Determining the complexity of computing Gr\"{o}bner bases is an important problem both in theory and in practice, and for that the solving degree plays a key role. In this paper, we study the solving degrees of 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 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 of 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 and signature schemes. From our experimental observation, we raise a conjecture and some questions related to this generalized semi-regularity. These new definitions and our results provide a theoretical formulation of (somehow heuristic) discussions done so far in the cryptographic community.

翻译：确定计算Gröbner基的复杂度在理论和实践上都是一个重要问题，其中求解次数起着关键作用。本文研究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仿射半正则序列生成理想的Gröbner基方法提供数学严谨的正确性证明。本文是作者前期工作的延续，针对求解次数及Gröbner基计算的重要行为给出了补充结果。我们还定义了齐次多项式序列的广义正则次数。对于仿射半正则序列的齐次化，我们将其广义正则次数与其最大Gröbner基次数（即齐次化序列的求解次数）建立联系。本文同时给出了广义（密码学）半正则序列的定义，并由此推导出用于评估密码系统和签名方案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该广义半正则性相关的问题。这些新定义及我们的研究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。