Hamming 理想与类 ISD 综合征解码的 Gröbner 基
Hamming Ideals and Grobner Bases for ISD-like Syndrome Decoding
- University of Bari（巴里大学）
- Technology Innovation Institute（技术创新研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种结合 Hamming 重量代数建模与类 ISD 解码策略的综合征解码方法，通过 GBDecode 和 MultiSolve 算法平衡组合搜索与 Gröbner 基求解，并在 Classic McEliece 参数下验证可行性。
AI中文摘要：
我们研究了一种基于 Hamming 重量约束重构并将其与信息集解码（ISD）范式相结合的代数方法来解决综合征解码问题。我们首先对 Hamming 簇进行系统分析，以初等对称函数的形式推导其定义方程。由于这些方程可能具有较高的次数，我们利用初等对称函数的卷积恒等式以及基于 Lucas 恒等式的因式分解，推导出一个带有辅助变量和有界次数方程的等价表述。基于此建模，我们通过一种类 ISD 解码策略推广了 ISD 范式，该策略由 GBDecode 算法实现，其中仅固定信息集的一个子集。这种方法以求解相关的多元非线性系统为代价，减小了组合搜索空间的大小。为了处理这一代数部分，我们采用 MultiSolve 算法，该算法将单个 Gröbner 基计算替换为对更简单系统的一组计算，这些系统通过在有限域上穷举分配不同数量的不定元获得。这在组合搜索与代数求解之间提供了可调的平衡。我们在随机二元线性码的综合征解码问题实例上对所提方法进行了实验评估，使用了与 Classic McEliece 密码系统 NIST 安全类别 1 参数集相对应的参数。实验评估了这种组合-代数方法的可行性，并深入了解了 Gröbner 基技术在类 ISD 解码框架中的实际行为。
英文摘要：
We investigate an algebraic approach to the Syndrome Decoding Problem, based on a reformulation of the Hamming weight constraint and its integration with the Information Set Decoding paradigm. We begin with a systematic analysis of the Hamming variety, deriving its defining equations in terms of elementary symmetric functions. Since these equations may have high degree, we exploit convolution identities for elementary symmetric functions, together with factorizations based on Lucas' identity, to derive an equivalent formulation with auxiliary variables and equations of bounded degree. Building on this modeling, we generalize the ISD paradigm through an ISD-like decoding strategy, implemented by the GBDecode algorithm, in which only a subset of an information set is fixed. This approach reduces the size of the combinatorial search space at the cost of solving the associated multivariate nonlinear systems. To handle this algebraic component, we employ the MultiSolve algorithm, which replaces a single Grobner basis computation with a collection of computations on simpler systems, obtained by exhaustively assigning a varying number of indeterminates over the finite field. This provides a tunable balance between combinatorial search and algebraic solving. We evaluate the resulting approach experimentally on instances of the Syndrome Decoding Problem for random binary linear codes, using parameters corresponding to the NIST Security Category 1 parameter set of the Classic McEliece cryptosystem. The experiments assess the feasibility of this combinatorial-algebraic approach and provide insights into the practical behavior of Grobner basis techniques within an ISD-like decoding framework.
