# Multiplication in finite fields with Chudnovsky-type algorithms on the projective line

Abstract : We propose a Recursive Polynomial Generic Construction (RPGC) of multiplication algorithms in any finite field $\mathbb{F}_{q^n}$ based on the method of D.V. and G.V. Chudnovsky specialized on the projective line. They are usual polynomial interpolation algorithms in small extensions and the Karatsuba algorithm is seen as a particular case of this construction. Using an explicit family of such algorithms, we show that their bilinear complexity is quasi-linear with respect to the extension degree n, and we give a uniform bound for this complexity. We also prove that the construction of these algorithms is deterministic and can be done in polynomial time. We give an asymptotic bound for the complexity of their construction.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-02911546
Contributor : Alexis Bonnecaze <>
Submitted on : Monday, August 3, 2020 - 9:31:35 PM
Last modification on : Tuesday, August 4, 2020 - 10:15:26 AM

### Identifiers

• HAL Id : hal-02911546, version 1
• ARXIV : 2007.16082

### Citation

Stéphane Ballet, Alexis Bonnecaze, Bastien Pacifico. Multiplication in finite fields with Chudnovsky-type algorithms on the projective line. 2020. ⟨hal-02911546⟩

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