Support-Guessing Decoding Algorithms in the Sum-Rank Metric

TL;DR

This paper introduces support-guessing decoding algorithms for sum-rank-metric codes, analyzes their average-case complexity, and extends decoding capabilities for Linearized Reed--Solomon codes beyond the unique decoding radius, with implications for cryptography.

Contribution

It provides the first average-case analysis of support-guessing algorithms, derives optimal support distributions, and proposes a randomized decoding algorithm for LRS codes that improves success probability and efficiency.

Findings

Optimal support-guessing distribution derived for asymptotic regime

Exact complexity estimates for unique decoding scenarios

Decoding algorithm extends beyond the unique radius with higher success probability

Abstract

The sum-rank metric generalizes the Hamming and rank metric by partitioning vectors into blocks and defining the total weight as the sum of the rank weights of these blocks, based on their matrix representation. In this work, we explore support-guessing algorithms for decoding sum-rank-metric codes. Support-guessing involves randomly selecting candidate supports and attempting to decode the error under the assumption that it is confined to these supports. While previous works have focused on worst-case scenarios, we analyze the average case and derive an optimal support-guessing distribution in the asymptotic regime. We show that this distribution also performs well for finite code lengths. Our analysis provides exact complexity estimates for unique decoding scenarios and establishes tighter bounds beyond the unique decoding radius. Additionally, we introduce a randomized decoding algorithm for Linearized Reed--Solomon (LRS) codes. This algorithm extends decoding capabilities beyond the unique decoding radius by leveraging an efficient error-and-erasure decoder. Instead of requiring the entire error support to be confined to the guessed support, the algorithm succeeds as long as there is sufficient overlap between the guessed support and the actual error support. As a result, the proposed method improves the success probability and reduces computational complexity compared to generic decoding algorithms. Our contributions offer more accurate complexity estimates than previous works, which are essential for understanding the computational challenges involved in decoding sum-rank-metric codes. This improved complexity analysis, along with optimized support-guessing distributions, provides valuable insights for the design and evaluation of code-based cryptosystems using the sum-rank metric. This is particularly important in the context of quantum-resistant cryptography.