Generalized rank weights of reducible codes, optimal cases and related properties

TL;DR

This paper investigates the security properties of reducible codes under the rank metric, providing bounds on their generalized rank weights, identifying optimal parameters, and analyzing their information leakage in network coding scenarios.

Contribution

It offers new bounds on generalized rank weights, characterizes when codes are MRD, and shows reducible codes often leak less information than worst-case bounds.

Findings

Bounds on generalized rank weights for reducible codes

Identification of parameters where codes are MRD

Reducible codes often leak less information than worst-case estimates

Abstract

Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes and can correct many rank errors beyond half of their minimum rank distance, which makes them suitable for error-correction in network coding. In this paper, we study their security behaviour against information leakage on networks when applied as coset coding schemes, giving the following main results: 1) we give lower and upper bounds on their generalized rank weights (GRWs), which measure worst-case information leakage to the wire-tapper, 2) we find new parameters for which these codes are MRD (meaning that their first GRW is optimal), and use the previous bounds to estimate their higher GRWs, 3) we show that all linear (over the extension field) codes whose GRWs are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence, and 4) we show that the information leaked to a wire-tapper when using reducible codes is often much less than the worst case given by their (optimal in some cases) GRWs. We conclude with some secondary related properties: Conditions to be rank equivalent to cartesian products of linear codes, conditions to be rank degenerate, duality properties and MRD ranks.