Four-dimensional Gallant-Lambert-Vanstone scalar multiplication

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

Volumn 7658 LNCS, Issue , 2012, Pages 718-739

  Longa, Patrick ^a   Sica, Francesco ^b  

^a MICROSOFT RESEARCH   (United States)

^b NAZARBAYEV UNIVERSITY   (Kazakhstan)

Author keywords

Elliptic curves; GLV GLS method; Multicore computation; Scalar multiplication; Side channel protection; Twisted Edwards curve

Indexed keywords

ELLIPTIC CURVE; GLV-GLS METHOD; MULTI CORE; SCALAR MULTIPLICATION; SIDE-CHANNEL; TWISTED EDWARDS CURVE;

CRYPTOGRAPHY; SECURITY OF DATA;

CURVE FITTING;

EID: 84871584771     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-34961-4_43     Document Type: Conference Paper

References (28)

1. Aranha, D.F., Faz-Hernández, A., López, J., Rodríguez-Henríquez, F.: Faster Implementation of Scalar Multiplication on Koblitz Curves. In: Hevia, A., Neven, G. (eds.) LatinCrypt 2012. LNCS, vol. 7533, pp. 177-193. Springer, Heidelberg (2012)
2. Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389-405. Springer, Heidelberg (2008)
3. Bernstein, D.J., Duif, N., Lange, T., Schwabe, P., Yang, B.-Y.: High-Speed High-Security Signatures. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 124-142. Springer, Heidelberg (2011)
4. Brumley, D., Boneh, D.: Remote timing attacks are practical. In: Mangard, S., Standaert, F.-X. (eds.) Proceedings of the 12th USENIX Security Symposium. LNCS, vol. 6225, pp. 80-94. Springer (2003)
5. Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer (1996)
6. h = P. Giornale di Mathematiche di Battaglini 46, 33-90 (1908)
7. Edwards, H.: A normal form for elliptic curves. Bulletin of the American Mathematical Society 44, 393-422 (2007)
8. Galbraith, S.D., Lin, X., Scott, M.: Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 518-535. Springer, Heidelberg (2009)
9. Galbraith, S.D., Lin, X., Scott, M.: Endomorphisms for faster elliptic curve cryptography on a large class of curves. J. Cryptology 24(3), 446-469 (2011)
10. Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190-200. Springer, Heidelberg (2001)
11. Hamburg, M.: Fast and compact elliptic-curve cryptography. Cryptology ePrint Archive, Report 2012/309 (2012), http://eprint.iacr.org/2012/309
12. Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Twisted Edwards Curves Revisited. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 326-343. Springer, Heidelberg (2008)
13. Hu, Z., Longa, P., Xu, M.: Implementing 4-dimensional GLV method on GLS elliptic curves with j-invariant 0. Designs, Codes and Cryptography 63(3), 331-343 (2012); also in Cryptology ePrint Archive, Report 2011/315, http://eprint.iacr.org/2011/315
14. Joye, M., Tunstall, M.: Exponent Recoding and Regular Exponentiation Algorithms. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 334-349. Springer, Heidelberg (2009)
15. Kasper, E.: Fast elliptic curve cryptography in openssl. In: 2nd Workshop on Real-Life Cryptographic Protocols and Standardization (2011)
16. Kocher, P.C.: Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104-113. Springer, Heidelberg (1996)
17. Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Ann. 261, 513-534 (1982)
18. Longa, P.: High-speed elliptic curve and pairing-based cryptography. PhD thesis, University of Waterloo (2011), http://hdl.handle.net/10012/5857
19. Longa, P., Gebotys, C.: Efficient Techniques for High-Speed Elliptic Curve Cryptography. In: Mangard, S., Standaert, F.-X. (eds.) CHES 2010. LNCS, vol. 6225, pp. 80-94. Springer, Heidelberg (2010)
20. Longa, P., Miri, A.: New Composite Operations and Precomputation Scheme for Elliptic Curve Cryptosystems over Prime Fields. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 229-247. Springer, Heidelberg (2008)
21. Longa, P., Sica, F.: Four-dimensional Gallant-Lambert-Vanstone scalar multiplication (full version). Cryptology ePrint Archive, Report 2011/608 (2012), http://eprint.iacr.org/2011/608
22. Morain, F.: Courbes elliptiques et tests de primalité. PhD thesis, Université de Lyon I (1990), http://www.lix.polytechnique.fr/Articles/ english.html; Chapter 2: On Cornacchia's algorithm (joint with J-L. Nicolas)
23. Nguyên, P.Q., Stehlé, D.: Low-Dimensional Lattice Basis Reduction Revisited. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 338-357. Springer, Heidelberg (2004)
24. Okeya, K., Takagi, T.: The Width-w NAF Method Provides Small Memory and Fast Elliptic Scalar Multiplications Secure against Side Channel Attacks. In: Joye, M. (ed.) CT-RSA 2003. LNCS, vol. 2612, pp. 328-342. Springer, Heidelberg (2003)
25. Sica, F., Ciet, M., Quisquater, J.-J.: Analysis of the Gallant-Lambert-Vanstone Method Based on Efficient Endomorphisms: Elliptic and Hyperelliptic Curves. In: Nyberg, K., Heys, H. (eds.) SAC 2002. LNCS, vol. 2595, pp. 21-36. Springer, Heidelberg (2003)
26. Stark, H.M.: Class-numbers of Complex Quadratic Fields. In: Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972). Lecture Notes in Mathematics, vol. 320, pp. 153-174. Springer, Berlin (1973)
27. Taverne, J., Faz-Hernández, A., Aranha, D.F., Rodríguez- Henríquez, F., Hankerson, D., López, J.: Software Implementation of Binary Elliptic Curves: Impact of the Carry-Less Multiplier on Scalar Multiplication. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 108-123. Springer, Heidelberg (2011)
28. Zhou, Z., Hu, Z., Xu, M., Song, W.: Efficient 3-dimensional GLV method for faster point multiplication on some GLS elliptic curves. Inf. Proc. Lett. 77(262), 1075-1104 (2010)