Speeding scalar multiplication over binary elliptic curves using the new carry-less multiplication instruction

Journal of Cryptographic Engineering

Volumn 1, Issue 3, 2011, Pages 187-199

  Taverne, Jonathan ^a   Faz Hernández, Armando ^b   Aranha, Diego F ^c   Rodríguez Henríquez, Francisco ^b   Hankerson, Darrel ^d   López, Julio ^c  

^a UNIVERSITÉ LYON 1   (France)

^b DEPARTAMENTO DE FÍSICA   (Mexico)

^c UNIVERSITY OF CAMPINAS   (Brazil)

^d AUBURN UNIVERSITY   (United States)

Author keywords

Efficient software implementation; Elliptic curve cryptography; Finite field arithmetic; Parallel algorithm

Indexed keywords

PARALLEL ALGORITHMS; PARALLEL PROCESSING SYSTEMS; PUBLIC KEY CRYPTOGRAPHY;

ELLIPTIC CURVE; ELLIPTIC CURVE CRYPTOGRAPHY; FINITE FIELD ARITHMETIC; MICRO ARCHITECTURES; PARALLELIZATION STRATEGIES; SCALAR MULTIPLICATION; SOFTWARE IMPLEMENTATION; STATE-OF-THE-ART PERFORMANCE;

GEOMETRY;

EID: 84857705252     PISSN: 21908508     EISSN: 21908516     Source Type: Journal    
DOI: 10.1007/s13389-011-0017-8     Document Type: Article

References (44)

