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On Hilbert-Poincaré series of affine semi-regular polynomial sequences and related Gröbner bases

Momonari Kudo, Kazuhiro Yokoyama

TL;DR

The paper investigates the complexity of computing Gröbner bases for input polynomials forming affine semi-regular sequences. It develops an explicit formula linking the Hilbert–Poincaré series of the homogenized ideal $igra m{F}^{h}igra$ in $R'=R[y]$ with that of the top-degree ideal $igra m{F}^{ m top}igra$ in $R$, yielding a precise truncation up to degree $D-1$ where $D=d_{ m reg}(igra m{F}^{ m top}igra)$. The results further provide exact degree-based descriptions of reduced GBs for $igra m{F}igra$, $igra m{F}^{h}igra$, and $igra m{F}^{ m top}igra$, along with rigorous proofs underpinning GB computation methods (notably F5-type approaches) in cryptographic settings. By relating Hilbert-series behavior to GB structure, the work both clarifies theoretical complexity and supports practical, provable strategies for solving multivariate polynomial systems arising in cryptography. Overall, it offers a principled framework to assess and bound GB-solution effort for affine semi-regular systems using homogenization and top-degree data.

Abstract

Gröbner bases are nowadays central tools for solving various problems in commutative algebra and algebraic geometry. A typical use of Gröbner bases is the multivariate polynomial system solving, which enables us to construct algebraic attacks against post-quantum cryptographic protocols. Therefore, the determination of the complexity of computing Gröbner bases is very important both in theory and in practice: One of the most important cases is the case where input polynomials compose an (overdetermined) affine semi-regular sequence. The first part of this paper aims to present a survey on Gröbner basis computation and its complexity. In the second part, we shall give an explicit formula on the (truncated) Hilbert-Poincaré series associated to the homogenization of an affine semi-regular sequence. Based on the formula, we also study (reduced) Gröbner bases of the ideals generated by an affine semi-regular sequence and its homogenization. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gröbner bases of the ideal generated by an affine semi-regular sequence.

On Hilbert-Poincaré series of affine semi-regular polynomial sequences and related Gröbner bases

TL;DR

The paper investigates the complexity of computing Gröbner bases for input polynomials forming affine semi-regular sequences. It develops an explicit formula linking the Hilbert–Poincaré series of the homogenized ideal in with that of the top-degree ideal in , yielding a precise truncation up to degree where . The results further provide exact degree-based descriptions of reduced GBs for , , and , along with rigorous proofs underpinning GB computation methods (notably F5-type approaches) in cryptographic settings. By relating Hilbert-series behavior to GB structure, the work both clarifies theoretical complexity and supports practical, provable strategies for solving multivariate polynomial systems arising in cryptography. Overall, it offers a principled framework to assess and bound GB-solution effort for affine semi-regular systems using homogenization and top-degree data.

Abstract

Gröbner bases are nowadays central tools for solving various problems in commutative algebra and algebraic geometry. A typical use of Gröbner bases is the multivariate polynomial system solving, which enables us to construct algebraic attacks against post-quantum cryptographic protocols. Therefore, the determination of the complexity of computing Gröbner bases is very important both in theory and in practice: One of the most important cases is the case where input polynomials compose an (overdetermined) affine semi-regular sequence. The first part of this paper aims to present a survey on Gröbner basis computation and its complexity. In the second part, we shall give an explicit formula on the (truncated) Hilbert-Poincaré series associated to the homogenization of an affine semi-regular sequence. Based on the formula, we also study (reduced) Gröbner bases of the ideals generated by an affine semi-regular sequence and its homogenization. Some of our results are considered to give mathematically rigorous proofs of the correctness of methods for computing Gröbner bases of the ideal generated by an affine semi-regular sequence.
Paper Structure (14 sections, 18 theorems, 56 equations)

This paper contains 14 sections, 18 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koszul complex and its homology
  4. Hilbert–Poincaré series and semi-regular sequences
  5. Cryptographic semi-regular sequences
  6. Quick review on the computation of Gröbner basis
  7. Overview of existing Gröbner basis algorithms
  8. Homogenization of polynomials and monomial orderings
  9. Signature and $F_5$ algorithm
  10. Complexity of the Gröbner basis computation
  11. Hilbert-Poincaré series of affine semi-regular sequence
  12. Application to Gröbner bases computation
  13. Gröbner basis elements of degree less than $D$
  14. Gröbner basis elements of degree not less than $D$

Key Result

theorem 1

With notation as above, assume that $\bm{F}$ is affine cryptographic semi-regular. Then ${\rm HF}_{R'/\langle \bm{F}^h \rangle}(d) = \sum_{i=0}^d{\rm HF}_{R/\langle \bm{F}^{\rm top} \rangle}(i)$ and $(\langle \mathrm{LM}(\langle \bm{F}^h \rangle) \rangle_{R'})_d = (\langle \mathrm{LM}(\langle \bm{F}

Theorems & Definitions (42)

  • theorem 1: Theorem \ref{['thm:main']}, Corollaries \ref{['cor:Dreg']} and \ref{['cor:LM']}
  • definition 1: Trivial syzygies
  • remark 1
  • definition 2: Hilbert–Poincaré series
  • theorem 2: cf. BW
  • definition 3: Semi-regular sequences, Pardue
  • lemma 1
  • proof
  • proposition 1: cf. Pardue
  • definition 4: BFS; see also Diem2
  • ...and 32 more