Secure Communication over Parallel Relay Channel

Secure Communication over Parallel Relay Channel

Abstract

We investigate the problem of secure communication over parallel relay channel in the presence of a passive eavesdropper. We consider a four terminal relay-eavesdropper channel which consists of multiple relay-eavesdropper channels as subchannels. For the discrete memoryless model, we establish outer and inner bounds on the rate-equivocation region. The inner bound allows mode selection at the relay. For each subchannel, secure transmission is obtained through one of two coding schemes at the relay: decoding-and-forwarding the source message or confusing the eavesdropper through noise injection. For the Gaussian memoryless channel, we establish lower and upper bounds on the perfect secrecy rate. Furthermore, we study a special case in which the relay does not hear the source and show that under certain conditions the lower and upper bounds coincide. The results established for the parallel Gaussian relay-eavesdropper channel are then applied to study the fading relay-eavesdropper channel. Analytical results are illustrated through some numerical examples.

1Introduction

In conventional point-to-point wired networks, security is facilitated by secret key sharing between relevant parties based on some common cryptographic algorithm. The premise is that only legitimate users have access to the encrypted messages and extraneous users (adversaries) are unable to access any useful information. The wireless channel is characterized by its inherit randomness and broadcast nature. Physical layer security exploits the basic attributes of the wireless channel for instance, difference of the fading gains between the legitimate channel (source to the legitimate receiver) and the channel to the adversary, to transmit information securely to the legitimate receiver. Thus, it eradicates the need of secret key sharing.

The wiretap channel introduced by Wyner is a basic information-theoretic model which incorporates physical layer attributes of the channel to transmit information securely [1]. Wyner’s basic model consists of a source, a legitimate receiver and an eavesdropper (wiretapper) under noisy channel conditions. Secrecy capacity is established when the eavesdropper channel (the channel from the source to the eavesdropper) is a degraded version of the main channel (the channel from the source to the legitimate receiver). The discrete memoryless (DM) channel studied by Wyner is further extended to study some other channels for which secrecy capacity is established, i.e., broadcast channels (BC) [2], multi-antenna channels [4], multiple access channels [7], fading channels [10] etc. The idea of cooperation between users in context of security was introduced by [12]. The intuition is that, when the main channel is more noisy than the channel to the eavesdropper, cooperation between users is utilized to achieve positive secrecy capacity. Secrecy is achieved by using the relay as a trusted node that facilitates the information decoding at the destination while concealing the information from the eavesdropper. A special case in which there is a physically degraded relay-eavesdropper channel was studied in [13]. The case in which the relay does not acts as a trusted node is studied in [14].

In this paper, we study a parallel relay-eavesdropper channel. A parallel relay-eavesdropper channel is a generalization of the setup in [12], in which each of the source-to-relay (S-R), source-to-destination (S-D), source-to-eavesdropper (S-E), relay-to-destination (R-D) and relay-to-eavesdropper (R-E) link is composed of several parallel channels as subchannels. The eavesdropper is passive in the sense that it just listens to the transmitted information without modifying it. We only focus on the perfect secrecy rate, i.e., the maximum achievable rate at which information is reliably sent to the legitimate receiver, and the eavesdropper is unable to decode it.

The parallel relay-eavesdropper channel considered in this paper relates to some of the channels studied previously. Compared to the parallel relay channel studied in [16], the parallel relay-eavesdropper channel requires an additional secrecy constraint. The parallel relay-eavesdropper channel without relay simplifies to a number of channels discussed previously. For example, the parallel wiretap channel studied in [17], the parallel broadcast channel with confidential messages (BCC) and no common message studied in [3].

Contributions. The main contributions of this paper are summarized as follows. For the discrete memoryless case, we establish inner and outer bounds on the rate-equivocation region for the parallel relay-eavesdropper channel. The inner bound is obtained through a coding scheme in which, for each subchannel, the relay operates either in decode-and-forward (DF) or in noise forwarding (NF) mode. We note that establishing our outer bound for DM case is not straightforward and it does not follow directly from the single-letter outer bound for the relay-eavesdropper channel developed in [12]. Therefore a converse is needed. The converse includes a re-definition of the involved auxiliary random variables, a technique much similar to the one used before in the context of secure transmission over broadcast channels [3].

