ECM on graphics cards

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

Volumn 5479 LNCS, Issue , 2009, Pages 483-501

  Bernstein, Daniel J ^a   Chen, Tien Ren ^b   Cheng, Chen Mou ^c   Lange, Tanja ^d   Yang, Bo Yin ^b  

^a 1120 Science and Engineering Offices   (United States)

^b INSTITUTE OF INFORMATION SCIENCE   (Taiwan)

^c NATIONAL TAIWAN UNIVERSITY   (Taiwan)

^d EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Edwards curves.; Elliptic curves; Elliptic curve method of factorization; Factorization; Graphics processing unit; Modular arithmetic

Indexed keywords

EDWARDS CURVES.; ELLIPTIC CURVES; ELLIPTIC-CURVE METHOD OF FACTORIZATION; GRAPHICS PROCESSING UNIT; MODULAR ARITHMETIC;

CRYPTOGRAPHY; ELECTRIC DISCHARGE MACHINING; FACTORIZATION;

GEOMETRY;

EID: 67650679679     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-01001-9_28     Document Type: Conference Paper

References (49)

1. 13th IEEE Symposium on Field-Programmable Custom Computing Machines (FCCM 2005), Napa, CA, USA, April 17-20, 2005. IEEE Computer Society, Los Alamitos (2005); ISBN 0-7695-2445-1. See [44]
2. Aoki, K., Franke, J., Kleinjung, T., Lenstra, A.K., Osvik, D.A.: A Kilobit Special Number Field Sieve Factorization. In: ASIACRYPT 2007 [31], pp. 1-12 (2007) (Cited in §1, §1)
3. Atkin, A.O.L., Morain, F.: Finding suitable curves for the elliptic curve method of factorization. Mathematics of Computation 60, 399-405 (1993); ISSN 0025-5718, MR 93k:11115, http://www.lix.polytechnique.fr/~morain/Articles/ articles.english.html (Cited in §2.2)
4. Bahr, F., Boehm, M., Franke, J., Kleinjung, T.: Subject: rsa200 (2005), http://www.crypto-world.com/announcements/rsa200.txt (Cited in §1)
5. Bahr, F., Franke, J., Kleinjung, T.: Discrete logarithms in GF(p) - 160 digits (2007), http://www.nabble.com/Discrete-logarithms-in-GF(p)--160 -digits-td8810595.html (Cited in §1)
6. Bernstein, D.J.: How to build the 2009.01.23 standard workstation, http://cr.yp.to/hardware/build-20090123.html (Cited in §6)
7. Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: AFRICACRYPT [47], pp. 389-405 (2008), http://eprint.iacr. org/2008/013 (Cited in §2.2)
8. Bernstein, D.J., Birkner, P., Lange, T., Peters, C.: ECM using Edwards curves (2008), http://eprint.iacr.org/2008/016 (Cited in §2, §2.2, §2.2, §2.2)
9. Bernstein, D.J., Lange, T.: Explicit-formulas database (2008), http://hyperelliptic.org/EFD (Cited in §2.2, §5)
10. Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: ASIACRYPT 2007 [31], pp. 29-50 (2007), http://cr.yp.to/papers. html#newelliptic (Cited in §2.2, §2.2)
11. Boneh, D. (ed.): CRYPTO 2003. LNCS, vol.2729. Springer, Heidelberg (2003); ISBN 3-540-40674- 3. See [43]
12. Cavallar, S., Dodson, B., Lenstra, A.K., Leyland, P.C., Lioen, W.M., Montgomery, P.L., Murphy, B., te Riele, H., Zimmermann, P.: Factorization of RSA-140 Using the Number Field Sieve. In: ASIACRYPT 1999 [33], pp. 195-207 (1999) (Cited in §1)
