A double large prime variation for small genus hyperelliptic index calculus

Mathematics of Computation

Volumn 76, Issue 257, 2007, Pages 475-492

  Gaudry, P ^a   Thome E ^a   Theriault N ^a   Diem, C ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 38049124867     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-06-01900-4     Document Type: Article

References (27)

1. L. M. Adleman, J. DeMarrais, and M.-D. Huang, A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields, ANTS-I (L. Adleman and M.-D. Huang, eds.), Lecture Notes in Comput. Sci., vol. 877, Springer-Verlag, 1994, pp. 28-40. MR1322708 (96b:11078)
2. A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The design and analysis of computer algorithms, Addison-Wesley, Reading, MA, 1974. MR0413592 (54:1706)
3. D. G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), no. 177, 95-101. MR0866101 (88f:11118)
4. F. Chung and L. Lu, The diameter of sparse random graphs, Adv. in Appl. Math. 26 (2001), 257-279. MR1826308 (2002c:05138)
5. D. Coppersmith, Solving linear equations over GF(2) via block Wiedemann algorithm, Math. Comp. 62 (1994), no. 205, 333-350. MR1192970 (94c:11124)
6. J.-M. Couveignes, Algebraic groups and discrete logarithm, Public-key cryptography and computational number theory, de Gruyter, 2001, pp. 17-27. MR1881624 (2003a:11160)
7. C. Diem, Index calculus algorithm for plane curves of small degree, ANTS-VII (F. Heß, S. Pauli and M. Pohst, eds.), Lecture Notes in Comput. Sci., vol. 4076, Springer-Verlag, 2006, pp. 543-557.
8. A. Enge, Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time, Math. Comp. 71 (2002), 729-742. MR1885624 (2003b:68083)
9. A. Enge and P. Gaudry, A general framework for subexponential discrete logarithm algorithms, Acta Arith. 102 (2002), 83-103. MR1884958 (2002k:11225)
10. P. Flajolet, D. Knuth, and B. Pittel, The first cycles in an evolving graph, Discrete Math. 75 (1989), 167-215. MR1001395 (90d:05184)
11. P. Gaudry, An algorithm for solving the discrete log problem on hyperelliptic curves, Advances in Cryptology, EUROCRYPT 2000 (B. Preneel, ed.), Lecture Notes in Comput. Sci., vol. 1807, Springer-Verlag, 2000, Proc. International Conference on the Theory and Application of Cryptographic Techniques, Brugge, Belgium, May 2000, Proceedings, pp. 19-34. MR1772021
12. _, Index calculus for abelian varieties and the elliptic curve discrete logarithm problem, Cryptology ePrint Archive: Report 2004/073, 2004.
13. F. Heß, Computing Riemann-Roch spaces in algebraic function fields and related topics, J. Symbolic Comput. 33 (2002), 425-445. MR1890579 (2003j:14032)
14. _, Computing relations in divisor class groups of algebraic curves over finite fields, Preprint, 2004.
15. M.-D. Huang and D. Ierardi, Counting points on curves over finite fields, J. Symbolic Comput. 25 (1998), 1-21. MR1600606 (98i:11040)
16. N. Koblitz, Hyperelliptic cryptosystems, J. of Cryptology 1 (1989), 139-150. MR1007215 (90k:11165)
17. A. K. Lenstra and M. S. Manasse, Factoring with two large primes, Math. Comp. 63 (1994), no. 208, 785-798. MR1250773 (95a:11107)
18. P. Leyland, A. K. Lenstra, B. Dodson, A. Muffett, and S. S. Wagstaff, Jr., MPQS with three large primes, ANTS-V (C. Fieker and D. R. Kohel, eds.), Lecture Notes in Comput. Sci., vol. 2369, Springer-Verlag, 2002, 5th Algorithmic Number Theory Symposium, Sydney, Australia, July 2002, Proceedings, pp. 448-462. MR2041103 (2005d:11178)
19. A. Menezes, Y.-H. Wu, and R. Zuccherato, An elementary introduction to hyperelliptic curves, Appendix to Algebraic aspects of cryptography, by N. Koblitz, pages 155-178, Springer-Verlag, 1998. MR1610535 (2000a:94012)
20. V. Müller, A. Stein, and C. Thiel, Computing discrete logarithms in real quadratic congruence function fields of large genus, Math. Comp. 68 (1999), no. 226, 807-822. MR1620235 (99i:11119)
21. J. Pila, Frobenius maps of abelian varieties and finding roots of unity in finite fields, Math. Comp. 55 (1990), no. 192, 745-763. MR1035941 (91a:11071)
22. S. Pohlig and M. Hellman, An improved algorithm for computing logarithms over GF(p) and its cryptographic significance, IEEE Trans. Inform. Theory IT-24 (1978), 106-110. MR0484737 (58:4617)
23. N. Thériault, Index calculus attack for hyperelliptic curves of small genus, Advances in Cryptology, ASIACRYPT 2003 (C. Laih, ed.), Lecture Notes in Comput. Sci., vol. 2894, Springer-Verlag, 2003, Proc. 9th International Conference on the Theory and Applications of Cryptology and Information Security, Nov. 30, Dec. 4, 2003, Taipei, Taiwan, Proceedings, pp. 75-92. MR2093253 (2006d:94068)
24. E. Thomé, Subquadratic computation of vector generating polynomials and improvement of the block Wiedemann algorithm, J. Symbolic Comput. 33 (2002), no. 5, 757-775. MR1919913 (2004b:11172)
25. _, Algorithmes de calcul de logarithme discret dans les corps finis, Thèse, École polytechnique, May, 2003.
26. E. Volcheck, Computing in the Jacobian of a plane algebraic curve, ANTS-I (L. Adleman and M.-D. Huang, eds.), Lecture Notes in Comput. Sci., vol. 877, Springer-Verlag, 1994, pp. 221-233. MR1322725 (96a:14033)
27. T. Wollinger, J. Pelzl, and C. Paar, Cantor versus Harley: Optimization and analysis of explicit formulae for hyperelliptic curve cryptosystem, Tech. report, Universität Bochum, 2004, Available at http://www.crypto.rub.de/.