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arXiv:2003.00240

Secure Polar Coding for Adversarial Wiretap Channel

Yizhi Zhao

Published Feb 29, 2020• 0 citations
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Abstract

The adversarial wiretap channel (AWTC) model is a secure communication model in which adversary can directly read and write the transmitted bits in legitimate communication with fixed fractions. In this paper we propose a secure polar coding scheme to provide secure and reliable communication over the AWTC model. For the adversarial reading and writing action, we present a $ρ$ equivalent channel block and study its transformation under the channel polarization operation. We find that the generated channels are polarized in the sense that part of them are full-noise channels with probability almost $1$ and the rest part of them are noiseless channels with probability almost $1$. Based on this result, we polarize both equivalent channel blocks of adversarial reading and writing, and then construct a secure polar coding scheme by applying the multi-block chaining structure on the polarized equivalent blocks. Theoretically we prove that when block length $N$ goes infinity, the proposed scheme achieves the secrecy capacity of the AWTC model under both reliability and strong security criterions. Then by simulations, we prove that the proposed scheme can provide secure and reliable communication over AWTC model.

DOI
10.48550/arxiv.2003.00240

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The adversarial wiretap channel (AWTC) model is a secure communication model in which adversary can directly read and write the transmitted bits in legitimate communication with fixed fractions. In this paper we propose a secure polar coding scheme to provide secure and reliable communication over the AWTC model. For the adversarial reading and writing action, we present a ρρρ equivalent channel block and study its transformation under the channel polarization operation. We find that the generated channels are polarized in the sense that part of them are full-noise channels with probability almost 111 and the rest part of them are noiseless channels with probability almost 111. Based on this result, we polarize both equivalent channel blocks of adversarial reading and writing, and then construct a secure polar coding scheme by applying the multi-block chaining structure on the polarized equivalent blocks. Theoretically we prove that when block length NNN goes infinity, the proposed scheme achieves the secrecy capacity of the AWTC model under both reliability and strong security criterions. Then by simulations, we prove that the proposed scheme can provide secure and reliable communication over AWTC model.
The problem of achieving the secrecy capacity of static adversarial wiretap channel with discrete memoryless channels is studied in this paper. To construct the explicit secure coding scheme, an irregular channel polarization operation is proposed which is the extension of Ar{\i}kan's channel polarization. As theoretically proofed, for NNN independent initial channels with different transition probabilities, channels generated by the operation GN\mathbf{G}_NGN​ are also polarized into full noise channels and noiseless channels, same as the regular channel polarization. Then for the adversarial behaviors including directly reading and rewriting, equivalent channels are constructed by channel cascading with full noise or noiseless binary erase channels. Finally by applying the irregular channel polarization to the equivalent channels, a secure polar coding scheme is constructed which successfully achieves the secrecy capacity of static adversarial wiretap channel with discrete memoryless channels under the reliability and strong security criterions.
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.
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.
With the emergence of wireless networks, cooperation for secrecy is recognized as an attractive way to establish secure communications. Departing from cryptographic techniques, secrecy can be provided by exploiting the wireless channel characteristics; that is, some error-correcting codes besides reliability have been shown to achieve information-theoretic security. In this paper, we propose a polar-coding-based technique for the primitive relay wiretap channel and show that this technique is suitable to provide information-theoretic security. Specifically, we integrate at the relay an additional functionality, which allows it to smartly decide whether it will cooperate or not based on the decoding detector result. In the case of cooperation, the relay operates in a decode-and-forward mode and assists the communication by transmitting a complementary message to the destination in order to correctly decode the initial source's message. Otherwise, the communication is completed with direct transmission from source to the destination. Finally, we first prove that the proposed encoding scheme achieves weak secrecy, then, in order to overcome the obstacle of misaligned bits, we implement a double-chaining construction, which achieves strong secrecy.
In this study, the authors investigate the secure coding issue for a wiretap channel model with fixed main channel and varying wiretap channel, by assuming that legitimate parties can obtain the wiretapping channel state information (CSI) with some delay. For the symmetric degraded delay CSI case, they present an explicit weak security scheme by constructing secure polar codes on a one‐time pad chaining structure, and prove its weak security, reliability and capability of approaching the secrecy capacity for perfect CSI case with delay CSI assumption. Further for the symmetric no‐degraded delay CSI case, they present a modified multi‐block chaining structure in which the original subset of frozen bit is designed for conveying functional random bits securely. Then they combine this modified multi‐block chaining structure with the weak security scheme to construct an explicit strong security polar coding scheme, and prove its strong security, reliability and also the capability of approaching the secrecy capacity for perfect CSI case with delay CSI assumption. At last, they carry out simulations to prove the performance of both secure schemes.