Generalization of semi-regular sequences: Maximal Gröbner basis degree, variants of genericness, and related conjectures
Momonari Kudo, Kazuhiro Yokoyama
TL;DR
This work analyzes the complexity of computing Gröbner bases for homogeneous polynomial systems by extending the notion of semi-regular sequences to generalized cryptographic semi-regular sequences in Krull-dimension-one settings. It derives a maximal Gröbner-basis degree bound $D^{(n,m)}$ and a corresponding improved complexity bound, showing optimality in key cases and tightness of Hashemi-Seiler-type inequalities under weakly reverse lexicographic initials. The paper also develops a robust genericness framework for generalized cryptographic semi-regular sequences via parametric Gröbner bases, linking to Fröberg's and Moreno-Socías' conjectures and establishing criteria under which these conjectures hold simultaneously. Additionally, it discusses how these results inform the genericness of cryptographic semi-regularity and the behavior of initial ideals, with implications for the security of multivariate cryptosystems and related algorithmic analyses. The inhomogeneous case is briefly addressed via homogenization, showing how the presented bounds extend to practical zero-dimensional and near-zero-dimensional instances.
Abstract
Nowadays, the notion of semi-regular sequences, originally proposed by Fröberg, becomes very important not only in Mathematics, but also in Information Science, in particular Cryptology. For example, it is highly expected that randomly generated polynomials form a semi-regular sequence, and based on this observation, secure cryptosystems based on polynomial systems can be devised. In this paper, we deal with a semi-regular sequence and its extension, named a generalized cryptographic semi-regular sequence, and give precise analysis on the complexity of computing a Gröbner basis of the ideal generated by such a sequence with help of several regularities of the ideal related to Lazard's bound on maximal Gröbner basis degree and other bounds. We also study the genericness of the property that a sequence is semi-regular, and its variants related to Fröberg's conjecture. Moreover, we discuss on the genericness of another important property that the initial ideal is weakly reverse lexicographic, related to Moreno-Socías' conjecture, and show some criteria to examine whether both Fröberg's conjecture and Moreno-Socías' one hold at the same time.