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IEEE Transactions on Information Theory

Polar Coding for the Multiple Access Wiretap Channel via Rate-Splitting and Cooperative Jamming

Rémi A. Chou, Aylin Yener

Published Aug 16, 2018• 28 citations
Publisher

Abstract

We consider strongly secure communication over a discrete memoryless multiple access wiretap channel with two transmitters. No degradation or symmetry assumptions are made on the channel. Our main result is that any rate pair known to be achievable with a random coding like proof, is also achievable with an explicit and low-complexity polar coding scheme. Moreover, if the rate pair is known to be achievable without time-sharing, then time-sharing is not needed in our polar coding scheme as well. Our proof technique relies on rate-splitting, which introduces two virtual transmitters, and cooperative jamming strategies implemented by these virtual transmitters. Specifically, our coding scheme combines point-to-point codes that either aim at secretly conveying a message to the legitimate receiver or at performing cooperative jamming. Each point-to-point code relies on block Markov encoding to be able to deal with an arbitrary channel and strong secrecy. Consequently, our coding scheme is the combination of inter-dependent block Markov constructions. We assess reliability and strong secrecy through a detailed analysis of the dependencies between the random variables involved in the scheme.

DOI
10.1109/tit.2018.2865741

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arXiv:1001.0210•Jan 1, 2010•0 citations
We consider strongly secure communication over a discrete memoryless multiple access wiretap channel with two transmitters. No degradation or symmetry assumptions are made on the channel. Our main result is that any rate pair known to be achievable with a random coding like proof, is also achievable with an explicit and low-complexity polar coding scheme. Moreover, if the rate pair is known to be achievable without time-sharing, then time-sharing is not needed in our polar coding scheme as well. Our proof technique relies on rate-splitting, which introduces two virtual transmitters, and cooperative jamming strategies implemented by these virtual transmitters. Specifically, our coding scheme combines point-to-point codes that either aim at secretly conveying a message to the legitimate receiver or at performing cooperative jamming. Each point-to-point code relies on block Markov encoding to be able to deal with an arbitrary channel and strong secrecy. Consequently, our coding scheme is the combination of inter-dependent block Markov constructions. We assess reliability and strong secrecy through a detailed analysis of the dependencies between the random variables involved in the scheme.
We consider strongly secure communication over a discrete memoryless multiple access wiretap channel with two transmitters - no degradation or symmetry assumptions are made on the channel. Our main result is that any rate pair known to be achievable with a random coding like proof, is also achievable with a low-complexity polar coding scheme. Moreover, if the rate pair is known to be achievable without time-sharing, then time-sharing is not needed in our polar coding scheme as well. Our proof technique relies on rate-splitting and different cooperative jamming strategies. Specifically, our coding scheme combines several point-to-point codes that either aim at secretly conveying a message to the legitimate receiver or at performing cooperative jamming. Each point-to-point code relies on a chaining construction to be able to deal with an arbitrary channel and strong secrecy. We assess reliability and strong secrecy through a detailed analysis of the dependencies between the random variables involved in the scheme.
This paper investigates an explicit polar coding scheme for a two-user discrete memoryless multiple access wiretap channel with partial rate-limited feedback (MAC-WT-PLF). Feedback is exploited to increase channel-input correlation, inject dummy messages, and encrypt messages using a one-time pad. Existing MAC-WT polar coding schemes assume independent channel inputs, which cannot support the correlation requirement. Therefore, we propose an explicit polar mapping that leverages feedback to introduce correlation between the channel inputs. This mapping allows both the transmitter and the receiver to restructure the correlation in opposite decoding directions. The proposed scheme relies on source polarization, block Markov coding, superposition coding, lossy source coding, and ratesplitting, without making symmetry or degradation assumptions on the channel model. Rigorous information-theoretic asymptotic analysis establishes that the proposed scheme ensures both reliability and strong secrecy, and attains the entire achievable secrecy rate region given in prior work.
In this paper, we consider the problem of polar coding for secure communications over the two-way wiretap channel, where two legitimate users communicate with each other simultaneously, while a passive eavesdropper overhears a combination of their exchanged signals. We design a low complexity polar coded cooperative jamming scheme that achieves the whole secrecy rate region of the general two-way wiretap channel under the strong secrecy criterion. The chaining method is used to make proper alignment of polar indices. The randomness required to be shared between two legitimate users is treated as a limited resource and we show that its rate can be made negligible by increasing the blocklength and the number of chained blocks. For the special case when the eavesdropper channel is degraded with respect to the legitimate ones, a simplified scheme is proposed which can simultaneously ensure reliability and weak secrecy within a single transmission block.
The authors study cooperative security in the physical layer of wireless systems based on the recently developed polar codes for multiple access channel (MAC). Specifically, the authors first consider the case of a m ‐user MAC with external eavesdropper. Using polar alignment, the authors formulate a discrete optimisation problem where security and reliability criteria can be handled separately over the bases of a set of deterministic binary matrices. A discrete algorithm with incrementally polynomial complexity is used to maximise the uniform sum secrecy rate of users. Moreover, the proposed coding scheme is shown to achieve strong security for any subset of users. The authors next examine the case of minimum number of cooperative helpers to fulfil a feasible secrecy rate requirement at the legitimate user. Here, a low‐complexity suboptimal algorithm is presented with at most one helper more than the optimal solution. Using Tal–Sharov–Vardy implementation of MAC polar codes, secure polar coding is implemented for a 2‐user Gaussian wiretap MAC channel. The upper and lower bound of block error probability are compared, respectively, at the legitimate receiver and the eavesdropper. The results demonstrate clearly that with sufficiently long block length, strong secrecy with respect to the eavesdropper is achieved, while block error probability approaches 0.5.
The wiretap channel with shared key (WTC-K) refers to the classic wiretap channel with a secret key of arbitrary rate shared between the transmitter and the legitimate receiver. While the secrecy capacity of WTC-K has been obtained using random coding arguments, in this paper, we propose a low-complexity polar coding scheme achieving the secrecy capacity under the strong secrecy condition. Our construction consists of choosing the sets of information indices and using the shared key properly. Specifically, we utilize the structure of superposition coding in the proposed scheme.
Suppose Alice wishes to send messages to Bob through a communication channel C_1, but her transmissions also reach an eavesdropper Eve through another channel C_2. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of Eve's equivocation ratio. Wyner showed that the situation is characterized by a single constant C_s, called the secrecy capacity, which has the following meaning: for all ε>0ε> 0ε>0, there exist coding schemes of rate R≥Cs−εR \ge C_s - εR≥Cs​−ε that asymptotically achieve both the reliability and the security objectives. However, his proof of this result is based upon a nonconstructive random-coding argument. To date, despite a considerable research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C_2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Herein, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C_1 and C_2 are symmetric and binary-input, and C_2 is degraded with respect to C_1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will be satisfied unless the main channel C_1 is noiseless, although we believe it can be always satisfied in practice.