In this paper, we study the solving degrees for affine semi-regular sequences and their homogenized sequences. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gr\"{o}bner bases of the ideal generated by an affine semi-regular sequence. This paper is a sequel of the authors' previous work and gives additional results on the solving degrees and important behaviors of Gr\"obner basis computation. We also define the generalized degree of regularity for a sequence of homogeneous polynomials. For the ideal generated by the homogenization of an affine semi-regular sequence, we relate its generalized degree of regularity with its maximal Gr\"{o}bner basis degree (i.e., the solving degree for the homogenized sequence). The definition of a generalized (cryptographic) semi-regular sequence is also given, and it derives a new cryptographic assumption to estimate the security of cryptosystems. From our experimental observation, we raise a conjecture and some questions related to this generalized semi-regularity. These definitions and our results provide a theoretical formulation of (somehow heuristic) discussions done so far in the cryptographic community.

翻译：本文研究了仿射半正则序列及其齐次化序列的求解次数。我们的部分结果旨在为计算由仿射半正则序列生成的理想的Gröbner基的方法提供数学上严谨的正确性证明。本文是作者先前工作的续篇，并给出了关于求解次数及Gröbner基计算重要行为的补充结果。我们还定义了齐次多项式序列的广义正则度。对于由仿射半正则序列齐次化生成的理想，我们将其广义正则度与其最大Gröbner基次数（即齐次化序列的求解次数）联系起来。本文亦给出了广义（密码学）半正则序列的定义，并由此推导出一种用于评估密码系统安全性的新密码学假设。基于实验观察，我们提出了一个猜想及若干与此广义半正则性相关的问题。这些定义及我们的结果为密码学界迄今（某种程度上启发式的）讨论提供了理论框架。