Generalization of semi-regular sequences: Maximal Gr""{o}bner basis degree, variants of genericness, and related conjectures

2024-10-30

AI Summary

This work addresses the theoretical generalization and computational properties of semi-regular sequences—particularly the precise degree bound of their Gröbner bases, generic behavior, and cryptographic generalizations. Method: Integrating tools from algebraic geometry, commutative algebra, and Lazard’s stratification theory, we systematically analyze initial ideals and regularity constraints under the weak reverse lexicographic order. Contribution/Results: We establish a tight upper bound framework for the maximal degree of Gröbner bases, introduce the novel notion of “generalized cryptographically semi-regular sequences,” and derive a verifiable criterion ensuring the simultaneous validity of the Fröberg and Moreno-Socías conjectures. Our results refine the probabilistic interpretation of semi-regularity in random polynomial systems, provide a more rigorous algebraic security foundation for MQ-based cryptosystems, and advance the synergistic resolution of two foundational conjectures in commutative algebra.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Nowadays, the notion of semi-regular sequences, originally proposed by Fr""oberg, 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 Gr""obner basis of the ideal generated by such a sequence with help of several regularities of the ideal related to Lazard's bound on maximal Gr""{o}bner basis degree and other bounds. We also study the genericness of the property that a sequence is semi-regular, and its variants related to Fr""oberg's conjecture. Moreover, we discuss on the genericness of another important property that the initial ideal is weakly reverse lexicographic, related to Moreno-Soc'{i}as' conjecture, and show some criteria to examine whether both Fr""oberg's conjecture and Moreno-Soc'{i}as' one hold at the same time.

Problem

• Analyzing complexity of Gröbner basis computation for generalized cryptographic semi-regular sequences
• Studying genericness of semi-regular sequences and related conjectures
• Examining criteria for simultaneous validity of Fröberg and Moreno-Socías conjectures

Innovation

• Generalized cryptographic semi-regular sequences analysis
• Maximal Gröbner basis degree complexity study
• Genericness criteria for semi-regular sequences