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Faster modular composition using two relation matrices

Vincent Neiger, Bruno Salvy, Éric Schost, Gilles Villard

TL;DR

The paper presents a faster modular composition algorithm by leveraging two polynomial relation matrices that encode minimal bases for modules of relations between input polynomials. Under generic input, it constructs a univariate-to-bivariate reduction and a truncated-powers framework to achieve a running time of $\tilde{O}\left(n^{(\omega+3)/4}\right)$, improving upon prior general algorithms and matching the best known matrix-multiplication exponents. The approach integrates minimal-basis computation in Popov form, bivariate modular composition via Kronecker-like substitutions, and simultaneous truncated product techniques, with extensions to bivariate inputs and multipoint evaluation. The work also establishes complexity equivalences between truncated powers, bivariate composition, and characteristic polynomials, highlighting fundamental connections among core subproblems in modular polynomial computation and offering potential Las Vegas variants in future work.

Abstract

Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved Brent and Kung's algorithm by computing and using a polynomial matrix that encodes a certain basis of algebraic relations between the polynomials. This is further improved here by making use of two polynomial matrices of smaller dimension. Under genericity assumptions on the input, this results in an algorithm using $\tilde{O}(n^{(ω+3)/4})$ arithmetic operations in the base field, where $ω$ is the exponent of matrix multiplication. With naive matrix multiplication, this is $\tilde{O}(n^{3/2})$, while with the best currently known exponent $ω$ this is $O(n^{1.343})$, improving upon the previously most efficient algorithms.

Faster modular composition using two relation matrices

TL;DR

The paper presents a faster modular composition algorithm by leveraging two polynomial relation matrices that encode minimal bases for modules of relations between input polynomials. Under generic input, it constructs a univariate-to-bivariate reduction and a truncated-powers framework to achieve a running time of , improving upon prior general algorithms and matching the best known matrix-multiplication exponents. The approach integrates minimal-basis computation in Popov form, bivariate modular composition via Kronecker-like substitutions, and simultaneous truncated product techniques, with extensions to bivariate inputs and multipoint evaluation. The work also establishes complexity equivalences between truncated powers, bivariate composition, and characteristic polynomials, highlighting fundamental connections among core subproblems in modular polynomial computation and offering potential Las Vegas variants in future work.

Abstract

Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved Brent and Kung's algorithm by computing and using a polynomial matrix that encodes a certain basis of algebraic relations between the polynomials. This is further improved here by making use of two polynomial matrices of smaller dimension. Under genericity assumptions on the input, this results in an algorithm using arithmetic operations in the base field, where is the exponent of matrix multiplication. With naive matrix multiplication, this is , while with the best currently known exponent this is , improving upon the previously most efficient algorithms.
Paper Structure (19 sections, 16 theorems, 27 equations, 1 figure)

This paper contains 19 sections, 16 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modules of relations and their bases
  3. Minimal bases
  4. Computing modulo K[x]-relations
  5. Computing modulo K[y]-relations
  6. Bivariate Modular Composition
  7. Algorithm and proof of Prop. \ref{['prop:bivmodcomp']} and Thm. \ref{['theo:bivmodcomp']}.
  8. Multipoint evaluation: proof of Thm. \ref{['theo:multipoint']}
  9. Truncated powers
  10. Simultaneous truncated products
  11. High part of a Euclidean remainder
  12. New TruncatedPowers algorithm
  13. Composition algorithm. Proof of Theorem \ref{['thm:modular_composition']}
  14. Complexity equivalences
  15. Truncated powers vs bivariate composition
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $f$ in $\mathbb{K}[x]$ of degree $n>0$ and $m,d$ in $\mathbb{N}_{>0}$ with $md\in \Theta(n)$, there exists a non-empty Zariski-open subset $\mathcal{V}_{f,d}$ of $\overline{\mathbb{K}}{}^{n}$ such that for $(a_0,\ldots,a_{n-1})$ in $\mathcal{V}_{f,d} \cap \mathbb{K}^n$, $a = \sum_{0 \le i <

Figures (1)

  • Figure 1: Subproblems involved in the new modular composition algorithm. The dotted lines indicate possible alternative paths.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • proposition 1: Minimal basis & reduction
  • corollary 1
  • lemma 1
  • proposition 2: Minimal basis NSSV24
  • lemma 2: Reduction
  • ...and 8 more