Arora & Ge introduced a noise-free polynomial system to compute the secret of a Learning With Errors (LWE) instance via linearization. Albrecht et al. later utilized the Arora-Ge polynomial model to study the complexity of Gr\"obner basis computations on LWE polynomial systems under the assumption of semi-regularity. In this paper we revisit the Arora-Ge polynomial and prove that it satisfies a genericity condition recently introduced by Caminata & Gorla, called being in generic coordinates. For polynomial systems in generic coordinates one can always estimate the complexity of DRL Gr\"obner basis computations in terms of the Castelnuovo-Mumford regularity and henceforth also via the Macaulay bound. Moreover, we generalize the Gr\"obner basis algorithm of Semaev & Tenti to arbitrary polynomial systems with a finite degree of regularity. In particular, existence of this algorithm yields another approach to estimate the complexity of DRL Gr\"obner basis computations in terms of the degree of regularity. In practice, the degree of regularity of LWE polynomial systems is not known, though one can always estimate the lowest achievable degree of regularity. Consequently, from a designer's worst case perspective this approach yields sub-exponential complexity estimates for general, binary secret, and binary error LWE. In recent works by Dachman-Soled et al. the hardness of LWE in the presence of side information was analyzed. Utilizing their framework we discuss how hints can be incorporated into LWE polynomial systems and how they affect the complexity of Gr\"obner basis computations.

翻译：Arora 与 Ge 引入了一个无噪声多项式系统，通过线性化方法计算带错误学习（LWE）实例的秘密值。Albrecht 等人随后利用 Arora-Ge 多项式模型，在半正则性假设下研究了 LWE 多项式系统上 Gröbner 基计算的复杂性。本文重新审视了 Arora-Ge 多项式，并证明其满足 Caminata 与 Gorla 近期提出的一种称为“处于一般坐标”的泛型条件。对于处于一般坐标的多项式系统，我们总可以通过 Castelnuovo-Mumford 正则性来估计 DRL 序 Gröbner 基计算的复杂性，进而也可通过 Macaulay 界进行估计。此外，我们将 Semaev 与 Tenti 的 Gröbner 基算法推广至具有有限正则度的任意多项式系统。该算法的存在性提供了另一种基于正则度估计 DRL 序 Gröbner 基计算复杂性的方法。实践中，LWE 多项式系统的正则度未知，但始终可估计其最低可达正则度。因此，从设计者最坏情况视角出发，该方法为一般 LWE、二进制秘密 LWE 以及二进制错误 LWE 提供了次指数复杂性估计。在 Dachman-Soled 等人近期工作中，分析了存在旁信息时 LWE 问题的难度。我们利用其框架讨论了如何将提示信息融入 LWE 多项式系统，以及这些信息对 Gröbner 基计算复杂性的影响。