Combinatorial Solution of the Syndrome Decoding Problem using Copula on Grassmann graph

TL;DR

This paper introduces a novel approach to the syndrome decoding problem using copula functions on Grassmann graphs, leveraging subspace dependencies to improve decoding performance in code-based cryptography.

Contribution

It proposes a new method exploiting subspace dependencies via copula functions on Grassmann graphs for syndrome decoding, enhancing solution approximation and performance.

Findings

Outperforms information set decoding in Bit Error Rate

Uses copula functions for dependency modeling and marginal distribution estimation

Employs maximum likelihood estimation for codeword search

Abstract

Computational hardness assumption from the syndrome decoding problem has been useful in designing the security of code based cryptosystem that are safe against quantum computing. Due to complexities in solution using high degree linearized polynomial equations modeled from subspaces, we proposed exploiting the dependency between subspaces in a Grassmann graph constructed from Boundary measurement maps by using copula functions. We also used copula functions to estimate the marginal distribution in these subspaces. Thereafter, the Maximum likelihood based estimation approach was used to search the codeword that maximizes the conditional distribution and in the process approximate a solution to the problem. Results of the Bit Error Rate performance obtained from simulation shows that the proposed solution performs better than the information set decoding method.