13. Cavallar, S., Dodson, B., Lenstra, A.K., Lioen, W.M., Montgomery, P.L., Murphy, B., te Riele, H., Aardal, K., Gilchrist, J., Guillerm, G., Leyland, P.C., Marchand, J., Morain, F., Muffett, A., Putnam, C., Putnam, C., Zimmermann, P.: Factorization of a 512-Bit RSA Modulus. In: EUROCRYPT 2000 [41], pp. 1-18 (2000) (Cited in §1, §1)
14. Cook, D.L., Ioannidis, J., Keromytis, A.D., Luck, J.: CryptoGraphics: Secret Key Cryptography Using Graphics Cards. In: CT-RSA 2005 [36], pp. 334-350 (2005) (Cited in §3)
15. Cook, D.L., Keromytis, A.D.: CryptoGraphics: Exploiting Graphics Cards For Security. In: Advances in Information Security, vol.20. Springer, Heidelberg (2006); ISBN 978-0- 387-29015-29017 (Cited in §3)
16. Cowie, J., Dodson, B., Elkenbracht-Huizing, R.M., Lenstra, A.K., Montgomery, P.L., Zayer, J.: A World Wide Number Field Sieve Factoring Record: On to 512 Bits. In: ASIACRYPT 1996 [28], pp. 382-394 (1996) (Cited in §1)
17. Dwork, C. (ed.): CRYPTO 2006. LNCS, vol.4117. Springer, Heidelberg (2006); ISBN 3-540- 37432-9. See [27]
18. Edwards, H.M.: A normal form for elliptic curves. Bulletin of the American Mathematical Society 44, 393-422 (2007), http://www.ams.org/bull/2007- 44-03/S0273-0979-07-01153-6/home.html (Cited in §2.2)
19. Franke, J., Kleinjung, T., Paar, C., Pelzl, J., Priplata, C., Stahlke, C.: SHARK: A Realizable Special Hardware Sieving Device for Factoring 1024-Bit Integers. In: CHES 2005 [42], pp. 119-130 (2005) (Cited in §1, §1)
20. Gaj, K., Kwon, S., Baier, P., Kohlbrenner, P., Le, H., Khaleeluddin, M., Bachimanchi, R.: Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware. In: CHES 2006 [23], pp. 119-133 (2006) (Cited in §1)
21. Galbraith, S.D. (ed.): Cryptography and Coding 2007. LNCS, vol.4887. Springer, Heidelberg (2007); ISBN 978-3-540-77271-2. See [38]
22. Geiselmann, W., Shamir, A., Steinwandt, R., Tromer, E.: Scalable Hardware for Sparse Systems of Linear Equations, with Applications to Integer Factorization. In: CHES 2005 [42], pp. 131-146 (2005) (Cited in §1)
23. Goubin, L., Matsui, M. (eds.): CHES 2006. LNCS, vol.4249. Springer, Heidelberg (2006); ISBN 3- 540-46559-6. See [20]
24. Hess, F., Pauli, S., Pohst, M.E. (eds.): ANTS 2006. LNCS, vol.4076. Springer, Heidelberg (2006); ISBN 3- 540-36075-1. See [48]
25. Hisil, H., Wong, K., Carter, G., Dawson, E.: Faster group operations on elliptic curves (2007), http://eprint.iacr.org/2007/441 (Cited in §2.2)
26. Joux, A., Lercier, R.: Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the Gaussian integer method, Mathematics of Computation 72, 953-967 (2003) (Cited in §1)
27. Joux, A., Lercier, R., Smart, N.P., Vercauteren, F.: The Number Field Sieve in the Medium Prime Case. In: CRYPTO 2006 [17], pp. 326-344 (2006) (Cited in §1)
28. Kim, K., Matsumoto, T. (eds.): ASIACRYPT 1996. LNCS, vol.1163. Springer, Heidelberg (1996); ISBN 3-540-61872-4. See [16]
29. Kleinjung, T.: Cofactorisation strategies for the number field sieve and an estimate for the sieving step for factoring 1024-bit integers. In: Proceedings of SHARCS 2006 (2006), http://www.math.uni-bonn.de/people/thor/cof. ps (Cited in §1, §1)
