Andrew Thangaraj

Applications of LDPC Codes to the Wiretap Channel

With the advent of quantum key distribution (QKD) systems, perfect (i.e., information-theoretic) security can now be achieved for distribution of a cryptographic key. QKD systems and similar protocols use classical error-correcting codes for both error correction (for the honest parties to correct errors) and privacy amplification (to make an eavesdropper fully ignorant). From a coding perspective, a good model that corresponds to such a setting is the wire tap channel introduced by Wyner in 1975. In this correspondence, we study fundamental limits and coding methods for wire tap channels. We provide an alternative view of the proof for secrecy capacity of wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel. We also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel and propose specific practical codes. In some cases our designs achieve the secrecy capacity and in others the codes provide security at rates below secrecy capacity. For the special case of a noiseless main channel and binary erasure channel, we consider encoder and decoder design for codes achieving secrecy on the wiretap channel; we show that it is possible to construct linear-time decodable secrecy codes based on low-density parity-check (LDPC) codes that achieve secrecy.

Strong Secrecy on the Binary Erasure Wiretap Channel Using Large-Girth LDPC Codes

For an arbitrary degree distribution pair (DDP), we construct a sequence of low-density parity-check (LDPC) code ensembles with girth growing logarithmically in block-length using Ramanujan graphs. When the DDP has minimum left degree at least three, we show using density evolution analysis that the expected bit-error probability of these ensembles, when passed through a binary erasure channel with erasure probability ϵ, decays as <i>O</i>(exp(-(<i>c</i>₁)<i>n(c</i>₂))) with the block-length <i>n</i> for positive constants <i>c</i>₁ and <i>c</i>₂, as long as ϵ is less than the erasure threshold ϵ_th of the DDP. This guarantees that the coset coding scheme using the dual sequence provides strong secrecy over the binary erasure wiretap channel for erasure probabilities greater than 1-ϵ_th.

Strong secrecy for erasure wiretap channels

We show that duals of certain low-density parity-check (LDPC) codes, when used in a standard coset coding scheme, provide strong secrecy over the binary erasure wiretap channel (BEWC). This result hinges on a stopping set analysis of ensembles of LDPC codes with block length n and girth ⋛ for some ⋛. We show that if the minimum left degree of the ensemble is l<inf>min</inf>, the expected probability of block error is O(1/n⌈^l min^k/2⌉ −k) when the erasure probability ∊ < ∊<inf>ef</inf>, where ∊<inf>ef</inf> depends on the degree distribution of the ensemble. As long as l<inf>min</inf> and k > 2, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than 1–∊<inf>ef</inf>.

Error-Control Coding for Physical-Layer Secrecy

The renewed interest for physical-layer security techniques has put forward a new role for error-control codes. In addition to ensuring reliability, carefully designed codes have been shown to provide a level of information-theoretic secrecy, by which the amount of information leaked to an adversary may be controlled. The ability to achieve information-theoretic secrecy relies on the study of alternative coding mechanisms, such as channel resolvability and privacy amplification, in which error-control codes are exploited as a means to shape the distribution of stochastic processes. This use of error-control codes, which goes much beyond that of correcting errors, creates numerous new design challenges. The objective of this paper is threefold. First, the paper aims at providing system engineers with explicit tools to build simple secrecy codes in order to stimulate interest and foster their integration in communication system prototypes. Second, it aims at providing coding and information theorists with a synthetic overview of the theoretical concepts and techniques for secrecy. Finally, it aims at highlighting the open challenges and opportunities faced for the integration of these codes in practical systems.

LDPC-based secret key agreement over the Gaussian wiretap channel

This paper investigates a practical secret key agreement protocol over the Gaussian wire-tap channel. The protocol is based on an efficient information reconciliation method which allows two parties having access to correlated continuous random variables to agree on a common bit string. We describe an explicit reconciliation method based on LDPC codes optimized with EXIT charts and density evolution. When used in conjunction with existing privacy amplification techniques our method allows secret key agreement over the Gaussian wire-tap channel close to the secrecy capacity

Confidential messages to a cooperative relay

We extend the broadcast channel with confidential messages to the situation where the receiver of the secret message also serves as a relay. We analyze the fundamental cooperation versus secrecy trade-offs for discrete memoryless channels and obtain the exact rate-equivocation region in this case. For the Gaussian channel, we consider various strategies leading to different levels of secrecy. Our study highlights the fundamental role of jamming as a means to increase secrecy rates, but also emphasizes the importance of carefully designed relaying strategies.

Thresholds and scheduling for LDPC-coded partial response channels

Low-density parity check (LDPC) codes exhibit a threshold phenomenon over a wide variety of channels. The threshold serves as a good parameter for system and code design, and is usually very close to channel capacity. There has been a great deal of recent work surrounding the use of LDPC and turbo codes with iterative decoding for partial response (PR) channels. In this paper, we calculate thresholds for LDPC codes over binary-input PR channels using a Gaussian approximation. We show that the threshold varies according to decoder schedule, and we identify schedules that result in good thresholds. Conversely, we use our results to guide the design of decoder schedules that minimize the number of computationally expensive Bahl-Cocke-Jelinek-Raviv algorithm steps.

Self-interference cancellation models for full-duplex wireless communications

In this paper, we study two models for self or loopback interference cancellation in full-duplex wireless communications. Both models are based on an underlying Z-channel with side information. We obtain achievable rate regions with suitable coding schemes under both models. Under model 1, where the self-interference channel gain is random, we employ training to estimate the unknown gain, and optimize the required training time. Under model 2, where the self-interference gain is exactly known, we show that the capacity of an ideal full-duplex node can be realized even when the side information is low rate and quantized. Our results show that loopback interference, rather than being treated as noise, can be effectively dealt with by suitable coding.

Quantum codes from cyclic codes over GF(4/sup m/)

We provide a construction for quantum codes (Hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4/sup m/). We also provide examples of these codes some of which meet the known bounds for quantum codes.

Secure Compute-and-Forward in a Bidirectional Relay

We consider the basic bidirectional relaying problem, in which two users in a wireless network wish to exchange messages through an intermediate relay node. In the compute-and-forward strategy, the relay computes a function of the two messages using the naturally occurring sum of symbols simultaneously transmitted by user nodes in a Gaussian multiple-access channel (MAC), and the computed function value is forwarded to the user nodes in an ensuing broadcast phase. In this paper, we study the problem under an additional security constraint, which requires that each users message be kept secure from the relay. We consider two types of security constraints: 1) perfect secrecy, in which the MAC channel output seen by the relay is independent of each users message and 2) strong secrecy, which is a form of asymptotic independence. We propose a coding scheme based on nested lattices, the main feature of which is that given a pair of nested lattices that satisfy certain goodness properties, we can explicitly specify probability distributions for randomization at the encoders to achieve the desired security criteria. In particular, our coding scheme guarantees perfect or strong secrecy even in the absence of channel noise. The noise in the channel only affects reliability of computation at the relay, and for Gaussian noise, we derive achievable rates for reliable and secure computation. We also present an application of our methods to the multihop line network in which a source needs to transmit messages to a destination through a series of intermediate relays.