Finding Optimal Chudnovsky-Chudnovsky Multiplication Algorithms

Conference paper

DOI: 10.1007/978-3-319-16277-5_3

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9061)
Cite this paper as:
Rambaud M. (2015) Finding Optimal Chudnovsky-Chudnovsky Multiplication Algorithms. In: Koç Ç., Mesnager S., Savaş E. (eds) Arithmetic of Finite Fields. WAIFI 2014. Lecture Notes in Computer Science, vol 9061. Springer, Cham

Abstract

The Chudnovsky-Chudnovsky method provides today’s best known upper bounds on the bilinear complexity of multiplication in large extension of finite fields. It is grounded on interpolation on algebraic curves: we give a theoretical lower threshold for the smallest bounds that one can expect from this method (with exceptions). This threshold appears often reachable: we moreover provide an explicit method for this purpose.

We also provide new bounds for the multiplication in small- algebras over \(\mathbf {F}_2\). Building on these ingredients, we:
  • explain how far elliptic curves can provide upper bounds for the multiplication over \(\mathbf {F}_2\);

  • using these curves, improve the bounds for the multiplication in the NIST-size extensions of \(\mathbf {F}_2\);

  • thus, turning to curves of higher genus, further improve these bounds with the well known family of classical modular curves.

Although illustrated only over \(\mathbf {F}_2\), the techniques introduced apply to all characteristics.

Keywords

Elliptic modular curves Finite field arithmetic Chudnovsky-Chudnovsky interpolation Tensor rank Optimal algorithms 

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Télécom ParisTechParisFrance

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