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).
Similar content being viewed by others
References
Arnaud N.: Évaluations dérivés, multiplication dans les corps finis et codes correcteurs. PhD thesis, Université de la Méditerranée, Institut de Mathématiques de Luminy (2006).
Baker R., Harman G., Pintz J.: The difference between consecutive primes, II. Proc. Lond. Math. Soc. 83(3), 532–562 (2001).
Ballet S.: Curves with many points and multiplication complexity in any extension of \(\mathbb{F}_q\). Finite Fields Their Appl. 5, 364–377 (1999).
Ballet S.: On the tensor rank of the multiplication in the finite fields. J. Number Theory 128, 1795–1806 (2008).
Ballet S., Le Brigand D.: On the existence of non-special divisors of degree \(g\) and \(g-1\) in algebraic function fields over \(\mathbb{F}_q\). J. Number Theory 116, 293–310 (2006).
Ballet S., Pieltant J.: Tower of algebraic function fields with maximal Hasse–Witt invariant and tensor rank of multiplication in any extension of \(\mathbb{F}_2\) and \(\mathbb{F}_3\). J. Pure Appl. Algebra 222(5), 1069–1086 (2018).
Ballet S., Rolland R.: Multiplication algorithm in a finite field and tensor rank of the multiplication. J. Algebra 272(1), 173–185 (2004).
Ballet S., Zykin A.: Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields. In: Proceedings of The Tenth International Workshop on Coding and Cryptography (2017). http://wcc2017.suai.ru/proceedings.html.
Ballet S., Pieltant J., Rambaud M., Sijsling J.: On some bounds for symmetric tensor rank of multiplication in finite fields. Contemp. Math. Am. Math. Soc. 686, 93–121 (2017).
Cenk M., Özbudak F.: On multiplication in finite fields. J. Complex. 26(2), 172–186 (2010).
Chudnovsky D., Chudnovsky G.: Algebraic complexities and algebraic curves over finite fields. J. Complex. 4, 285–316 (1988).
Dudek A.: An explicit result for primes between cubes. Funct. Approx. Commmentarii Math. 55(2), 177–197 (2016).
Randriambololona H.: Divisors of the form 2d-g without sections and bilinear complexity of multiplication in finite fields. ArXiv e-prints (2011).
Randriambololona H.: Bilinear complexity of algebras and the Chudnovsky–Chudnovsky interpolation method. J. Complex. 28(4), 489–517 (2012).
Shparlinski I., Tsfasman M., Vlăduţ S.: Curves with many points and multiplication in finite fields. In: Stichtenoth H., Tsfasman M.A. (eds.) Coding Theory and Algebraic Geometry, vol. 1518, pp. 145–169. Lectures Notes in MathematicsSpringer, Berlin (1992).
Tsfasman M., Vlăduţ S.: Algebraic-Geometric Codes. Kluwer Academic Publishers, Dordrecht (1991).
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’.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-018-0560-8
Keywords
- Algebraic function field
- Tower of function fields
- Tensor rank
- Algorithm
- Finite field
- Modular curve
- Shimura curve