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Traps to the BGJT-algorithm for discrete logarithms

  • Qi Cheng (a1), Daqing Wan (a2) and Jincheng Zhuang (a3)

Abstract

In the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thomé, a quasi-polynomial time algorithm is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular when the field is not a Kummer extension. We propose a fix to the algorithm in non-Kummer cases, without altering the heuristic quasi-polynomial time complexity. Further study is required in order to fully understand the effectiveness of the new approach.

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References

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1.Adleman, L. M., ‘A subexponential algorithm for the discrete logarithm problem with applications to cryptography’, Proc. 20th IEEE Symp. on Foundations of Comp. Science (IEEE, 1979) 5560.
2.Adleman, L. M., ‘The function field sieve’, Algorithmic number theory, Lecture Notes in Computer Science 877 (eds Adleman, L. M. and Huang, M. D. A.; Springer, 1994) 108121.
3.Barbulescu, R., Gaudry, P., Joux, A. and Thomé, E., ‘A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic’, Cryptology ePrint Archive, Report 2013/400, 2013.
4.Coppersmith, D., ‘Fast evaluation of logarithms in fields of characteristic two’, IEEE Trans. Inform. Theory 30 (1984) no. 4, 587594.
5.Diffie, W. and Hellman, M. E., ‘New directions in cryptography’, IEEE Trans. Inform. Theory 6 (1976) 644654.
6.ElGamal, T., ‘A public key cryptosystem and a signature scheme based on discrete logarithms’, IEEE Trans. Inform. Theory 33 (1985) 469472.
7.Enge, A., ‘A general framework for subexponential discrete logarithm algorithms in groups of unknown order’, Finite geometries, Developments in Mathematics 3 (eds Blokhuis, A., Hirschfeld, J. W. P., Jungnickel, D. and Thas, J. A.; Kluwer, 2001) 133146.
8.Göloglu, F., Granger, R., McGuire, G. and Zumbrägel, J., ‘On the function field sieve and the impact of higher splitting probabilities’, Advances in cryptology – CRYPTO 2013, Lecture Notes in Computer Science 8043 (eds Canetti, R. and Garay, J. A.; Springer, 2013) 109128.
9.Gordon, D. M., ‘Discrete logarithms in GF(p) using the number field sieve’, SIAM J. Discrete Math. 6 (1993) no. 1, 124138.
10.Huang, M.-D. and Narayanan, A. K., ‘Finding primitive elements in finite fields of small characteristic’, CoRR (2013) Preprint, 2013, arXiv:1304.1206 [cs.DM].
11.Joux, A., ‘Faster index calculus for the medium prime case application to 1175-bit and 1425-bit finite fields’, Advances in cryptology – EUROCRYPT 2013, Lecture Notes in Computer Science 7881 (eds Johansson, T. and Nguyen, P. Q.; Springer, 2013) 177193.
12.Joux, A., ‘A new index calculus algorithm with complexity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{L}(1/4+o(1))$ in very small characteristic’, Cryptology ePrint Archive, Report 2013/095, 2013.
13.Joux, A. and Lercier, R., ‘The function field sieve in the medium prime case’, Advances in cryptology – EUROCRYPT 2006, Lecture Notes in Computer Science 4004 (ed. Vaudenay, S.; Springer, 2006) 254270.
14.Joux, A., Lercier, R., Smart, N. and Vercauteren, F., ‘The number field sieve in the medium prime case’, Advances in cryptology – CRYPTO 2006, Lecture Notes in Computer Science 4117 (Springer, 2006) 326344.
15.Merkle, R., ‘Secrecy, authentication, and public key systems’, PhD Thesis, Stanford University, 1979.
16.Panario, D., Gourdon, X. and Flajolet, P., ‘An analytic approach to smooth polynominals over finite fields’, Algorithmic number theory, Lecture Notes in Computer Science 1423 (ed. Buhler, J.; Springer, 1998) 226236.
17.Pollard, J., ‘Monte Carlo methods for index computations (mod p)’, Math. Comp. 32 (1978) no. 143, 918924.
18.Wan, D., ‘Generators and irreducible polynomials over finite fields’, Math. Comp. 66 (1997) no. 219, 11951212.
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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
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