots using a scheme similar to the SISO scheme of [2, 16]. Thus, the transmitted signals at time are:

(37)
(38)

where is an full rank matrix with , is a dimensional vector whose entries are drawn in an i.i.d. fashion from , and and are precoding matrices that will be fixed later. The channel outputs are:

(39)
(40)

We now choose to be any matrix with full column rank, and choose

(41)

where . It can be verified that this selection aligns with , , at the eavesdropper, and this guarantees that the information leakage is . On the other hand, the legitimate receiver decodes the desired signals , and the aligned artificial noise symbols , i.e., symbols using observations in time slots, to within noise variance. This completes the scheme for the regime .

5.3

In this regime, the optimal sum s.d.o.f. is . Therefore, transmitter in our scheme sends Gaussian symbols, , in time slots. The transmitted signals in time slot are given by

(42)
(43)

where the , , and are precoding matrices to be designed. Let us define

(44)

Further, if we define

(45)

and similarly, we can compactly represent the channel outputs over all time slots as

(46)
(47)

where , , and is defined similarly. To ensure secrecy, we impose the following conditions

(48)
(49)

We rewrite the conditions in (48)-(49) as

(50)

where

(51)

Note that has a nullity . Since in this regime, we can choose vectors of dimension randomly such that they are linearly independent and lie in the nullspace of . We can then assign to , and , the top, the middle and the bottom rows of the matrix comprising the chosen vectors. This guarantees secrecy of the message symbols at the eavesdropper.

To see the decodability, we rewrite the received signal at the legitimate receiver as

(52)

where . We note that is and full rank almost surely; thus, the desired signals and can be decoded at the legitimate receiver within noise distortion at high SNR.

5.4

The optimal s.d.o.f. in this regime is . To achieve this s.d.o.f., the first transmitter sends Gaussian symbols , while the second transmitter sends Gaussian symbols , in one time slot. The scheme is as follows. The transmitted signals are

(53)
(54)

where and are artificial noise vectors, whose entries are drawn in an i.i.d. fashion from . The precoding matrices , and will be chosen later. The channel outputs are

(55)
(56)

To ensure secrecy, we want to impose the following conditions:

(57)
(58)
(59)

To satisfy (57), we choose and to be the first and the last rows of a matrix whose columns consist of any linearly independent vectors drawn randomly from the nullspace of . This is possible since, in this regime. To satisfy (58)-(59), we let , and to be the first, the second and the last rows of a matrix whose columns are randomly chosen to span the dimensional nullspace of the matrix given by

(60)

To see the decodablity, we can rewrite the observation at the legitimate receiver as

(61)

where is the matrix defined as

(62)

Since is full rank almost surely, the legitimate receiver can decode its desired symbols , and .

5.5

In this regime, it is clear from Fig. 2 that the multiple access wiretap channel has the same optimal sum s.d.o.f. as the optimal s.d.o.f. of the wiretap channel with one helper. Thus, an optimal achievable scheme for the wiretap channel with one helper suffices as the scheme for the multiple access wiretap channel as well. Such an optimal scheme, based on real interference alignment, is provided in [4] for the wiretap channel with one helper with fixed channel gains. Here, we provide a scheme based on vector space alignment.

In order to achieve the optimal sum s.d.o.f. of in this regime, the first transmitter sends independent Gaussian symbols securely, in one time slot. The second transmitter just transmits artificial noise symbols , whose entries are drawn in an i.i.d. fashion from . The transmitted signals are

(63)
(64)

where and are precoding matrices to be fixed later. The received signals are

(65)
(66)

To ensure security, we wish to ensure that

(67)

This can be done by choosing and to be the top and the bottom rows of a matrix whose linearly independent columns are drawn randomly from the nullspace of . The decodability is ensured by noting that the matrix is full column rank and in this regime.

6 Achievable Schemes for Fixed Channel Gains

We note that the achievable schemes proposed for the fading channel gains in the regimes and are single time-slot schemes and suffice for the fixed channel gains case. However, in the regime , the schemes for the fading channel gains exploit the diversity of channel gains over three time slots; thus, these schemes cannot be used in the fixed channel gains case. Therefore, we now propose new achievable schemes for this regime. In this regime, the optimal sum s.d.o.f. is of the form , where is an integer. When , the sum s.d.o.f. is an integer and carefully precoded Gaussian signaling suffices. However, when , the s.d.o.f. has a fractional part, and Gaussian signaling alone is not optimal, since Gaussian signals with full power cannot carry fractional d.o.f. of information.

The general structure of our schemes is as follows: We decompose the channel input at each transmitter into two parts: a Gaussian signaling part carrying (the integer part) d.o.f. of information securely, and a structured signaling part carrying (the fractional part) d.o.f. of information securely. The structure of the Gaussian signals carrying the integer s.d.o.f.  are the same as that of the corresponding schemes for the fading channel gains. This ensures security at the eavesdropper as well as decodability at the legitimate receiver as long as the structured signals carrying the fractional s.d.o.f.  from both transmitters can be decoded at the legitimate receiver. The design of the structured signals is motivated from the SISO scheme of [2]. In fact, when , we use the signal structure of the scheme in [2], where real interference alignment is used to transmit sum s.d.o.f. on the SISO multiple access wiretap channel. However, when , a new scheme is required to achieve sum s.d.o.f. on the MIMO multiple access wiretap channel with two antennas at every terminal. To that end, we first provide a novel scheme, based on asymptotic real interference alignment [17, 5], for the canonical MIMO multiple access wiretap channel.

6.1 Scheme for the System

The optimal sum s.d.o.f. is . Since the legitimate receiver has 2 antennas, we achieve s.d.o.f. on each antenna. The scheme is as follows.

Let be a large integer. Define , where will be specified later. The channel inputs are given by

(68)
(69)

where are dimensional precoding vectors which will be fixed later, and are independent random variables drawn uniformly from the same PAM constellation given by

(70)

where is a positive integer and is a real number used to normalize the transmission power. The exact values of and will be specified later. The variables denote the information symbols of transmitter , while are the cooperative jamming signals being transmitted from transmitter .

The channel outputs are given by

(71)
(72)

where and . Note that the information symbols are buried in the cooperative jamming signals , where , at the eavesdropper. Intuitively, this ensures security of the information symbols at the eavesdropper. At the legitimate receiver, we can express the received signal more explicitly as

(73)

We define

(74)
(75)

Letting , we note that

(76)

We choose to be the dimensional vector that has all the elements of . We note that all elements in are rationally independent, since the channel gains are drawn independently from a continuous distribution. Also, the elements of can be verified to be rationally independent of the elements of , if . With the above selections, let us analyze the structure of the received signal at the legitimate receiver.

At the first antenna, and arrive along the dimensions of . The signals and