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

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).

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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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Correspondence to Stéphane Ballet.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Ballet, S., Zykin, A. Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields. Des. Codes Cryptogr. 87, 517–525 (2019). https://doi.org/10.1007/s10623-018-0560-8

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Keywords

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

Mathematics Subject Classification

  • 14Q05
  • 14Q20
  • 11Y16
  • 12Y05