Quadratic Modelings of Syndrome Decoding

TL;DR

This paper improves quadratic modeling techniques for syndrome decoding problems over finite fields, analyzing their algebraic complexity and demonstrating their effectiveness with experimental results using Gröbner bases.

Contribution

It introduces enhanced quadratic reductions for SDP over _2 and a novel method to transform SDP over _q into polynomial systems, with detailed complexity analysis.

Findings

Improved quadratic models for SDP over _2

New transformation method for SDP over _q

Experimental evaluation of solving SDP with Gröbner bases

Abstract

This paper presents enhanced reductions of the bounded-weight and exact-weight Syndrome Decoding Problem (SDP) to a system of quadratic equations. Over $\mathbb{F}_2$, we improve on a previous work and study the degree of regularity of the modeling of the exact weight SDP. Additionally, we introduce a novel technique that transforms SDP instances over $\mathbb{F}_q$ into systems of polynomial equations and thoroughly investigate the dimension of their varieties. Experimental results are provided to evaluate the complexity of solving SDP instances using our models through Gr\"obner bases techniques.