Faster polynomial multiplication over finite fields

TL;DR

This paper introduces a new complexity bound for polynomial multiplication over finite fields, achieving faster algorithms by extending F"urer-type techniques to F_p[X], especially for large degrees.

Contribution

It establishes the first F"urer-type complexity bound for polynomial multiplication over finite fields, improving previous bounds significantly.

Findings

New bound: M_p(n) = O(n log n 8^(log^* n) log p)

First F"urer-type complexity result for F_p[X]

Improves previous complexity bounds for large n

Abstract

Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two polynomials in F_p[X] of degree less than n. For n large compared to p, we establish the bound M_p(n) = O(n log n 8^(log^* n) log p), where log^* is the iterated logarithm. This is the first known F\"urer-type complexity bound for F_p[X], and improves on the previously best known bound M_p(n) = O(n log n log log n log p).