Hessam Mahdavifar

Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes

Suppose that Alice wishes to send messages to Bob through a communication channel C1, but her transmissions also reach an eavesdropper Eve through another channel C2. This is the wiretap channel model introduced by Wyner in 1975. 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 Bobs probability of error in recovering the message, while security is measured in terms of the mutual information between the message and Eves observations. Wyner showed that the situation is characterized by a single constant Cs, called the secrecy capacity, which has the following meaning: for all ε >; 0, there exist coding schemes of rate R ≥ Cs-ε that asymptotically achieve the reliability and security objectives. However, his proof of this result is based upon a random-coding argument. To date, despite consider able research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eves channel C2 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. In this paper, 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 C1 and C2 are symmetric and binary-input, and C2 is degraded with respect to C1. 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 also be satisfied unless the main channel C1 is noiseless, although we believe it can be always satisfied in practice.

Achieving the secrecy capacity of wiretap channels using Polar codes

Suppose that 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 . This is the wiretap channel model introduced by Wyner in 1975. 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 Bobs probability of error in recovering the message, while security is measured in terms of the ratio of Eves equivocation about the message to its a priori entropy. Wyner showed that the situation is characterized by a single constant C s , called the secrecy capacity, which has the following meaning: for all e > 0, there exist coding schemes of rate R ⩾ C s - e 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 codes that achieve secrecy capacity is when Eves channel C 2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently introduced by Arikan. They achieve the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. In this paper, we use polar codes to construct a coding scheme that achieves the secrecy capacity of general wiretap channels. Our construction works for any instantiation of the wiretap channel model, as originally defined by Wyner, as long as both C 1 and C 2 are symmetric and binary-input.

A nearly optimal construction of flash codes

Flash memory is a non-volatile computer memory comprised of blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing an entire block. Since block erasures are highly undesirable, coding schemes—known as floating codes or flash codes—have been designed in order to maximize the number of times that information stored in a flash memory can be written (and re-written) prior to incurring a block erasure. An (n, k, t)q flash code ℂ is a coding scheme for storing k information bits in n cells in such a way that any sequence of up to t writes (where a write is a transition 0 → 1 or 1 → 0 in any one of the k bits) can be accommodated without a block erasure. The total number of available level transitions in n cells is n(q−1), and the write deficiency of ℂ, defined as δ(ℂ) = n(q−1)−t, is a measure of how close the code comes to perfectly utilizing all these transitions. For k ≫ 6 and large n, the best previously known construction of flash codes achieves a write defficiency of O(qk2). On the other hand, the best known lower bound on write deficiency is Ω(qk). In this paper, we present a new construction of flash codes that approaches this lower bound to within a factor logarithmic in k. To this end, we first improve upon the so-called “indexed” flash codes, due to Jiang and Bruck, by eliminating the need for index cells in the Jiang-Bruck construction. Next, we further increase the number of writes by introducing a new multi-stage (recursive) indexing scheme. We then show that the write defficiency of the resulting flash codes is O(qk log k) if q ⩾ log2 k, and at most O(k log2 k) otherwise.

Algebraic list-decoding on the operator channel

The operator channel was introduced by Koetter and Kschischang as a model of errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). The input and output of the operator channel are vector subspaces of the ambient space; thus error-correcting codes for this channel are collections of such subspaces. Koetter and Kschischang also constructed a remarkable family of codes for the operator channel. The Koetter-Kschischang codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang codes on the operator channel. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer L, our list-L decoder guarantess successful recovery of the message subspace provided the normalized dimension of the error is at most L − L2(L + 1) over 2 R where R is the normalized rate of the code. Just as in the case of Sudans list-decoding algorithm, this exceeds the previously best-known error-correction radius 1 - R, demonstrated by Koetter and Kschischang, for low rates R.

Performance Limits and Practical Decoding of Interleaved Reed-Solomon Polar Concatenated Codes

A scheme for concatenating the recently invented polar codes with non-binary MDS codes, as Reed-Solomon codes, is considered. By concatenating binary polar codes with interleaved Reed-Solomon codes, we prove that the proposed concatenation scheme captures the capacity-achieving property of polar codes, while having a significantly better error-decay rate. We show that for any ε > 0, and total frame length N, the parameters of the scheme can be set such that the frame error probability is less than 2-N1-ε, while the scheme is still capacity achieving. This improves upon 2-N0.5-ε, the frame error probability of Arikans polar codes. The proposed concatenated polar codes and Arikans polar codes are also compared for transmission over channels with erasure bursts. We provide a sufficient condition on the length of erasure burst which guarantees failure of the polar decoder. On the other hand, it is shown that the parameters of the concatenated polar code can be set in such a way that the capacity-achieving properties of polar codes are preserved. We also propose decoding algorithms for concatenated polar codes, which significantly improve the error-rate performance at finite block lengths while preserving the low decoding complexity.

