On the Rank Decoding Problem Over Finite Principal Ideal Rings

TL;DR

This paper investigates the complexity of the rank decoding problem over finite principal ideal rings, establishing its difficulty relative to finite fields and providing algorithms for finite chain rings.

Contribution

It demonstrates the equivalence in complexity of rank decoding over finite rings and fields and introduces combinatorial algorithms for finite chain rings.

Findings

Rank decoding over finite principal ideal rings is at least as hard as over finite fields.

Computing minimum rank distance over finite rings is equivalent to the problem over finite fields.

Provides combinatorial algorithms with average complexities for finite chain rings.

Abstract

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings. In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.