# Fast multiplication for skew polynomials

## TL;DR

This paper introduces a new, faster algorithm for multiplying skew polynomials that leverages evaluation and interpolation techniques, achieving optimal asymptotic complexity for large degrees and improving existing methods.

## Contribution

The authors develop a novel algorithm for skew polynomial multiplication based on evaluation and interpolation, improving complexity bounds and extending efficiency to small degrees.

## Key findings

- Achieves optimal asymptotic complexity for large degree skew polynomial multiplication
- Provides an efficient algorithm for small degree skew polynomial multiplication
- Improves complexity bounds for various arithmetic problems involving skew polynomials

## Abstract

We describe an algorithm for fast multiplication of skew polynomials. It is based on fast modular multiplication of such skew polynomials, for which we give an algorithm relying on evaluation and interpolation on normal bases. Our algorithms improve the best known complexity for these problems, and reach the optimal asymptotic complexity bound for large degree. We also give an adaptation of our algorithm for polynomials of small degree. Finally, we use our methods to improve on the best known complexities for various arithmetics problems.