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Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 627–645 | Cite as

Gaps between prime numbers and tensor rank of multiplication in finite fields

  • Hugues RandriamEmail author
Article
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Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field \(\mathbb {F}_{p^2}\) of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal construction of auxiliary divisors for multiplication algorithms by evaluation-interpolation on curves. Most of this material dates back to a 2011 unpublished work of the author, but it still provides the best results on this topic at the present time. Then a few updates are given in order to take recent developments into account, including comparison with a similar work of Ballet and Zykin, generalization to classical bilinear complexity over \(\mathbb {F}_p\), and to short multiplication of polynomials, as well as a discussion of open questions on gaps between prime numbers or more generally values of certain arithmetic functions.

Keywords

Finite fields Bilinear complexity Tensor rank Prime numbers Algebraic curves 

Mathematics Subject Classification

12Y05 (main) 11A41 11T71 14Q05 

Notes

Acknowledgements

H. Randriam was supported by ANR-14-CE25-0015 Project Gardio and ANR-15-CE39-0013 Project Manta.

References

  1. 1.
    Baker R.C., Harman G., Pintz J.: The difference between consecutive primes, II. Proc. London Math. Soc. 83, 532–562 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ballet S.: Curves with many points and multiplication complexity in any extension of \(\mathbb{F}_q\). Finite Fields Appl. 5, 364–377 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ballet S.: Low increasing tower of algebraic function fields and bilinear complexity of multiplication in any extension of \(\mathbb{F}_q\). Finite Fields Appl. 9, 472–478 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ballet S.: On the tensor rank of the multiplication in the finite fields. J. Number Theory 128, 1795–1806 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ballet S., Rolland R.: Multiplication algorithm in a finite field and tensor rank of the multiplication. J. Algebra 272, 173–185 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ballet S., Zykin A.: Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields, preprint (June 2017). arXiv:1706.09139.
  7. 7.
    Brocket R.W., Dobkin D.: On the optimal evaluation of a set of bilinear forms. Lin. Alg. Appl. 19, 624–628 (1978).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bshouty N.: A lower bound for the multiplication of polynomials modulo a polynomial. Inform. Process. Lett. 41, 321–326 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cascudo I.: On asymptotically good strongly multiplicative linear secret sharing. Ph.D. dissertation. University of Oviedo (2010).Google Scholar
  10. 10.
    Cascudo I., Cramer R., Xing C.: Torsion limits and Riemann-Roch systems for function fields and applications. IEEE Trans. Inform. Theory 60, 3871–3888 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cenk M., Özbudak F.: On multiplication in finite fields. J. Complex. 26, 172–186 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chudnovsky D.V., Chudnovsky G.V.: Algebraic complexities and algebraic curves over finite fields. Proc. Natl. Acad. Sci. USA 84, 1739–1743 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chudnovsky D.V., Chudnovsky G.V.: Algebraic complexities and algebraic curves over finite fields. J. Complex. 4, 285–316 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cramer H.: On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2, 23–46 (1936).CrossRefzbMATHGoogle Scholar
  15. 15.
    Dudek A.: An explicit result for primes between cubes. Funct. Approx. Comment. Math. 55, 177–197 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dusart P.: Estimates of some functions over primes without R.H., preprint (February 2010). arXiv:1002.0442.
  17. 17.
    Ford K.: The distribution of totients. Ramanujan J. 2, 67–151 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kadiri H.: Short effective intervals containing primes in arithmetic progressions and the seven cubes problem. Math. Comput. 77, 1733–1748 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lempel A., Seroussi G., Winograd S.: On the complexity of multiplication in finite fields. Theoret. Comput. Sci. 22, 285–296 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lempel A., Winograd S.: A new approach to error-correcting codes. IEEE Trans. Inform. Theory 23, 503–508 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Miyake T.: Modular Forms. Springer, Tokyo (1989).CrossRefzbMATHGoogle Scholar
  22. 22.
    Ramaré O., Saouter Y.: Short effective intervals containing primes. J. Number Theory 98, 10–33 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Randriambololona H.: \((2,1)\)-separating systems beyond the probabilistic bound. Israel J. Math. 195, 171–186 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Randriambololona H.: Diviseurs de la forme \(2D-G\) sans sections et rang de la multiplication dans les corps finis, preprint (March 2011). arXiv:1103.4335.
  25. 25.
    Randriambololona H.: Bilinear complexity of algebras and the Chudnovsky–Chudnovsky interpolation method. J. Complex. 28, 489–517 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Randriambololona H.: “On products and powers of linear codes under componentwise multiplication. In: Algorithmic arithmetic, geometry, and coding theory, Contemporary Mathematics, vol. 637. American Mathematical Society, pp. 3–78 (2015).Google Scholar
  27. 27.
    Schoenfeld L.: Sharper bounds for the Chebyshev functions \(\theta (x)\) and \(\psi (x)\), II. Math. Comput. 30, 337–360 (1976).MathSciNetzbMATHGoogle Scholar
  28. 28.
    Shokrollahi M.A.: Optimal algorithms for multiplication in certain finite fields using elliptic curves. SIAM J. Comput. 21, 1193–1198 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shparlinski I., Tsfasman M., Vladut S.: Curves with many pointsand multiplication in finite fields. In: Stichtenoth H., Tsfasman M.A. (eds.) Coding Theory and Algebraic Geometry (Luminy, 1991). Lecture Notes in Mathematics, vol. 1518, pp. 145–169. Springer, Berlin (1992).CrossRefGoogle Scholar
  30. 30.
    Stichtenoth H.: Algebraic Function Fields and Codes, Universitext. Springer, Berlin (1993).zbMATHGoogle Scholar
  31. 31.
    Tsfasman M.A., Vladut S.G.: Algebraic-Geometric Codes. Kluwer Academic Publishers, Norwell (1991).CrossRefzbMATHGoogle Scholar
  32. 32.
    Winograd S.: Some bilinear forms whose multiplicative complexity depends on the field of constants. Math. Syst. Theory 10, 169–180 (1977).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.TELECOM ParisTechParisFrance

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