List-decoding of subspace codes and rank-metric codes up to Singleton bound

Subspace codes and rank-metric codes can be used to correct errors and erasures in network, with linear network coding. Both types of codes have been extensively studied in the past five years. Subspace codes were introduced by Koetter and Kschischang to correct errors and erasures in networks where topology is unknown (the non-coherent case). In this model, the codewords are vector subspaces of a fixed ambient space; thus codes for this model are collections of such subspaces. In a previous work, we have developed a family of subspace codes, based upon the Koetter-Kschichang construction, which are efficiently list decodable. Using these codes, we achieved a better decoding radius than Koetter-Kschischang codes at low rates. Herein, we introduce a new family of subspace codes based upon a different approach which leads to a linear-algebraic list-decoding algorithm. The resulting error-correction radius can be expressed as follows: for any integer s, our list-decoder using s + 1-variate interpolation polynomials guarantees successful recovery of the message sub-space provided the normalized dimension of errors is at most s(1 - sR). The same list-decoding algorithm can be used to correct erasures as well as errors. The size of output list is at most Qs - 1, where Q is the size of the field that message symbols are chosen from. Rank-metric codes are suitable for error correction in the case where the network topology and the underlying network code are known (the coherent case). Gabidulin codes are a well-known class of algebraic rank-metric codes that meet the Singleton bound on the minimum rank-distance of a code. In this paper, we introduce a folded version of Gabidulin codes analogous to the folded Reed-Solomon codes of Guruswami and Rudra along with a list-decoding algorithm for such codes. Our list-decoding algorithm makes it possible to recover the message provided that the normalized rank of error is at most 1 - R - ϵ, for any ϵ >; 0. Notably this achieves the information theoretic bound on the decoding radius of a rank-metric code.

Compound polar codes

A capacity-achieving scheme based on polar codes is proposed for reliable communication over multi-channels which can be directly applied to bit-interleaved coded modulation schemes. We start by reviewing the ground-breaking work of polar codes and then discuss our proposed scheme. Instead of encoding separately across the individual underlying channels, which requires multiple encoders and decoders, we take advantage of the recursive structure of polar codes to construct a unified scheme with a single encoder and decoder that can be used over the multi-channels. We prove that the scheme achieves the capacity over this multi-channel. Numerical analysis and simulation results for BICM channels at finite block lengths shows a considerable improvement in the probability of error comparing to a conventional separated scheme.

Polar Coding for Bit-Interleaved Coded Modulation

A polar coding scheme is proposed for reliable communication over channels with bit-interleaved coded modulation (BICM). In an ideal information-theoretic model, BICM schemes are modeled as a set of multiple binary-input channels that experience different reliabilities. This model is referred to as multichannel. The conventional scheme for encoding information over multi-channels is to encode separately across the individual constituent channels, which requires multiple encoders and decoders. As opposed to the conventional scheme, we take advantage of the recursive structure of polar codes to construct a unified compound polar code. The proposed scheme has a single encoder and decoder and can be used over multi-channels, in particular over channels with BICM. We prove that the scheme achieves the multi-channel capacity with the same error decay rate as in Arıkans polar codes. Furthermore, due to more levels of channel polarization, we obtain a better finite block length performance compared with the separated polarization scheme. This is confirmed through various simulations over the binary erasure channel and with transmissions over additive white Gaussian noise (AWGN) and fast fading channels with BICM and different modulation orders.

Achieving the Uniform Rate Region of General Multiple Access Channels by Polar Coding

We consider the problem of polar coding for transmission over m-user multiple access channels. In the proposed scheme, all users encode their messages using a polar encoder, while a multiuser successive cancellation decoder is deployed at the receiver. The encoding is done separately across the users and is independent of the target achievable rate. For the code construction, the positions of information bits and frozen bits for each of the users are decided jointly. This is done by treating the polar transformations across all the m users as a single polar transformation with a certain polarization base. We characterize the resolution of achievable rates on the dominant face of the uniform rate region in terms of the number of users m and the length of the polarization base L. In particular, we prove that for any target rate on the dominant face, there exists an achievable rate, also on the dominant face, within the distance at most (m-1)√m/L from the target rate. We then prove that the proposed L MAC polar coding scheme achieves the whole uniform rate region with fine enough resolution by changing the decoding order in the multiuser successive cancellation decoder, as L and the code block length N grow large. The encoding and decoding complexities are O(N log N) and the asymptotic block error probability of O(2-N0.5-ϵ) is guaranteed. Examples of achievable rates for the 3-user multiple access channel are provided.

Algebraic List-Decoding of Subspace Codes

Subspace codes are collections of subspaces of a certain ambient vector space over a finite field. Koetter and Kschischang introduced subspace codes in order to correct errors and erasures in noncoherent (random) linear network coding. They have also studied a remarkable family of subspace codes obtained by evaluating certain linearized polynomials. The Koetter-Kschischang subspace codes are widely regarded as the counterpart of Reed-Solomon codes in the domain of network error-correction. Koetter and Kschischang have furthermore devised an algebraic decoding algorithm for these codes, analogous to the Berlekamp-Welch decoding algorithm for Reed-Solomon codes. In this paper, we develop list-decoding algorithms for subspace codes, thereby providing, for certain code parameters, a better tradeoff between rate and error-correction capability than that of the Koetter-Kschischang codes. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer, our list-L decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error satisfies τ ≤ L - L(L + 1)/2 R* where R* is the packet rate, which is a new parameter introduced in this paper. As in the case of Sudans list-decoding algorithm, this exceeds the previously best-known error-correction radius 1 - R*, established by Koetter and Kschischang, only for low rates R*.