On Polynomial Multiplication in Chebyshev Basis

In a recent paper, Lima, Panario, and Wang have provided a new method to multiply polynomials expressed in Chebyshev basis which reduces the total number of multiplication for small degree polynomials. Although their method uses Karatsuba´s multiplication, a quadratic number of operations are still needed. In this paper, we extend their result by providing a complete reduction to polynomial multiplication in monomial basis, which therefore offers many subquadratic methods. Our reduction scheme does not rely on basis conversions and we demonstrate that it is efficient in practice. Finally, we show a linear time equivalence between the polynomial multiplication problem under monomial basis and under Chebyshev basis.