On Polynomial Multiplication in Chebyshev Basis

TL;DR

This paper introduces a new reduction scheme for polynomial multiplication in Chebyshev basis that leverages algorithms from the monomial basis, achieving the same asymptotic complexity and improved practical performance for large degrees.

Contribution

It extends previous methods by providing a direct reduction scheme that avoids conversions, enabling efficient polynomial multiplication in Chebyshev basis with asymptotic and practical benefits.

Findings

Reduction scheme allows direct use of monomial algorithms in Chebyshev basis

Achieves the same asymptotic complexity as monomial basis multiplication

Outperforms existing algorithms for large degree polynomials

Abstract

In a recent paper Lima, Panario and Wang have provided a new method to multiply polynomials in Chebyshev basis which aims at reducing the total number of multiplication when polynomials have small degree. Their idea is to use Karatsuba's multiplication scheme to improve upon the naive method but without being able to get rid of its quadratic complexity. In this paper, we extend their result by providing a reduction scheme which allows to multiply polynomial in Chebyshev basis by using algorithms from the monomial basis case and therefore get the same asymptotic complexity estimate. Our reduction allows to use any of these algorithms without converting polynomials input to monomial basis which therefore provide a more direct reduction scheme then the one using conversions. We also demonstrate that our reduction is efficient in practice, and even outperform the performance of the best known algorithm for Chebyshev basis when polynomials have large degree. Finally, we demonstrate a linear time equivalence between the polynomial multiplication problem under monomial basis and under Chebyshev basis.