30. Koblitz, N., Menezes, A.: Pairing-Based Cryptography at High Security Levels. In: Coding and Cryptography [45], pp. 13-36 (2005) (Cited in §1) (Pubitemid 43778715)
31. Kurosawa, K. (ed.): ASIACRYPT 2007. LNCS, vol.4833. Springer, Heidelberg (2007); See [2], [10]
32. Laih, C.-S. (ed.): ASIACRYPT 2003. LNCS, vol.2894. Springer, Heidelberg (2003); ISBN 3-540-20592-6. See [35]
33. Lam, K.-Y., Okamoto, E., Xing, C. (eds.): ASIACRYPT 1999. LNCS, vol.1716. Springer, Heidelberg (1999); ISBN 3-540-66666-4. See [12]
34. Lenstra Jr., H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649-673 (1987); ISSN0003-486X,MR89g:11125, http://links.jstor. org/ sici?sici=0003-486X(198711)2:126:32.0.CO;2-V (Cited §1)
35. Lenstra, A.K., Tromer, E., Shamir, A., Kortsmit, W., Dodson, B., Hughes, J., Leyland, P.C.: Factoring Estimates for a 1024-Bit RSA Modulus. In: ASIACRYPT 2003 [32], pp. 55-74 (2003) (Cited in §1)
36. Menezes, A.J. (ed.): CT-RSA 2005. LNCS, vol.3376. Springer, Heidelberg (2005); ISBN 3- 540-24399-2. See [14]
37. Montgomery, P.L.: Modular multiplication without trial division. Mathematics of Computation 44, 519-521 (1985), http://www.jstor.org/pss/2007970 (Cited in §4.1)
38. Moss, A., Page, D., Smart, N.P.: Toward Acceleration of RSA Using 3D Graphics Hardware. In: Cryptography and Coding 2007 [21], pp. 364-383 (2007) (Cited in §3)
39. Oswald, E., Rohatgi, P. (eds.): CHES 2008. LNCS, vol.5154. Springer, Heidelberg (2008); ISBN 978-3-540-85052-6. See [46]
40. Pelzl, J., Simka, M., Kleinjung, T., Franke, J., Priplata, C., Stahlke, C., Drutarovsky, M., Fischer, V., Paar, C.: Area-time efficient hardware architecture for factoring integers with the elliptic curve method. IEE Proceedings on Information Security 152, 67-78 (2005) (Cited in §1)
41. Preneel, B. (ed.): EUROCRYPT 2000. LNCS, vol.1807. Springer, Heidelberg (2000); ISBN 3-540-67517-5. See [13]
42. Rao, J.R., Sunar, B. (eds.): CHES 2005. LNCS, vol.3659. Springer, Heidelberg (2005); ISBN 3-540-28474-5. See [19], [22]
43. Shamir, A., Tromer, E.: Factoring Large Numbers with the TWIRL Device. In: CRYPTO 2003 [11], pp. 1-26 (2003) (Cited in §1)
44. Simka, M., Pelzl, J., Kleinjung, T., Franke, J., Priplata, C., Stahlke, C., Drutarovsky, M., Fischer, V.: Hardware Factorization Based on Elliptic Curve Method. In: FCCM 2005 [1], pp. 107-116 (2005) (Cited in §1)
45. Smart, N.P. (ed.): Cryptography and Coding 2005. LNCS, vol.3796. Springer, Heidelberg (2005); See [30]
46. Szerwinski, R., Güneysu, T.: Exploiting the Power of GPUs for Asymmetric Cryptography. In: CHES 2008 [39], pp. 79-99 (2008) (Cited in §3.1, §6, §2)
47. Vaudenay, S. (ed.): AFRICACRYPT 2008. LNCS, vol.5023. Springer, Heidelberg (2008); ISBN 978-3- 540-68159-5. See [7]
48. Zimmermann, P., Dodson, B.: 20 Years of ECM. In: ANTS 2006 [24], pp. 525-542 (2006) (Cited in §2)
49. Zimmermann, P.: 50 largest factors found by ECM, http://www.loria.fr/ ~zimmerma/records/top50.html (Cited in §1)