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Estimates for Discrete Logarithm Computations in Finite Fields of Small Characteristic

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Cryptography and Coding (Cryptography and Coding 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2898))

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Abstract

We give estimates for the running-time of the function field sieve (FFS) to compute discrete logarithms in \(\mathbb F{^X_{p^n}}\) for small p. Specifically, we obtain sharp probability estimates that allow us to select optimal parameters in cases of cryptographic interest, without appealing to the heuristics commonly relied upon in an asymptotic analysis. We also give evidence that for any fixed field size some may be weaker than others of a different characteristic or field representation, and compare the relative difficulty of computing discrete logarithms via the FFS in such cases.

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References

  1. Adleman, L.M.: The function field sieve. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 108–121. Springer, Heidelberg (1994)

    Google Scholar 

  2. Adleman, L.M., Huang, M.A.: Function field sieve method for discrete logarithms over finite fields. Inform. and Comput. 151(1-2), 5–16 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bertoni, G., Guajardo, J., Kumar, S., Orlando, G., Paar, C., Wollinger, T.: Efficient GF(p m ) arithmetic architectures for cryptographic applications. In: Joye, M. (ed.) CT-RSA 2003. LNCS, vol. 2612, pp. 158–175. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  4. Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  5. Coppersmith, D.: Evaluating logarithms in GF(2n). In: 16th ACM Symp. Theory of Computing, pp. 201–207 (1984)

    Google Scholar 

  6. Coppersmith, D.: Fast evaluation of logarithms in fields of characteristic two. IEEE Transactions in Information Theory 30(4), 587–594 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Galbraith, S.D.: Supersingular curves in cryptography. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 495–513. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  8. Gao, S., Howell, J.: A general polynomial sieve. Designs, Codes and Cryptography 18, 149–157 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gordon, D.M., McCurley, K.S.: Massively parallel computation of discrete logarithms. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 312–323. Springer, Heidelberg (1993)

    Google Scholar 

  10. 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)

    Chapter  Google Scholar 

  11. Lenstra, A.K., Lenstra, H.W.: The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993)

    Book  MATH  Google Scholar 

  12. Lenstra Jr., H.W.: Finding isomorphisms between finite fields. Mathematics of Computation 56(193), 329–347 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lidl, R., Niederreiter, H.: Finite Fields. Enclyclopedia of Mathematics and its Applications, vol. 20. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  14. Meletiou, G.C.: Explicit form for the discrete logarithm over the field GP(p,k). Archivum Mathematicum (BRNO) 29, 25–28 (1993)

    MATH  MathSciNet  Google Scholar 

  15. 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)

    Chapter  Google Scholar 

  16. Page, D., Smart, N.: Hardware implementation of finite fields of characteristic three. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 529–539. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  17. Pohlig, S., Hellman, M.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Transactions on Information Theory 24, 106–110 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  18. Schirokauer, O.: The special function field sieve. SIAM Journal on Discrete Mathematics 16(1), 81–98 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Thomé, E.: Computation of discrete logarithms in \(\rm \mathbb{F}_{2^{607}}\). In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 107–124. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol, BS8 1UB, UK

    Robert Granger

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  1. Robert Granger
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Editor information

Editors and Affiliations

  1. Information Security Group, Royal Holloway, University of London, TW20 0EX, Egham, Surrey, U.K.

    Kenneth G. Paterson

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Granger, R. (2003). Estimates for Discrete Logarithm Computations in Finite Fields of Small Characteristic. In: Paterson, K.G. (eds) Cryptography and Coding. Cryptography and Coding 2003. Lecture Notes in Computer Science, vol 2898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40974-8_16

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  • DOI: https://doi.org/10.1007/978-3-540-40974-8_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20663-7

  • Online ISBN: 978-3-540-40974-8

  • eBook Packages: Springer Book Archive

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