An Algorithm to Solve the Discrete Logarithm Problem with the Number Field Sieve

  • An Commeine
  • Igor Semaev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3958)

Abstract

Recently, Shirokauer’s algorithm to solve the discrete logarithm problem modulo a prime p has been modified by Matyukhin, yielding an algorithm with running time \(L_{p}[\frac{1}{3},1.09018...]\), which is, at the present time, the best known estimate of the complexity of finding discrete logarithms over prime finite fields and which coincides with the best known theoretical running time for factoring integers, obtained by Coppersmith. In this paper, another algorithm to solve the discrete logarithm problem in \(\mathbb{F}^{*}_{p}\) for p prime is presented. The global running time is again \(L_{p}[\frac{1}{3},1.09018...]\), but in contrast with Matyukhins method, this algorithm enables us to calculate individual logarithms in a separate stage in time \(L_{p}[\frac{1}{3},3^{1/3}]\), once a \(L_{p}[\frac{1}{3},1.09018...]\) time costing pre-computation stage has been executed. We describe the algorithm as derived from [6] and estimate its running time to be \(L_{p}[\frac{1}{3},(\frac{64}{9})^{1/3}]\), after which individual logarithms can be calculated in time \(L_{p}[\frac{1}{3},3^{1/3}]\).

Keywords

Discrete Logarithms Number Field Sieve 

References

  1. 1.
    Canfield, E., Erdös, P., Pomerance, C.: On a problem of Oppenheim concerning factorisatio umerorum. J.Number Theory 17, 1–28 (1983)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Coppersmith, D.: Fast Evaluation of Logarithms in Fields of Characteristic Two. IEEE Transactions on Information Theory IT 30, 587–594 (1984)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Coppersmith, D.: Modifications to the Number Field Sieve. J. Cryptology 6, 169–180 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Coppersmith, D., Odlyzko, A., Schroeppel, R.: Discrete logarithms in GF(p). Algorithmica 1, 1–15 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gordon, D.: Discrete logarithms in GF(p) using the number field sieve. SIAM Journal of Discrete Mathematics 6, 124–138 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Joux, A., Lercier, R.: Improvements to the general Number Field Sieve for discrete logarithms in prime fields. Mathematics of Computation 72, 953–967 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Joux, A., Lercier, R.: Calcul de logarithmes discrets dans GF(p) — 130 chiffres. In: CRYPTO Mailing List (June 2005)Google Scholar
  8. 8.
    Lenstra, A., Lenstra, H. (eds.): The Development of the Number Field Sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993)MATHGoogle Scholar
  9. 9.
    Lenstra, H.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Matyukhin, D.: On asymptotic complexity of computing discrete logarithms over GF(p). Discrete Mathematics and Applications 13, 27–50 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    McCurley, K.: The discrete logarithm problem. In: Pomerance, C. (ed.) Cryptography and Computational Number Theory. Proc. Symp.Appl.Math., vol. 42, Amer. Math. Soc (1990)Google Scholar
  12. 12.
    Odlyzko, A.: Discrete logarithms: The past and the future. Designs, Codes and Cryptography 19, 129–145 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Odlyzko, A.: Discrete Logarithms and Smooth Polynomials. In: Mullen, G., Shiue, P. (eds.) Finite Fields: Theory, Applications and Algorithms. Contemporary Math, vol. 168, pp. 269–278. Amer. Math. Soc (1994)Google Scholar
  14. 14.
    Odlyzko, A.M.: Discrete logarithms in finite fields and their cryptographic significance. In: Beth, T., Cot, N., Ingemarsson, I. (eds.) EUROCRYPT 1984. LNCS, vol. 209, pp. 224–314. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  15. 15.
    Odlyzko, A.: On the complexity of Computing Discrete Logarithms and Factoring Integers. In: Cover, T., Gopinath, B. (eds.) Open Problems in Communication and Computation, pp. 113–116. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  16. 16.
    Pollard, J.: Monte Carlo methods for index computations mod p. Mathematics of Computation 32, 918–924 (1978)MathSciNetMATHGoogle Scholar
  17. 17.
    Pollard, J.: Factoring with cubic integers. In: [8], pp. 4–10. Springer, Heidelberg (1993)Google Scholar
  18. 18.
    Pomerance, C.: Fast, rigorous factorization and discrete logarithm algorithms. In: Nozaki, N., Johnson, D., Nishizaki, T., Wilf, H. (eds.) Discrete Algorithms and Complexity, pp. 119–143. Academic Press, London (1987)CrossRefGoogle Scholar
  19. 19.
    Schirokauer, O.: Discrete logarithms and local units. Philosophical Transactions of the Royal Society of London (A) 345, 409–423 (1993)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Schirokauer, O.: Virtual Logarithms. Journal of Algorithms 57, 140–147 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Semaev, I.: Special prime numbers and discrete logs in prime finite fields. Mathematics of Computation 71, 363–377 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Shoup, V.: Searching for primitive roots in finite fields. Mathematics of Computation 58, 918–924 (1992)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    van Oorschot, P., Wiener, M.: Parallel collision search with cryptanalytic applications. J. Cryptology 12, 1–28 (1999)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wiedemann, D.: Solving sparse linear equations over finite fields. IEEE Trans.Inform. Theory 32, 54–62 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • An Commeine
    • 1
  • Igor Semaev
    • 2
  1. 1.Departement Wiskunde, Afdeling AlgebraKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Institutt for informatikk, HIBUniversitetet i BergenBergenNorway

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