Faster modular composition using two relation matrices

TL;DR

This paper introduces a faster algorithm for modular composition of univariate polynomials by utilizing two smaller polynomial matrices, achieving improved complexity under generic conditions.

Contribution

It advances modular composition algorithms by employing two polynomial matrices of smaller dimension, reducing computational complexity compared to prior methods.

Findings

Achieves $ ilde{O}(n^{(\omega+3)/4})$ complexity under generic assumptions.

Improves upon previous algorithms with complexity $O(n^{1.343})$ using the best known matrix multiplication exponent.

Provides a practical approach for faster polynomial composition in algebraic computations.

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.