Solving a 676-bit discrete logarithm problem in GF(36n)

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 6056 LNCS, Issue , 2010, Pages 351-367

  Hayashi, Takuya ^a   Shinohara, Naoyuki ^b   Wang, Lihua ^b   Matsuo, Shin'Ichiro ^b   Shirase, Masaaki ^c   Takagi, Tsuyoshi ^a  

^a KYUSHU UNIVERSITY   (Japan)

^b NATIONAL INSTITUTE OF INFORMATION AND COMMUNICATIONS TECHNOLOGY   (Japan)

^c FUTURE UNIVERSITY HAKODATE   (Japan)

Author keywords

discrete logarithm problem; function field sieve; pairing based cryptosystems

Indexed keywords

BIT SIZE; CRYPTOGRAPHIC SCHEMES; CRYPTOSYSTEMS; DISCRETE LOGARITHM PROBLEMS; ELLIPTIC CURVES OVER FINITE FIELDS; FUNCTION FIELDS; OKAMOTO; PAIRING-BASED CRYPTOSYSTEMS; PRACTICAL IMPLEMENTATION; SUPERSINGULAR CURVES;

PUBLIC KEY CRYPTOGRAPHY; SIEVES;

ALGEBRA;

EID: 79955530924     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-13013-7_21     Document Type: Conference Paper

References (30)

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)
2. Adleman, L.M., Huang, M.-D.A.: Function field sieve method for discrete logarithms over finite fields. Inform. and Comput. 151, 5-16 (1999) (Pubitemid 129608053)
3. Aoki, K., Shimoyama, T., Ueda, H.: Experiments on the linear algebra step in the number field sieve. In: Miyaji, A., Kikuchi, H., Rannenberg, K. (eds.) IWSEC 2007. LNCS, vol. 4752, pp. 58-73. Springer, Heidelberg (2007)
4. Aoki, K., Ueda, H.: Sieving using bucket sort. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 92-102. Springer, Heidelberg (2004) (Pubitemid 39747748)
5. Barreto, P.S.L.M., Galbraith, S., ÓhÉigeartaigh, C., Scott., M.: Efficient pairing computation on supersingular abelian varieties. Des. Codes Cryptogr. 42(3), 239-271 (2007) (Pubitemid 46384645)
6. Beuchat, J.-L., Brisebarre, N., Detrey, J., Okamoto, E., Shirase, M., Takagi, T.: Algorithms and arithmetic operators for computing the ?T pairing in characteristic three. IEEE Trans. Comput. 57(11), 1454-1468 (2008)
7. Boneh, D., Crescenzo, D., Ostrovsky, R., Persiano, G.: Public key encryption with keyword search. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 506-522. Springer, Heidelberg (2004)
8. Boneh, D., Franklin, M.: Identity based encryption from the Weil pairing. SIAM J. Comput. 32(3), 586-615 (2003)
9. Gordon, D.M.: Discrete logarithms in GF(p) using the number field sieve. SIAM J. Discrete Math. 6(1), 124-138 (1993)
10. 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)
11. Granger, R.: Estimates for discrete logarithm computations in finite fields of small characteristic. In: Paterson, K.G. (ed.) Cryptography and Coding 2003. LNCS, vol. 2898, pp. 190-206. Springer, Heidelberg (2003)
12. Granger, R., Holt, A.J., Page, D., Smart, N.P., Vercauteren, F.: Function field sieve in characteristic three. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 223-234. Springer, Heidelberg (2004)
13. Granger, R., Page, D., Stam, M.: Hardware and software normal basis arithmetic for pairing-based cryptography in characteristic three. IEEE Trans. Comput. 54(7), 852-860 (2005) (Pubitemid 41055318)
14. Hankerson, D., Menezes, A., Scott, M.: Software implementation of pairings. In: Identity Based Cryptography, pp. 188-206 (2009)
15. Joux, A., et al.: Discrete logarithms in GF(2607) and GF(2613). Posting to the Number Theory List (2005), http://listserv.nodak.edu/cgi-bin/wa.exe?A2= ind0509&L=nmbrthry&T=0&P=3690
16. 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)
17. 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. Comp. 72(242), 953-967 (2002) (Pubitemid 36344905)
18. 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) (Pubitemid 44072244)
19. Joux, A., Lercier, R., Naccache, D., Thome, E.: Oracle-assisted static Diffie-Hellman is easier than discrete logarithms. In: Parker, M.G. (ed.) IMACC 2009. LNCS, vol. 5921, pp. 351-367. Springer, Heidelberg (2009)
20. Joux, A., Lercier, R., Smart, N.P., Vercauteren, F.: The number field sieve in the medium prime case. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 326-344. Springer, Heidelberg (2006) (Pubitemid 44532124)
21. Kleinjung, T., et al.: Discrete logarithms in GF(p) - 160 digits. Posting to the Number Theory List (2007), http://listserv.nodak.edu/cgi-bin/wa.exe?A2= ind0702&L=nmbrthry&T=0&P=194
22. 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)
23. Matsumoto, R.: Using Cab curves in the function field sieve. IEICE Trans. Fundamentals E82-A, 551-552 (1999)
24. Menezes, A.J., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inform. Theory 39(5), 1639-1646 (1993)
25. Page, D., Smart, N.P., Vercauteren, F.: A comparison of MNT curves and supersingular curves. Appl. Algebra Engrg. Comm. Comput. 17(5), 379-392 (2006) (Pubitemid 44569382)
26. Pollard, J.: The lattice sieve. The Development of the Number Field Sieve, 43-49 (1991)
27. Pomerance, C., Smith, J.W.: Reduction of huge, sparse matrices over finite fields via created catastrophes. Experiment. Math. 1(2), 89-94 (1992)
28. Schirokauer, O.: The special function field sieve. SIAM J. Discrete Math. 16(1), 81-98 (2003)
29. Wambach, G., Wettig, H.: Block sieving algorithms. Technical Report 190, Informatik, Universität zu Köln (1995)
30. Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory 32(1), 54-62 (1986)