Secrecy Coding for the Binary Symmetric Wiretap Channel via Linear Programming

In this paper, we use a linear programming (LP) optimization approach to evaluate the equivocation for a wiretap channel where the main channel is noiseless, and the wiretap channel is a binary symmetric channel (BSC). Using this technique, we present an analytical limit for the achievable secrecy rate in the finite blocklength regime that is tighter than traditional fundamental limits. We also propose a secrecy coding technique that outperforms random binning codes. When there is one overhead bit, this coding technique is optimum and achieves the analytical limit. For cases with additional bits of overhead, our coding scheme can achieve equivocation rates close to the new limit. Furthermore, we evaluate the patterns of the generator matrix and the parity-check matrix for linear codes and we present binning techniques for both linear and non-linear codes using two different approaches: recursive and non-recursive. To our knowledge, this is the first optimization solution for secrecy coding obtained through linear programming.Comment: Submitted for possible Journal publicatio

Subspace Decomposition of Coset Codes

A new method is explored for analyzing the performance of coset codes over the binary erasure wiretap channel (BEWC) by decomposing the code over subspaces of the code space. This technique leads to an improved algorithm for calculating equivocation loss. It also provides a continuous-valued function for equivocation loss, permitting proofs of local optimality for certain finite-blocklength code constructions, including a code formed by excluding from the generator matrix all columns which lie within a particular subspace. Subspace decomposition is also used to explore the properties of an alternative secrecy code metric, the chi squared divergence. The chi squared divergence is shown to be far simpler to calculate than equivocation loss. Additionally, the codes which are shown to be locally optimal in terms of equivocation are also proved to be globally optimal in terms of chi squared divergence.Comment: 36 pages, 2 figures, submitted to Transactions on Information Theor

On Dual Relationships of Secrecy Codes

International audienceWe investigate properties of finite blocklength codes and their duals when used for coset coding over the binary erasure wiretap channel (BEWC). We identify sufficient conditions, related to the ranks of sub-matrices of a generator matrix that codes may satisfy to achieve the maximum equivocation among all codes with given blocklength and dimension, irrespective of the eavesdropper's channel erasure probability. We point out that binary maximum distance separable (MDS) codes are optimal for secrecy and we also show that simplex codes (and Hamming codes) have higher equivocation than families of codes with a single repeated column in the generator matrix (parity-check matrix). We conjecture that simplex and Hamming codes are optimal when used as the base linear code in a coset coding scheme for secrecy over the BEWC