Faster modular composition using two relation matrices

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.