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IEEE Transactions on Information Forensics and Security

Polar Coding for the Wiretap Channel With Shared Key

Haowei Wang, Xiaofeng Tao, Na Li, Zhu Han

Published Nov 16, 2017• 22 citations
Publisher

Abstract

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.

DOI
10.1109/tifs.2017.2774499

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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.
Channel state information (CSI) is differently available at each terminal in state-dependent wiretap channels (SD-WTCs). Considering a random channel state non-causally available only at the encoder, this paper investigates an explicit polar coding scheme for the discrete and memoryless SD-WTC, where Alice aims to transmit a secret message (SM) and a secret key (SK) to Bob while concealing them from Eve. Based on a two-layer superposition coding, which includes an inner layer and an outer layer, the proposed polar coding scheme can achieve the inner bound of the current optimal SM-SK capacity region for the discrete and memoryless SD-WTC with CSI non-causally available only at the encoder. A cross-layer construction is proposed to address the issue where the regular chaining construction used in wiretap polar codes cannot be applied in the inner layer. Results show that the decoding error probability (DEP) of the legitimate decoder vanishes to zero as the block length increases. In addition, the proposed scheme is proven to satisfy strong secrecy.
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.
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.
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.
The wiretap channel with secret keys models the problem of secure communication in the presence of a non-legitimate eavesdropper, where the transmitter and legitimate receiver share a secret key of finite rate. This communication scenario is studied for compound channels. An achievable strong secrecy rate is presented and a multi-letter upper bound is derived. For degraded compound wiretap channels with secret keys the strong secrecy capacity is established. Finally, the case of channel state information at the transmitter is discussed and the secrecy capacity is established for this case as well.
Wyner's work on wiretap channels and the recent works on information theoretic security are based on random codes. Achieving information theoretical security with practical coding schemes is of definite interest. In this note, the attempt is to overcome this elusive task by employing the polar coding technique of Arıkan. It is shown that polar codes achieve non-trivial perfect secrecy rates for binary-input degraded wiretap channels while enjoying their low encoding-decoding complexity. In the special case of symmetric main and eavesdropper channels, this coding technique achieves the secrecy capacity. Next, fading erasure wiretap channels are considered and a secret key agreement scheme is proposed, which requires only the statistical knowledge of the eavesdropper channel state information (CSI). The enabling factor is the creation of advantage over Eve, by blindly using the proposed scheme over each fading block, which is then exploited with privacy amplification techniques to generate secret keys.