Quadratic Modelings of Syndrome Decoding

TL;DR

This work advances algebraic cryptanalysis of the Syndrome Decoding Problem by presenting enhanced quadratic reductions to polynomial systems over $\mathbb{F}_2$ and generalizing to $\mathbb{F}_q$, supplemented by a thorough complexity and experimental analysis. The core contributions include a refined MPS-based modeling that reduces the number of variables and equations, a construction that preserves equivalence while remaining quadratic, and a new $\mathbb{F}_q$-oriented framework using linear recurrences to control weight-encoding complexity. The authors derive explicit degree-of-regularity results for the quadratic components, demonstrate that solving-degree behavior can be markedly smaller than the worst-case bounds, and provide experimental evidence that, for small parameters, Gröbner-basis methods can succeed on these reformulations, with practical software provided. The findings offer both theoretical insight into the hardness landscape of SDP variants and practical modeling techniques that inform parameter selection and potential hybrid strategies in code-based cryptanalysis.

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 Gröbner bases techniques.

Theorems & Definitions (42)

• remark 1
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• proposition 1
• proof
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• remark 6
• lemma 1
• proof