A Model for Adversarial Wiretap Channel

In wiretap model of secure communication the goal is to provide (asymptotic) perfect secrecy and reliable communication over a noisy channel that is eavesdropped by an adversary with unlimited computational power. This goal is achieved by taking advantage of the channel noise and without requiring a shared key. The model has attracted attention in recent years because it captures eavesdropping attack in wireless communication. The wiretap adversary is a passive eavesdropping adversary at the physical layer of communication. In this paper we propose a model for adversarial wiretap (AWTP) channel that models active adversaries at this layer. We consider a $(\rho_r, \rho_w)$ wiretap adversary who can see a fraction $\rho_r$, and modify a fraction $\rho_w$, of the sent codeword. The code components that are read and/or modified can be chosen adaptively, and the subsets of read and modified components in general, can be different. AWTP codes provide secrecy and reliability for communication over these channels. We give security and reliability definitions and measures for these codes, and define secrecy capacity of an AWTP channel that represents the secrecy potential of the channel. The paper has two main contributions. First, we prove a tight upper bound on the rate of AWTP codes with perfect secrecy for $(\rho_r, \rho_w)$-AWTP channels, and use the bound to derive the secrecy capacity of the channel. We prove a similar bound for $\epsilon$-secure codes also, but in this case the bound is not tight. Second, we give an explicit construction for a capacity achieving AWTP code family, and prove its security and efficiency properties. We show that AWTP model is a natural generalization of Wyner's wiretap models and somewhat surprisingly, also provides a direct generalization for a seemingly unrelated cryptographic primitive, Secure Message Transmission (SMT)

Codes That Achieve Capacity on Symmetric Channels

Transmission of information reliably and efficiently across channels is one of the fundamental goals of coding and information theory. In this respect, efficiently decodable deterministic coding schemes which achieve capacity provably have been elusive until as recent as 2008, even though schemes which come close to it in practice existed. This survey tries to give the interested reader an overview of the area. Erdal Arikan came up with his landmark polar coding shemes which achieve capacity on symmetric channels subject to the constraint that the input codewords are equiprobable. His idea is to convert any B-DMC into efficiently encodable-decodable channels which have rates 0 and 1, while conserving capacity in this transformation. An exponentially decreasing probability of error which independent of code rate is achieved for all rates lesser than the symmetric capacity. These codes perform well in practice since encoding and decoding complexity is O(N log N). Guruswami et al. improved the above results by showing that error probability can be made to decrease doubly exponentially in the block length. We also study recent results by Urbanke et al. which show that 2-transitive codes also achieve capacity on erasure channels under MAP decoding. Urbanke and his group use complexity theoretic results in boolean function analysis to prove that EXIT functions, which capture the error probability, have a sharp threshold at 1-R, thus proving that capacity is achieved. One of the oldest and most widely used codes - Reed Muller codes are 2-transitive. Polar codes are 2-transitive too and we thus have a different proof of the fact that they achieve capacity, though the rate of polarization would be better as found out by Guruswami.Comment: Survey done under the guidance of Prof. Prahladh Harsha as part of the Visiting Students' Research Programme 2015 at the School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai. Keywords : capacity achieving codes, polar codes, reed muller code

Using Reed-Solomon codes in the (U∣U+V)\left( U\mid U+V\right) construction and an application to cryptography

In this paper we present a modification of Reed-Solomon codes that beats the Guruwami-Sudan $1-\sqrt{R}$ decoding radius of Reed-Solomon codes at low rates $R$. The idea is to choose Reed-Solomon codes $U$ and $V$ with appropriate rates in a $\left( U\mid U+V\right)$ construction and to decode them with the Koetter-Vardy soft information decoder. We suggest to use a slightly more general version of these codes (but which has the same decoding performances as the $\left( U\mid U+V\right)$-construction) for code-based cryptography, namely to build a McEliece scheme. The point is here that these codes not only perform nearly as well (or even better in the low rate regime) as Reed-Solomon codes, their structure seems to avoid the Sidelnikov-Shestakov attack which broke a previous McEliece proposal based on generalized Reed-Solomon codes

How to beat the sphere-packing bound with feedback

The sphere-packing bound $E_{sp}(R)$ bounds the reliability function for fixed-length block-codes. For symmetric channels, it remains a valid bound even when strictly causal noiseless feedback is allowed from the decoder to the encoder. To beat the bound, the problem must be changed. While it has long been known that variable-length block codes can do better when trading-off error probability with expected block-length, this correspondence shows that the {\em fixed-delay} setting also presents such an opportunity for generic channels. While $E_{sp}(R)$ continues to bound the tradeoff between bit error and fixed end-to-end latency for symmetric channels used {\em without} feedback, a new bound called the ``focusing bound'' gives the limits on what can be done with feedback. If low-rate reliable flow-control is free (ie. the noisy channel has strictly positive zero-error capacity), then the focusing bound can be asymptotically achieved. Even when the channel has no zero-error capacity, it is possible to substantially beat the sphere-packing bound by synthesizing an appropriately reliable channel to carry the flow-control information.Comment: 9 pages, 3 figures, corrected typos and increased font size. Submitted to IT Transaction

