Faster modular composition using two relation matrices

2026-01-24

AI Summary

This work addresses the problem of efficient modular composition of univariate polynomials modulo a third polynomial. By introducing two low-dimensional polynomial relation matrices to replace the traditional single large relation matrix, the method encodes the algebraic dependencies among input polynomials using a more compact structure. Leveraging fast polynomial modular arithmetic and advanced matrix multiplication techniques, the algorithm achieves a computational complexity of Õ(n^{(ω+3)/4}) under standard input assumptions. With the current best-known matrix multiplication exponent (ω ≈ 2.373), this yields a field operation complexity of O(n^{1.343}), significantly improving upon existing approaches.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $\tilde{O}(n^{(\omega+3)/4})$ arithmetic operations in the base field, where $\omega$ is the exponent of matrix multiplication. With naive matrix multiplication, this is $\tilde{O}(n^{3/2})$, while with the best currently known exponent $\omega$ this is $O(n^{1.343})$, improving upon the previously most efficient algorithms.

Problem

• modular composition
• polynomial composition
• algebraic relations
• univariate polynomials
• matrix multiplication

Innovation

• modular composition
• polynomial matrices
• algebraic relations
• matrix multiplication exponent
• algorithmic complexity