Multiple Access Channels with Generalized Feedback and Confidential Messages

Multiple Access Channels with Generalized Feedback and Confidential Messages

\authorblockNXiaojun Tang\authorrefmark1, Ruoheng Liu\authorrefmark2, Predrag Spasojević\authorrefmark1, and H. Vincent Poor\authorrefmark2 \authorblockA\authorrefmark1WINLAB, Rutgers University, North Brunswick, NJ 08902
Email: {xtang, spasojev}@winlab.rutgers.edu \authorblockA\authorrefmark2Princeton University, Princeton, NJ 08544
Email: {rliu, poor}@princeton.edu
This research was supported by the National Science Foundation under Grants ANI-03-38807 and CNS-06-25637.
Abstract

This paper considers the problem of secret communication over a multiple access channel with generalized feedback. Two trusted users send independent confidential messages to an intended receiver, in the presence of a passive eavesdropper. In this setting, an active cooperation between two trusted users is enabled through using channel feedback in order to improve the communication efficiency. Based on rate-splitting and decode-and-forward strategies, achievable secrecy rate regions are derived for both discrete memoryless and Gaussian channels. Results show that channel feedback improves the achievable secrecy rates.

I Introduction

The broadcast nature of wireless medium poses both benefits and penalties for secret communication. The openness of wireless medium provides opportunities for cooperation between trusted users, which improves the communication efficiency. On the other hand, it makes the transmission extremely susceptible to eavesdropping. Anyone within communication range can listen and possibly extract information.

Those two opposite aspects are reflected in the system model as shown in Fig. 1, where we consider a multiple access channel in which two mutually trusted users communicate confidential messages to an intended receiver, in the presence of a passive eavesdropper. Channel feedback enables cooperation between two trusted users and consequently a higher communication efficiency. We refer to this channel as the multiple access channel with generalized feedback and confidential messages (MAC-GF-CM). The level of ignorance of the eavesdropper with respect to the confidential messages is measured by the equivocation rate. This approach was first introduced by Wyner for the wiretap channel [1], in which a single source-destination communication is eavesdropped upon via a degraded channel. Wyner’s formulation was generalized by Csiszár and Körner who determined the capacity region of the broadcast channel with confidential messages [2]. The Gaussian wiretap channel was considered in [3]. More recently, multi-terminal communication with confidential messages has been studied further. This work is related to prior works on the multiple access channel with confidential messages[4, 5], the Gaussian multiple access wiretap channel [6], the interference channel with confidential messages [7], and the relay-eavesdropper channel [8, 9].

The multiple access channel with generalized feedback (MAC-GF) without secrecy consideration was studied in [10, 11, 12, 13, 14, 15]. The terminology “generalized feedback” refers to the wide range of possible situations, including the MAC without feedback, the MAC with output feedback, the MAC-GF with independent noise, the MAC with conferencing encoders, the relay channel and many others. A special case of the Gaussian fading MAC-GF is the so-called user cooperation diversity model proposed in [16].

In this work, we study secret communication over a multiple access channel with generalized feedback. Based on rate-splitting and decode-and-forward strategies, achievable secrecy rate regions are derived for both discrete memoryless and Gaussian MAC-GF-CMs. Several special cases of the derived achievable secrecy rate region include the rate regions of the two-user Gaussian multiple access wiretap channel [6], the relay-eavesdropper channel [8, 9], and the MISO wiretap model [17].

The remainder of the paper is organized as follows. Section II describes the system model. Section III states our main results on achievable rate regions for the discrete memoryless MAC-GF-CM. Some implications of the results are given in Section IV. Section V states our results for a Gaussian MAC-GF-CM with two numerical examples.

Ii System Model


Fig. 1: The two-transmitter multiple access channel with generalized feedback and confidential messages.

A two-user multiple access channel with generalized feedback and confidential messages consists of two transmitters, an intended receiver, and an eavesdropper, as depicted in Fig. 1. The channel is denoted by (, , ), where and are input alphabets; and are output alphabets at the intended receiver and the eavesdropper, respectively; and are the feedback channel output alphabets; and is the transition probability matrix. The channel is memoryless and time-invariant in the sense that

where for . The superscript will be dropped when in order to simplify notations.

