Quadratic Modelings of Syndrome Decoding

2024-12-06

IACR Cryptology ePrint Archive

AI Summary

This work investigates the algebraic modeling and solving complexity of the Weighted Syndrome Decoding Problem (SDP) over finite fields—particularly $mathbb{F}_2$ and $mathbb{F}_q$. For both exact-weight and bounded-weight SDP variants, we establish, for the first time, a unified polynomial system model and introduce a novel algebraic transformation from SDP over $mathbb{F}_q$ to polynomial systems. We further conduct the first rigorous analysis of the degree of regularity for exact-weight SDP modeling and precisely determine the dimension of the associated algebraic variety. Integrating Gröbner basis techniques with tools from algebraic geometry, we significantly improve the efficiency of algebraic attack modeling. Experimental results demonstrate that our models yield lower computational complexity in Gröbner basis computations compared to prior approaches. This provides a more efficient and theoretically grounded framework for cryptanalysis—especially for assessing the security of post-quantum code-based cryptosystems such as McEliece-type schemes.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Syndrome Decoding Problem
• Digital Weight
• F_2 Mathematical Environment

Innovation

• Quadratic Equations
• Syndrome Decoding Problem (SDP)
• Polynomial Transformation