Faster Modular Composition

TL;DR

This paper introduces a faster Las Vegas algorithm for polynomial composition modulo a third polynomial over any field, breaking the previous exponent barrier and improving computational complexity.

Contribution

The paper presents the first algorithm with an exponent below 1.5 for polynomial composition, surpassing the longstanding 3/2 barrier in algebraic complexity.

Findings

Uses $O(n^{1.43})$ field operations, breaking the 3/2 barrier.

Runs in $n^{5/3+o(1)}$ operations with cubic-time matrix multiplication.

Achieves faster composition over arbitrary fields with randomized approach.

Abstract

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by $n$, the algorithm uses $O(n^{1.43})$ field operations, breaking through the $3/2$ barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require $O(n^{1.63})$ field operations in general, and ${n^{3/2+o(1)}}$ field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in ${n^{5/3+o(1)}}$ operations, while previous ones run in $O(n^2)$ operations. Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation.