Classical and Quantum algorithms for generic Syndrome Decoding problems and applications to the Lee metric

TL;DR

This paper extends syndrome decoding algorithms to various weight functions, including the Lee metric, analyzing their classical and quantum complexities to inform cryptographic security.

Contribution

It introduces a generalized ISD framework using Wagner's algorithm for diverse weight functions and computes their asymptotic complexities in classical and quantum settings.

Findings

Wagner's algorithm can be integrated into ISD for various weight functions.

Classical and quantum complexities of SD are characterized for these new settings.

Decoding in the Lee metric can be made computationally hard, impacting cryptosystem security.

Abstract

The security of code-based cryptography usually relies on the hardness of the syndrome decoding (SD) problem for the Hamming weight. The best generic algorithms are all improvements of an old algorithm by Prange, and they are known under the name of Information Set Decoding (ISD) algorithms. This work aims to extend ISD algorithms' scope by changing the underlying weight function and alphabet size of SD. More precisely, we show how to use Wagner's algorithm in the ISD framework to solve SD for a wide range of weight functions. We also calculate the asymptotic complexities of ISD algorithms both in the classical and quantum case. We then apply our results to the Lee metric, which currently receives a significant amount of attention. By providing the parameters of SD for which decoding in the Lee weight seems to be the hardest, our study could have several applications for designing code-based cryptosystems and their security analysis, especially against quantum adversaries.