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Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 517–525 | Cite as

Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields

  • Stéphane BalletEmail author
  • Alexey Zykin
Article
  • 66 Downloads
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field \(\mathbb {F}_p\) or \(\mathbb {F}_{p^2}\) where p denotes a prime number \(\ge 5\). In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on sufficiently dense families of modular curves defined over \(\mathbb {F}_{p^2}\) attaining the Drinfeld–Vladuts bound and on the descent of these families to the definition field \(\mathbb {F}_p\). These families are obtained thanks to prime number density theorems of type Hoheisel, in particular a result due to Dudek (Funct Approx Commmentarii Math, 55(2):177–197, 2016).

Keywords

Algebraic function field Tower of function fields Tensor rank Algorithm Finite field Modular curve Shimura curve 

Mathematics Subject Classification

14Q05 14Q20 11Y16 12Y05 

Notes

Acknowledgements

The first author wishes to thank Sary Drappeau, Olivier Ramaré, Hugues Randriambololona, Joël Rivat and Serge Vladuts for valuable discussions. The second author, tragically deceased in April 2017, was partially supported by ANR Globes ANR-12-JS01-0007-01 and by the Russian Academic Excellence Project ‘5-100’.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aix Marseille Univ, CNRS, Centrale Marseille, I2MMarseilleFrance
  2. 2.Institut de Mathématiques de MarseilleMarseille Cedex 9France
  3. 3.Laboratoire GAATIUniversité de la Polynésie françaiseFaa’aFrench Polynesia
  4. 4.National Research University Higher School of Economics, AG Laboratory NRU HSEMoscowRussia
  5. 5.Institute for Information Transmission Problems of the Russian Academy of SciencesMoscowRussia

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