A strategy to optimize the complexity of Chudnovsky-type algorithms over the projective line

Chudnovsky-type algorithms of multiplication in finite fields are well known for their good bilinear complexity. Recently, two advances have been obtained in the study of these algorithms: a strategy to optimize the scalar complexity of the original algorithm and the development of a generic recursive construction over the projective line. The construction of recursive Chudnodvsky-type algorithms over the projective line makes possible an efficient generic strategy to optimize their complexity (number of scalar and bilinear multiplications and additions in the base field). Then, several examples are given. In particular, considering Baum-Shokrollahi’s experiment (1992), this constructive method provides a Chudnovsky-type algorithm of multiplication in F 256 / F 4 \mathbb {F}_{256}/\mathbb {F}_4 with the best known complexity, while being much more efficient than existing optimization methods.