Generic Decoding in the Cover Metric

TL;DR

This paper investigates the decoding difficulty of codes in the cover metric, establishing its NP-hardness and proposing a generic decoding algorithm with exponential complexity in the matrix dimensions.

Contribution

It proves NP-hardness of cover metric decoding via reduction and introduces a generic decoder inspired by information set decoding.

Findings

Decoding in the cover metric is NP-hard.

The proposed decoder has exponential complexity in matrix dimensions.

Cover metric decoding complexity differs from Hamming metric decoding.

Abstract

In this paper, we study the hardness of decoding a random code endowed with the cover metric. As the cover metric lies in between the Hamming and rank metric, it presents itself as a promising candidate for code-based cryptography. We give a polynomial-time reduction from the classical Hamming-metric decoding problem, which proves the NP-hardness of the decoding problem in the cover metric. We then provide a generic decoder, following the information set decoding idea from Prange's algorithm in the Hamming metric. A study of its cost then shows that the complexity is exponential in the number of rows and columns, which is in contrast to the behaviour in the Hamming metric, where the complexity grows exponentially in the number of code symbols.