On the Function Field Sieve and the Impact of Higher Splitting Probabilities

Application to Discrete Logarithms in \(\mathbb{F}_{2^{1971}}\) and \(\mathbb{F}_{2^{3164}}\)
  • Faruk Göloğlu
  • Robert Granger
  • Gary McGuire
  • Jens Zumbrägel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8043)

Abstract

In this paper we propose a binary field variant of the Joux-Lercier medium-sized Function Field Sieve, which results not only in complexities as low as \(L_{q^n}(1/3,(4/9)^{1/3})\) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one and two elements when the field has a subfield of an appropriate size. To illustrate the efficiency of the method, we have successfully solved the DLP in the finite fields with 21971 and 23164 elements, setting a record for binary fields.

Keywords

Discrete logarithm problem function field sieve 

References

  1. 1.
    Adleman, L.M., Huang, M.D.A.: Function field sieve method for discrete logarithms over finite fields. Inform. and Comput. 151(1-2), 5–16 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bailey, D.V., Paar, C., Sarkozy, G., Hofri, M.: Computation in optimal extension fields. In: Conference on The Mathematics of Public Key Cryptography, The Fields Institute for Research in the Mathematical Sciences, pp. 12–17 (2000)Google Scholar
  3. 3.
    Bluher, A.W.: On xq + 1 + ax + b. Finite Fields and Their Applications 10(3), 285–305 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chung, F.R.K.: Diameters and eigenvalues. J. Amer. Math. Soc. 2(2), 187–196 (1989)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Coppersmith, D.: Fast evaluation of logarithms in fields of characteristic two. IEEE Trans. Inform. Theory 30(4), 587–593 (1984)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Faugère, J.-C., Otmani, A., Perret, L., Tillich, J.-P.: Algebraic cryptanalysis of mcEliece variants with compact keys. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 279–298. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Faugère, J.-C., Perret, L., Petit, C., Renault, G.: Improving the complexity of index calculus algorithms in elliptic curves over binary fields. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 27–44. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Gaudry, P., Hess, F., Smart, N.P.: Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptology 15(1), 19–46 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Göloğlu, F., Granger, R., McGuire, G., Zumbrägel, J.: Discrete Logarithms in GF(21971). NMBRTHRY list (February 19, 2013)Google Scholar
  10. 10.
    Granger, R., Vercauteren, F.: On the discrete logarithm problem on algebraic tori. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 66–85. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Granlund, T.: The GMP development team: GNU MP: The GNU Multiple Precision Arithmetic Library, 5.0.5 edn (2012), http://gmplib.org/
  12. 12.
    Helleseth, T., Kholosha, A.: \(x^{{2^l}+1}+x+a\) and related affine polynomials over GF(2k). Cryptogr. Commun. 2(1), 85–109 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Joux, A.: Faster index calculus for the medium prime case application to 1175-bit and 1425-bit finite fields. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 177–193. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. 14.
    Joux, A., Lercier, R.: The function field sieve is quite special. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 431–445. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Joux, A., Lercier, R.: Improvements to the general number field sieve for discrete logarithms in prime fields: a comparison with the gaussian integer method. Math. Comput. 72(242), 953–967 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Joux, A., Lercier, R.: The function field sieve in the medium prime case. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 254–270. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Joux, A., Lercier, R., Naccache, D., Thomé, E.: Oracle-assisted static diffie-hellman is easier than discrete logarithms. In: Parker, M.G. (ed.) Cryptography and Coding 2009. LNCS, vol. 5921, pp. 351–367. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    LaMacchia, B.A., Odlyzko, A.M.: Solving large sparse linear systems over finite fields. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 109–133. Springer, Heidelberg (1991)Google Scholar
  19. 19.
    Lenstra, A.K., Lenstra Jr., H.W.: The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993)CrossRefMATHGoogle Scholar
  20. 20.
    Lenstra Jr., H.W.: Finding isomorphisms between finite fields. Math. Comp. 56(193), 329–347 (1991)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Misoczki, R., Barreto, P.S.L.M.: Compact McEliece keys from Goppa codes. In: Jacobson Jr., M.J., Rijmen, V., Safavi-Naini, R. (eds.) SAC 2009. LNCS, vol. 5867, pp. 376–392. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Pollard, J.M.: Monte carlo methods for index computation (mod p). Math. Comp. 32(143), 918–924 (1978)MathSciNetMATHGoogle Scholar
  23. 23.
    Rubin, K., Silverberg, A.: Torus-based cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  24. 24.
    Shoup, V.: NTL: A library for doing number theory, 5.5.2 edn. (2009), http://www.shoup.net/ntl/
  25. 25.
    Wagstaff, S., et al.: The Cunningham Project, http://homes.cerias.purdue.edu/~ssw/cun/index.html
  26. 26.
    Wan, D.: Generators and irreducible polynomials over finite fields. Math. Comp. 66(219), 1195–1212 (1997)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2013

Authors and Affiliations

  • Faruk Göloğlu
    • 1
  • Robert Granger
    • 1
  • Gary McGuire
    • 1
  • Jens Zumbrägel
    • 1
  1. 1.Complex & Adaptive Systems Laboratory and, School of Mathematical SciencesUniversity College DublinIreland

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