Paper 2022/967 Original: eprint.iacr.org/2022/967 ↗

On the Computational Hardness of the Code Equivalence Problem in Cryptography

Abstract

Code equivalence is a well-known concept in coding theory. Recently, literature saw an increased interest in this notion, due to the introduction of protocols based on the hardness of finding the equivalence between two linear codes. In this paper, we analyze the security of code equivalence, with a special focus on the hardest instances, in the interest of cryptographic usage. Our work stems from a thorough review of existing literature, identifies the various types of solvers for the problem, and provides a precise complexity analysis, where previously absent. Furthermore, we are able to improve on the state of the art, providing more efficient algorithm variations, for which we include numerical simulation data. Our results include also a dedicated method for solving code equivalence with a quantum algorithm, as well as a refinement of quantum Information-Set Decoding (ISD) algorithms. In the end, the goal of this paper is to provide a complete, single point of access, which can be used as a tool for designing schemes that rely on the code equivalence problem.