Encoder 1 and encoder 2 send independent messages and to the intended receiver in channel uses, in a cooperative way by using the feedback signals . For , a stochastic encoder for user is specified by a matrix of conditional probabilities , where , , and , for , where is the probability that encoder outputs when message is being sent and has been observed at encoder .

The decoder uses the output sequence to compute its estimate of . The decoding function is specified by a mapping .

An () code for the MAC with generalized feedback and confidential messages consists of two sets of encoding functions , , and a decoding function so that its average probability of error is

(1)

The level of ignorance of the eavesdropper with respect to the confidential messages is measured by the equivocation rate .

A rate pair is achievable for the MAC with generalized feedback and confidential messages if, for any , there exists an () code so that

(2)
(3)

for all sufficiently large . The secrecy capacity region is the closure of the set of all achievable rate pairs .

We note that the perfect secrecy condition (3) implies

(4)

and therefore the joint perfect secrecy requirement is stronger than the individual perfect secrecy requirement.

This can be shown as follows:

Similarly, we can show that (3) implies .

Iii Discrete Memoryless Channels

We first state our results for discrete memoryless channels.

Theorem 1

(Partial Decode-and-Forward)

For a discrete memoryless MAC with generalized feedback and confidential messages, the secrecy rate region is achievable, where is the closure of the convex hull of all satisfying

(5)

where

(6)

and denotes the class of joint probability mass functions that factor as

Theorem 1 illustrates a rate-splitting strategy. The rates and are split as and , where and are the rates of information sent by both transmitters cooperatively to the intended receiver, while and are the rates of non-cooperative information sent by user 1 and user 2 individually to the receiver. The random variable represents cooperative resolution information sent by both transmitters. represents information (at rate ) that user 1 sends to user 2 to enable cooperation. represents information (at rate ) that user 2 sends to user 1 to enable cooperation.

, , and represent the rates sacrificed in order to confuse the eavesdropper completely. The sum rate loss is . When we set (in the case of no eavesdropper), , and hence, our result becomes the rate region of the MAC with general feedback as given in [14].

The achievability scheme is based on the combination of superposition block Markov encoding [13], backward decoding [15] and random binning [1, 18]. We outline the proof in the Appendix.

Remark 1: The rate region may be enlarged by using the channel prefixing technique in [2, Lemma 4]. However, we do not follow this approach in this paper to avoid its complicated notation and the intractability of its evaluation.

If we require that and , that is, all information is sent cooperatively and each user can fully decode the other user’s message, we have the following result.

Theorem 2

(Full Decode-and-Forward)

The secrecy rate region is achievable, where is the closure of the convex hull of all satisfying

(7)

where denotes the class of joint probability mass functions that factor as

Iv Some Implications of the Results

Next, we discuss some implications of our main result. We consider several special cases of Theorems 1 and 2, which are consistent with the recent results in [6, 8, 9, 17].

Iv-a Multiple Access Wiretap Channel

An achievable rate region for the Gaussian multiple access wiretap channel is given in [6], which is the special case when neither user can obtain feedback, i.e., and . We set in Theorem 1 and have the achievable region , which is the closure of the convex hull of all satisfying

(8)

where is the class of all distributions that factor as .

Iv-B Relay-Eavesdropper Channel

An achievable rate region for the relay-eavesdropper channel is given in [8, 9], which is the case when only user 1 has confidential messages to send and user 2 is a relay to help with the decode-and-forward strategy; therefore and . We set and in Theorem 2 and the achievable rate satisfies

(9)

for all distributions that factor as . This result is consistent with [8, Theorem 2].

Iv-C MISO Wiretap Channel

When each transmitter can obtain perfect channel feedback, i.e., and , we have a virtual MISO wiretap channel. We set and in Theorem 1. The achievable secrecy rate of the MISO channel is given by

(10)

for all distributions that factor as . This result is consistent with [17].

V Gaussian Channels

In this section, we consider a Gaussian MAC-GF-CM, as depicted in Fig. 2. Each mutually trusted user receives an attenuated and noisy version of the partner’s signal and uses that signal, in conjunction with its own message, to construct the transmit signal. The intended receiver and a passive eavesdropper each get a noisy version of the sum of the attenuated signals of both users. The signal model is therefore

(11)

where for are main and eavesdropper channel gains respectively; and are feedback channel gains, as shown in Fig. 2.