For the Gaussian memoryless model, we establish lower and upper bounds on the perfect secrecy rate. The lower bound established for the Gaussian model follows directly from the DM case. However, we note that establishing a computable upper bound on the secrecy rate for the Gaussian model is non-trivial, and it does not follow directly from the DM case. In part, this is because the upper bound established for the DM case involves auxiliary random variables, the optimal choice of which is difficult to obtain. In this work, we develop a new upper bound on the secrecy rate for the parallel Gaussian relay-eavesdropper channel. Our converse proof uses elements from converse techniques developed in [5] in context of multi-antenna wiretap channel; and in a sense, can be viewed as an extension of these results to the parallel relay-eavesdropper channel. This upper bound is especially useful when the multiple access part of the channel is the bottleneck. We show that, in contrast to upper bounding techniques for our model that can be obtained straightforwardly by applying recent results on multi-antenna wiretap channels [4], our upper bound shows some degree of separability for the different subchannels.

We also study a special case in which the relay does not hear the source, for example due to very noisy source-to-relay links. In this case we show that under some specific conditions noise-forwarding on all links achieves the secrecy capacity. The converse proof follows from a new genie-aided upper bound that assumes full cooperation between the relay and the destination, and a constrained eavesdropper. The eavesdropper is constrained in the sense that it has to treat the relay’s transmission as unknown noise for all subchannels, an idea used previously in context of a class of classic relay-eavesdropper channel with orthogonal components [18]. These assumptions turn the parallel Gaussian relay-eavesdropper channel into a parallel Gaussian wiretap channel, the secrecy capacity of which is established in [3].

Furthermore, we study an application of the results established for the parallel Gaussian relay-eavesdropper channel to the fading relay-eavesdropper channel. We assume that perfect non-causal channel state information (CSI) is available at all nodes. The fading relay-eavesdropper channel is a special case of the parallel Gaussian relay-eavesdropper channel in which each realization of a fading state corresponds to one subchannel. We illustrate our results through some numerical examples.

The rest of the paper is organized as follows. In section II, we establish outer and inner bounds on the rate-equivocation region for the DM channel. In section III, we establish lower and upper bounds on the perfect secrecy rate for the Gaussian model, and consider a special case in which under some specific conditions secrecy capacity is achieved. In section IV, we present an application of the results established in section III to the fading model. We illustrate these results with some numerical examples in section V. Section VI concludes the paper by summarizing its contribution.

Notations. In this paper, the notation is used as a shorthand for , the notation is used as a shorthand for where for , , the notation is used as a shorthand for , the notation is used as a shorthand for , denotes the expectation operator, denotes the cardinality of set , denotes the number of subchannels, the boldface letter denotes the covariance matrix. We denote the entropy of a discrete and continuous random variable by and respectively. We define the functions and . Throughout the paper the logarithm function is taken to the base 2.

2Discrete memoryless channel

In this section, we establish outer and inner bounds on the rate-equivocation region for the discrete memoryless parallel relay-eavesdropper channel.

The parallel relay-eavesdropper channel.
The parallel relay-eavesdropper channel.

2.1Channel Model

Due to the openness of the wireless medium, the eavesdropper listens for free to what the source and relay transmit. It then tries to guess the information being transmitted. The equivocation rate per channel use is defined as . Perfect secrecy for the channel is obtained when the eavesdropper gets no information about the confidential message from . That is, the equivocation rate is equal to the unconditional source entropy.

2.2Outer Bound

The following theorem provides an outer bound on the rate-equivocation region for the parallel relay-eavesdropper channel.

2.3Achievable Rate-Equivocation Region

In this subsection we establish an achievable rate-equivocation region for the parallel relay-eavesdropper channel. The achievable region is established by the combination of two different coding schemes, namely decode-and-forward and noise forwarding. In DF scheme, for each message source associates a number of confusion codewords, the relay after receiving the source codewords, decode it and re-transmits it towards the legitimate receiver and eavesdropper (see [12] for details). In the NF scheme the relay does not decode the source codewords, but transmits confusion codewords independent from the source codewords, towards the legitimate receiver and the eavesdropper (see [12] for details).

for some distribution for and for , are achievable.

3Gaussian Channel

In this section we study a parallel Gaussian relay-eavesdropper channel. Figure 1 depicts the studied model. We only focus on the perfectly secure achievable rates, i.e., .

Figure 1: The parallel Gaussian relay-eavesdropper channel.
Figure 1: The parallel Gaussian relay-eavesdropper channel.

