CONSTRUCTION OF ASYMMETRIC CHUDNOVSKY ALGORITHMS WITHOUT DERIVATED EVALUATION FOR MULTIPLICATION IN FINITE FIELDS

Stéphane Ballet (1) , Nicolas Baudru (2) , Alexis Bonnecaze (1) , Mila Tukumuli (1)

1 I2M - Institut de Mathématiques de Marseille
2 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)

The Chudnovsky and Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear with respect to the degree of the extension. Recently, Ran-driambololona has generalized the method, allowing asymmetry in the interpolation procedure and leading to new upper bounds on the bilinear complexity. In this article, we first translate this generalization into the language of algebraic function fields. Then, we propose a strategy to effectively construct asymmetric algorithms using places of higher degrees and without derivated evaluation. Finally, we provide examples of three multiplication algorithms along with their Magma implementation: in F 16 13 using only rational places, in F 4 5 using also places of degree two, and in F 2 5 using also places of degree four.

Mots clés: Multiplication algorithm; bilinear complexity; interpolation on algebraic curve; finite field

Domaines: Géométrie algébrique [math.AG]

Stéphane Ballet, Nicolas Baudru, Alexis Bonnecaze, Mila Tukumuli. CONSTRUCTION OF ASYMMETRIC CHUDNOVSKY ALGORITHMS WITHOUT DERIVATED EVALUATION FOR MULTIPLICATION IN FINITE FIELDS. Designs, Codes and Cryptography, In press. ⟨hal-02115213⟩