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arXiv:1001.0210

Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes

Hessam Mahdavifar, Alexander Vardy

Published Jan 1, 2010• 0 citations
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Abstract

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 $ε> 0$, there exist coding schemes of rate $R \ge C_s - ε$ 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.

DOI
10.48550/arxiv.1001.0210

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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.
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 Arikan. It is shown that polar codes achieve nontrivial 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.
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.
Reliable communication imposes an upper limit on the achievable rate, namely the Shannon capacity. Wyner's wiretap coding, which ensures a security constraint also in addition to reliability, results in decrease of the achievable rate. To mitigate the loss in the secrecy rate, we propose a coding scheme paper where we use sufficiently old messages as key and for this scheme prove that multiple messages are secure with respect to (w.r.t.) all the information possessed by the eavesdropper. We also show that we can achieve security in the strong sense. Next we consider a fading wiretap channel with full channel state information of the eavesdropper's channel and use our coding/decoding scheme to achieve secrecy capacity close to the Shannon capacity of the main channel (in the ergodic sense). Finally we also consider the case where the transmitter does not have the instantaneous information of the channel state of the eavesdropper, but only its distribution.
This paper presents a polar coding scheme to achieve secrecy in block fading binary symmetric wiretap channels without the knowledge of instantaneous channel state information (CSI) at the transmitter. For this model, a coding scheme that hierarchically utilizes polar codes is presented. In particular, on polarization of different binary symmetric channels over different fading blocks, each channel use is modeled as an appropriate binary erasure channel over fading blocks. Polar codes are constructed for both coding over channel uses for each fading block and coding over fading blocks for certain channel uses. In order to guarantee security, random bits are introduced at appropriate places to exhaust the observations of the eavesdropper. It is shown that this coding scheme, without instantaneous CSI at the transmitter, is secrecy capacity achieving for the simultaneous fading scenario. For the independent fading case, the capacity is achieved when the fading realizations for the eavesdropper channel is always degraded with respect to the receiver. For the remaining cases, the gap is analyzed by comparing lower and upper bounds. Remarkably, for the scenarios where the secrecy capacity is achieved, the results imply that instantaneous CSI does not increase the secrecy capacity.
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.
This paper presents a polar coding scheme to achieve secrecy in block fading binary symmetric wiretap channels without the knowledge of instantaneous channel state information (CSI) at the transmitter. For this model, a coding scheme that hierarchically utilizes polar codes is presented. In particular, on the polarization of different binary symmetric channels over different fading blocks, each channel use is modeled as an appropriate binary erasure channel over fading blocks. Polar codes are constructed for both coding over channel uses for each fading block and coding over fading blocks for certain channel uses. In order to guarantee security, random bits are introduced at appropriate places to exhaust the observations of the eavesdropper. It is shown that this coding scheme, without instantaneous CSI at the transmitter, is secrecy capacity achieving for the simultaneous fading scenario. For the independent fading case, the capacity is achieved when the fading realizations for the eavesdropper channel are always degraded with respect to the receiver. For the remaining cases, the gap between lower and upper bounds is analyzed. Remarkably, for the scenarios where the secrecy capacity is achieved, the results imply that the instantaneous CSI does not increase the secrecy capacity.