3.1Channel Model

For a parallel Gaussian relay-eavesdropper channel, the received signals at the relay, destination and eavesdropper are given by

where is the time index, and are noise processes, independent and identically distributed (i.i.d) with the components being zero mean Gaussian random variables with variances , and respectively, for . We assume that the source and relay know the noise variances present at the receivers. For the subchannel , and are inputs from the source and relay nodes respectively. The parameter indicates the ratio of the R-D link signal-to-noise (SNR) to the S-D link SNR and indicates the ratio of the R-E link SNR to the S-E link SNR for subchannel respectively. The source and relay input sequences are subject to separate power constraints and , i.e.,

Figure 2: An example of a deterministic parallel relay-eavesdropper channel with two subchannels.
Figure 2: An example of a deterministic parallel relay-eavesdropper channel with two subchannels.

3.2Lower Bound on the Perfect Secrecy Rate

For the parallel Gaussian relay-eavesdropper channel , we apply Theorem ? to obtain a lower bound on the perfect secrecy rate.1

The parameters and indicate the source and relay power allocated for transmission over the -th subchannel. In , after some straightforward algebra, the contribution to the equivocation of information sent through NF (set in Theorem ?) can be condensed by observing that we only need to consider , to get higher secrecy rate. A simplified expression for is given by

This remark is elucidated by the following example.

Example: We consider a deterministic parallel relay-eavesdropper channel with two subchannels, i.e., , as shown in Figure 2. For subchannel 1, the link capacities to the relay, legitimate receiver and eavesdropper are given by and respectively. For subchannel 2, the link capacities to the relay, legitimate receiver and eavesdropper are given by and respectively. For this channel, achievable rate obtained by coding across subchannels is given by

Similarly achievable rate obtained by coding separately over each subchannel is given by

which is clearly smaller than . This shows the usefulness of coding across subchannels.

3.3Upper Bound on the Perfect Secrecy Rate

The following theorem provides an upper bound on the secrecy rate for the parallel Gaussian relay-eavesdropper channel.

The computation of the upper bound is given in Appendix Section 8.

Example Application: We consider a parallel relay channel with interference at the eavesdropper. The received signals at the relay, destination and eavesdropper are given by

This model can represent the equivalent channel model of a MIMO relay-eavesdropper channel with the interference at the relay and legitimate receiver avoided through singular-value decomposition; as the source can always get some feedback from both the relay and legitimate receiver, and the relay from the legitimate receiver, which then transforms the MIMO transmission into one on parallel channels among the source, relay and legitimate receiver. The eavesdropper however does not feedback information on his channel, and so is subjected to cross-antenna interference. Constraining the eavesdropper to treat the cross-antenna interference as independent noise, one can obtain an upper bound on the secrecy capacity of the model with constrained eavesdropper by direct application of . Straightforward algebra gives

3.4Secrecy Capacity in Some Special Cases

We now study the case in which the S-R links are very noisy, i.e., the relay does not hear the source.

4Example Application

In this section we apply the results which we established for the Gaussian memoryless model in section III to study a fading relay-eavesdropper channel.

For a fading relay-eavesdropper channel, the received signals at the relay, legitimate receiver and eavesdropper are given by

where is the time index, , , , and are the fading gain coefficients associated with S-D, R-D, S-E, R-E and S-R links, given by complex Gaussian random variables with zero mean and unit variance respectively. The noise processes are zero mean i.i.d complex Gaussian random variables with variances , and respectively. The source and relay input sequences are subject to an average power constraint, i.e., , . We define and assume that perfect non-causal channel state information (CSI) is available at all nodes. For a given fading state realization , the fading relay-eavesdropper channel is a Gaussian relay-eavesdropper channel. Therefore, for a given channel state with fading state realizations, i.e., , the fading relay-eavesdropper channel can be seen as a parallel Gaussian relay-eavesdropper channel with subchannels. The power allocation vectors at the source and relay are denoted by and respectively. The ergodic achievable secrecy rate of the fading relay-eavesdropper channel , which follows from is given by

5Numerical Results

In this section we provide numerical examples to illustrate the performance of fading relay-eavesdropper channel. We consider a fading relay-eavesdropper channel with L realizations of fading state. It is assumed that perfect channel state information is available at all nodes. We can consider this channel as a Gaussian relay-eavesdropper channel with L subchannels. Alternatively, this model can be seen as an OFDM system with sub-carriers. We model channel gain between node and as distance dependent Rayleigh fading, that is, , where is the path loss exponent, is the distance between the node and , and is a complex Gaussian random variable with zero mean and variance one. Each subchannel is corrupted by additive white Gaussian noise with zero mean and variance one. Furthermore, for each symbol transmission same subchannel is used on S-R and R-D links to make the optimization tractable. The objective function for both lower and upper bounds are optimized numerically using AMPL with a commercially available solver, for instance SNOPT.