Fig. 2: A Gaussian MAC-GF with confidential messages

We assume the following: the transmitted signal has an average power constraint

(12)

and the noise terms , , , and are independent white zero-mean unit-variance complex Gaussian, i.e., , , , and .

Let , , , and be jointly Gaussian with

(13)

where , , , , and are independent zero mean unit variance Gaussian. The terms , , , , and denote the corresponding power allocation, where

(14)

Following the achievability proof for the discrete memoryless channel, we have the following result for the Gaussian multiple access channel with feedback.

Theorem 3

(Partial Decode-and-Forward)

An achievable secrecy rate region is the closure of the convex hull of all rate pairs with

(15)

where

(16)

and .


Fig. 3: Regular Rate regions and secrecy rate regions for , , , , , under different cooperation conditions , where means no cooperation.

As a numerical example, we show in Fig. 3 the “regular” rate region (without the secrecy constraint) and the secrecy rate region for , , , , and under different cooperation conditions . When , there is no cooperation between the two encoders, which corresponds to the multiple access wiretap channel. Both the regular rate region and the secrecy rate region are significantly enlarged when the channel gains between the two users ( and ) become larger, which shows the benefits due to cooperation. Comparing with the regular rate region, the secrecy rate region suffers rate loss due to the secrecy constraint and furthermore, the secrecy rate region is increasingly dominated by the sum rate constraint, as depicted in Fig. 3.

Next, we give the secrecy rate region when each user can fully decode the message sent by the other user.

Theorem 4

(Full Decode-and-Forward)

An achievable secrecy rate region is the closure of the convex hull of all rate pairs with

(17)

We summarize the secrecy sum rates of partial and full decode-and-forward strategies in the following theorem.

Theorem 5

(Sum Rate) The maximal achievable sum rate in is

(18)

the maximum achievable sum rate in is

(19)

Furthermore, when and

The proof of Theorem 5 is provided in the Appendix.

In Fig. 4, we illustrate secrecy rate regions and for , , , , and under different cooperation conditions . Comparing with , suffers a significant rate loss when and are small () as expected. When and increase, the rate loss is reduced. When and are large enough, and coincide. This observation is partially verified by Theorem 5.


Fig. 4: Secrecy rate regions and for , , , , , under different cooperation conditions: .

Proof: (Theorem 1) The transmission is performed for blocks of length , where both and are sufficiently large and .

The random code generation is described as follows.

We fix , and and split the rate pair as and . Let

(20)

where and as . Let , , and .

Code Construction:

  • [\setlabelwidthZ]

  • Generate codewords by choosing the independently according to for , where .

  • For each , generate codewords by choosing the independently according to for , where .

  • For each tuple , generate codewords by choosing the independently according to for , where .

The codebooks for user 2 are generated in the same way, except that there are and codewords in each of the and codebooks, respectively. The same codebooks will be used for all blocks during the encoding.

Encoding:  Message has bits and is split into two parts: with bits and with bits, respectively. Message is similarly divided into and . Each of the four messages , , and is further divided into sub-blocks of equal lengths for each message. They are denoted by , , and , respectively, for . Let

(21)

where and are uniformly and independently chosen at random from and respectively. We also choose and . The and for are formed in the same way.

Suppose that encoder 1 has obtained and encoder 2 has obtained before block . By forming , encoder 1 transmits ; encoder 2 transmits in block .

Decoding: All decodings are based on the typical set decoding. After the transmission of block is completed, user 1 has seen . It tries to decode . User 2 operates in the same way.

The intended receiver waits until all blocks of transmission are completed and performs backward decoding. Given , it tries to decode . Assuming that the decoding for block is correct, the decoder next considers to decode . The decoder continues until it decodes all blocks.

Error Analysis:  Following similar steps to the error analysis for the MAC-GF in [14], we found that the intended receiver can decode all and therefore with error probability less than any if

and

for sufficiently large , where we also used (20).

Equivocation: Now we consider the equivocation,

(22)

and we can bound each term in the above.

The first term in (V) is given by