Hilbert series and degrees of regularity of Oil & Vinegar and mixed quadratic systems

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2025-05-24 DOI:10.1007/s10623-025-01644-6

Antonio Corbo Esposito, Rosa Fera, Francesco Romeo

引用次数: 0

Abstract

In this paper, we analyze the algebraic invariants for two classes of multivariate quadratic systems: systems made by oil and vinegar quadratic polynomials and systems made by both oil and vinegar polynomials and fully-quadratic ones. For such systems, we explicitly compute the Hilbert series in the homogeneous case, and we also give bounds on the degree of regularity, solving degree and first fall degree. Such degrees can be relevant to compute the complexity of solving those systems and to estimate their cryptographic security.

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油醋混合二次系统的希尔伯特级数和正则度

本文分析了两类多元二次系统的代数不变量：油醋多项式系统和油醋多项式与全二次多项式系统。对于这样的系统，我们显式地计算了齐次情况下的Hilbert级数，并给出了正则度、解度和一阶降度的界。这种程度可能与计算解决这些系统的复杂性和估计其加密安全性相关。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.