Figure 3: Achievable perfect secrecy rate of a parallel relay-eavesdropper channel.
Figure 3: Achievable perfect secrecy rate of a parallel relay-eavesdropper channel.

To illustrate the system performance, we set the source and relay power to 64 Watt each. We consider a network geometry in which the source is located at the point (0,0), the relay is located at the point (,0), the destination is located at the point (1,0) and the eavesdropper is located at the point (0,1), where is the distance between the source and the relay. In all numerical results we set path loss exponent :=2 and . For all numerical examples, secrecy rate is given by bits per channel use. For each subchannel the selection of the coding scheme at the relay is based on the relative strength of the S-D link w.r.t the S-R link, i.e., we use NF scheme (set ) when and DF scheme (set ) when . Figure 3 shows the power allocation for a fading channel with 64 subchannels where the relay is located at (0.5,0), and marker ’’ denotes NF on a particular subchannel while marker ’’ denotes DF on a particular subchannel. It can be seen from Figure 3 that, achievable perfect secrecy rate is zero for some subchannels. Roughly speaking, this happens when the condition is violated.

Figure 4: Comparison of achievable perfect secrecy rate of the lower bound with optimized power allocation and with uniform power allocation over all subchannels.
Figure 4: Comparison of achievable perfect secrecy rate of the lower bound with optimized power allocation and with uniform power allocation over all subchannels.

Figure 4 compares the average perfect secrecy rate of the lower bound, with optimized power allocation and with uniform power allocation, i.e., allocating same power at the source and relay for all subchannels in and in . It can be seen that for separate source and relay powers, optimized power allocation scheme outperforms uniform power allocation scheme. This fact follows because optimized power allocation scheme maximizes the achievable perfect secrecy rate and hence enhances the system performance.

Figure 5: Comparison of achievable perfect secrecy rate of some schemes with the lower bound.
Figure 5: Comparison of achievable perfect secrecy rate of some schemes with the lower bound.

Mode selection at the relay by only considering the relative strength of the S-D and the S-R link in the lower bound is suboptimal because the achievable secrecy rate also depends on the gain of other link. We now consider the case in which the relay selects the scheme which maximizes the rate for each subchannel. We plot the lower bound with this criteria and compare it with the case in which same scheme is used on all subchannels. As a reference we consider the case in which there is no relay, i.e., a parallel wiretap channel. Figure 5 shows the achievable average perfect secrecy rate of different schemes. It can be seen that when the relay is close to the source, DF scheme on all subchannels gives higher secrecy rate. Similarly when the relay is close to the destination, NF scheme on all subchannels offers better rate. The region when the relay is between is of particular interest. In this region the relay selects between DF scheme and NF scheme for each subchannel and utilizes the gain from both schemes. It is interesting to note that when the relay is close to the destination, use of DF scheme on all subchannels does not offer any gain because in this case the relay is unable to decode the source codewords and hence the average secrecy rate decreases. The lower bound always perform better than the wiretap channel which shows the usefulness of the relay.

Figure 6: Bounds on perfect secrecy rate.
Figure 6: Bounds on perfect secrecy rate.

In Figure 6 we compare the lower bound obtained in Figure 5, with the upper bound on the secrecy capacity for the fading relay-eavesdropper channel. It can be seen that when the relay is close to the source, the lower and upper bounds coincide. This is achieved by using DF scheme on all subchannels.

6Conclusions

We studied the problem of secure communication over parallel relay channel. Outer and inner bounds on the rate-equivocation are established for the DM case. Developing an outer bound on the parallel relay-eavesdropper channel is non-trivial and it does not follow directly from the one established in [12]. For the Gaussian memoryless case, lower and upper bounds on the perfect secrecy rate are established. The computable upper bound for the Gaussian model shows some separability over subchannels. In the case in which the relay does not hear the source, under some specific conditions the lower and upper bounds coincide and secrecy capacity is established. We apply the results established for the Gaussian memoryless model to a more practical fading relay-eavesdropper channel. Numerical examples showed that power adjustment among parallel channels results in higher secrecy rate.

7Proof of Theorem 1

The proof generalizes the results of Theorem 1 in [12] and uses elements from a similar proof in the context of parallel BCC in [3].
a) We first bound the equivocation rate as follows.

where as ; follows from Fano’s inequality; and and follows from lemma 7 in [2].

We introduce a random variable uniformly distributed over and set, , and . We define , for . Note that satisfies the following Markov chain condition

Thus, we have

where and follow by using the above definition.
We can also bound the equivocation rate as follows. We continue from to get

where and follow from the above definition.

b)

We now bound the rate as follows.