Polar-Coded Non-Orthogonal Multiple Access

Non-orthogonal multiple access (NOMA) is one of the key techniques to address the high spectral efficiency and massive connectivity requirements for the fifth generation (5G) wireless system. To efficiently realize NOMA, we propose a joint design framework combining the polar coding and the NOMA transmission, which deeply mines the generalized polarization effect among the users. In this polar coded NOMA (PC-NOMA) framework, the original NOMA channel is decomposed into multiple bit polarized channels by using a three-stage channel transform, that is, user$\to$signal$\to$bit partitions. Specifically, for the first-stage channel transform, we design two schemes, namely sequential user partition (SUP) and parallel user partition (PUP). For the SUP, a joint successive cancellation detecting and decoding scheme is developed, and a search algorithm is proposed to schedule the NOMA detecting order which improves the system performance by enhanced polarization among the user synthesized channels. The "worst-goes-first" idea is employed in the scheduling strategy, and its theoretic performance is analyzed by using the polarization principle. For the PUP, a corresponding parallel detecting scheme is exploited to reduce the latency. The block error ratio performances over the additive white Gaussian noise channel and the Rayleigh fading channel indicate that the proposed PC-NOMA obviously outperforms the state-of-the-art turbo coded NOMA scheme due to the advantages of joint design between the polar coding and NOMA.Comment: First versio

Binary Polar Codes are Optimized Codes for Bitwise Multistage Decoding

Polar codes are considered the latest major breakthrough in coding theory. Polar codes were introduced by Ar{\i}kan in 2008. In this letter, we show that the binary polar codes are the same as the optimized codes for bitwise multistage decoding (OCBM), which have been discovered before by Stolte in 2002. The equivalence between the techniques used for the constructions and decodings of both codes is established.Comment: Accepted at Electronics Letter

Algebraic Soft-Decision Decoding of Reed-Solomon Codes Using Bit-level Soft Information

The performance of algebraic soft-decision decoding of Reed-Solomon codes using bit-level soft information is investigated. Optimal multiplicity assignment strategies of algebraic soft-decision decoding with infinite cost are first studied over erasure channels and the binary symmetric channel. The corresponding decoding radii are calculated in closed forms and tight bounds on the error probability are derived. The multiplicity assignment strategy and the corresponding performance analysis are then generalized to characterize the decoding region of algebraic softdecision decoding over a mixed error and bit-level erasure channel. The bit-level decoding region of the proposed multiplicity assignment strategy is shown to be significantly larger than that of conventional Berlekamp-Massey decoding. As an application, a bit-level generalized minimum distance decoding algorithm is proposed. The proposed decoding compares favorably with many other Reed-Solomon soft-decision decoding algorithms over various channels. Moreover, owing to the simplicity of the proposed bit-level generalized minimum distance decoding, its performance can be tightly bounded using order statistics.Comment: 32 pages, 12 figures, to appear in IEEE Trans. on Information Theor

On the capacity of the binary adversarial wiretap channel

New bounds on the semantic secrecy capacity of the binary adversarial wiretap channel are established . Against an adversary which reads a $\rho_r$ fraction of the transmitted codeword and modifies a $\rho_w$ fraction of the codeword, we show an achievable rate of $1-h(\rho_w)-\rho_r$, where $h(\cdot)$ is the binary entropy function. We also give an upper bound which is nearly matching when $\rho_r$ is small.Comment: v2: Capacity upper bound has been correcte

Does Gaussian Approximation Work Well for The Long-Length Polar Code Construction?

Gaussian approximation (GA) is widely used to construct polar codes. However when the code length is long, the subchannel selection inaccuracy due to the calculation error of conventional approximate GA (AGA), which uses a two-segment approximation function, results in a catastrophic performance loss. In this paper, new principles to design the GA approximation functions for polar codes are proposed. First, we introduce the concepts of polarization violation set (PVS) and polarization reversal set (PRS) to explain the essential reasons that the conventional AGA scheme cannot work well for the long-length polar code construction. In fact, these two sets will lead to the rank error of subsequent subchannels, which means the orders of subchannels are misaligned, which is a severe problem for polar code construction. Second, we propose a new metric, named cumulative-logarithmic error (CLE), to quantitatively evaluate the remainder approximation error of AGA in logarithm. We derive the upper bound of CLE to simplify its calculation. Finally, guided by PVS, PRS and CLE bound analysis, we propose new construction rules based on a multi-segment approximation function, which obviously improve the calculation accuracy of AGA so as to ensure the excellent performance of polar codes especially for the long code lengths. Numerical and simulation results indicate that the proposed AGA schemes are critical to construct the high-performance polar codes

Delay-Sensitive Communication over Fading Channel: Queueing Behavior and Code Parameter Selection

This article examines the queueing performance of communication systems that transmit encoded data over unreliable channels. A fading formulation suitable for wireless environments is considered where errors are caused by a discrete channel with correlated behavior over time. Random codes and BCH codes are employed as means to study the relationship between code-rate selection and the queueing performance of point-to-point data links. For carefully selected channel models and arrival processes, a tractable Markov structure composed of queue length and channel state is identified. This facilitates the analysis of the stationary behavior of the system, leading to evaluation criteria such as bounds on the probability of the queue exceeding a threshold. Specifically, this article focuses on system models with scalable arrival profiles, which are based on Poisson processes, and finite-state channels with memory. These assumptions permit the rigorous comparison of system performance for codes with arbitrary block lengths and code rates. Based on the resulting characterizations, it is possible to select the best code parameters for delay-sensitive applications over various channels. The methodology introduced herein offers a new perspective on the joint queueing-coding analysis of finitestate channels with memory, and it is supported by numerical simulations