1. Ahmadi O., Hankerson D., Rodríguez-Henríquez F.: Parallel formulations of scalar multiplication on Koblitz curves. J. UCS 14(3), 481-504 (2008).
2. Aranha, D. F., López, J., Hankerson, D.: Efficient software implementation of binary field arithmetic using vector instruction sets. In: Abdalla, M., Barreto, P. S. L. M. (eds.) The First International Conference on Cryptology and Information Security (LATINCRYPT 2010). Lecture Notes in Computer Science, vol. 6212, pp. 144-161 (2010).
3. Avanzi, R. M.: Another look at square roots (and other less common operations) in fields of even characteristic. In: Adams, C. M., Miri, A., Wiener, M. J. (eds.) 14th International Workshop on Selected Areas in Cryptography (SAC 2007). Lecture Notes in Computer Science, vol. 4876, pp. 138-154. Springer (2007).
4. Bellare, M. (ed.): Advances in Cryptology-CRYPTO 2000. Lecture Notes in Computer Science, vol. 1880. Springer (2000).
5. Bernstein, D., Lange, T.: Analysis and optimization of elliptic-curve single-scalar multiplication. In: Proceedings 8th International Conference on Finite Fields and Applications (Fq8), vol. 461, pp. 1-20. AMS (2008).
6. Bernstein, D. J.: Batch Binary Edwards. In: Halevi, S. (ed.) Advances in Cryptology-CRYPTO 2009. Lecture Notes in Computer Science, vol. 5677, pp. 317-336. Springer (2009).
7. Bernstein, D. J., Lange, T. (eds.) eBACS: ECRYPT Benchmarking of Cryptographic Systems. http://bench. cr. yp. to. Accessed 25 Aug 2011.
8. Beuchat, J.-L., Díaz, J., Mitsunari, S., Okamoto, E., Rodríguez-Henríquez, F., Teruya, T.: High-speed software implementation of the optimal ate pairing over Barreto-Naehrig curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing-Based Cryptography-Pairing 2010. Lecture Notes in Computer Science, vol. 6487, pp. 21-39 (2010).
9. Blake I. F., Murty V. K., Xu G.: A note on window τ-NAF algorithm. Inf. Process. Lett. 95(5), 496-502 (2005).
10. Bodrato, M.: Towards optimal Toom-Cook multiplication for univariate and multivariate polynomials in characteristic 2 and 0. In: Carlet, C., Sunar, B. (eds.) Arithmetic of Finite Fields (WAIFI 2007). Lecture Notes in Computer Science, vol. 4547, pp. 116-133. Springer (2007).
11. Bos, J. W., Kleinjung, T., Niederhagen, R., Schwabe, P.: ECC2K-130 on Cell CPUs. In: Bernstein, D. J., Lange, T. (eds.) 3rd International Conference on Cryptology in Africa (AFRICACRYPT 2010). Lecture Notes in Computer Science, vol. 6055, pp. 225-242. Springer (2010).
12. Comba P. G.: Exponentiation cryptosystems on the IBM PC. IBM Syst. J. 29(4), 526-538 (1990).
13. Dahmen, E., Okeya, K., Schepers, D.: Affine precomputation with sole inversion in elliptic curve cryptography. In: Pieprzyk, J., Ghodosi, H., Dawson, E. (eds.) Information Security and Privacy (ACISP 2007). Lecture Notes in Computer Science, vol. 4586, pp. 245-258. Springer (2007).
14. Firasta, N., Buxton, M., Jinbo, P., Nasri, K., Kuo, S.: Intel AVX: new frontiers in performance improvement and energy efficiency. White paper. http://software. intel. com/.
15. Fog, A.: Instruction tables: list of instruction latencies, throughputs and micro-operation breakdowns for Intel, AMD and VIA CPUs. http://www. agner. org/optimize/instruction_tables. pdf. Accessed 01 Mar 2011.
16. Fong K., Hankerson D., López J., Menezes A.: Field inversion and point halving revisited. IEEE Trans. Comput. 53(8), 1047-1059 (2004).
17. Grabher, P., Großschädl, J., Page, D.: On software parallel implementation of cryptographic pairings. Cryptology ePrint Archive, Report 2008/205. http://eprint. iacr. org/ (2008).
18. Guajardo J., Paar C.: Itoh-Tsujii inversion in standard basis and its application in cryptography and codes. Des. Codes Cryptogr. 25(2), 207-216 (2002).
19. Gueron, S.: Intel Advanced Encryption Standard (AES) Instructions Set. White paper. http://software. intel. com/.
20. Gueron, S., Kounavis, M. E.: Carry-less multiplication and its usage for computing the GCM mode. White paper. http://software. intel. com/.
21. Hankerson D., Menezes A. J., Vanstone S.: Guide to Elliptic Curve Cryptography. Springer, Secaucus (2004).
22. Hu, Z., Longa P., Xu, M.: Implementing 4-dimensional GLV method on GLS elliptic curves with j-invariant 0. Des. Codes Cryptogr. (to appear).
23. Intel.: Intel SSE4 Programming Reference. Technical Report. http://software. intel. com/.
24. m) using normal bases. Inf. Comput. 78(3), 171-177 (1988).
25. Järvinen, K., Optimized FPGA-based elliptic curve cryptography processor for high-speed applications. Integr. VLSI J. (to appear).
26. Järvinen K. U., Skyttä J.: On parallelization of high-speed processors for elliptic curve cryptography. IEEE Trans. VLSI Syst. 16(9), 1162-1175 (2008).
27. Järvinen K. U., Skyttä J.: Fast point multiplication on Koblitz curves: Parallelization method and implementations. Microprocess. Microsyst. Embedded Hardware Des. 33(2), 106-116 (2009).
28. Karatsuba, A., Ofman, Y.: Multiplication of many-digital numbers by automatic computers. Doklady Akad. Nauk SSSR 145, 293-294 (1962). Translation in Physics-Doklady 7, 595-596 (1963).
29. Kim, K. H., Kim, S. I.: A new method for speeding up arithmetic on elliptic curves over binary fields. Cryptology ePrint Archive, Report 2007/181. http://eprint. iacr. org/ (2007).
30. King, B. Rubin, B.: Improvements to the point halving algorithm. In: Wang, H., Pieprzyk, J., Varadharajan, V. (eds.) 9th Australasian Conference on Information Security and Privacy (ACISP 2004). Lecture Notes in Computer Science, vol. 3108, pp. 262-276. Springer (2004).
31. Knudsen, E.: Elliptic scalar multiplication using point halving. In: Lam, K., Okamoto, E. (eds.) Advances in Cryptology-ASIACRYPT '99. Lecture Notes in Computer Science, vol. 1716, pp. 135-149. Springer (1999).
32. Koblitz, N.: CM-curves with good cryptographic properties. In: Feigenbaum, J. (ed.) Advances in Cryptology-CRYPTO '91. Lecture Notes in Computer Science, vol. 576, pp. 279-287. Springer (1992).
33. Lomont, C.: Introduction to Intel advanced vector extensions. Intel Software Network. http://software. intel. com/file/37205 (2011).
34. Longa, P., Gebotys, C. H.: Efficient techniques for high-speed elliptic curve cryptography. In: Mangard, S., Standaert, F.-X. (eds.) Cryptographic Hardware and Embedded Systems (CHES 2010). Lecture Notes in Computer Science, vol. 6225, pp. 80-94. Springer (2010).
35. m)without precomputation. In: Koç, Ç. K., Paar, C. (eds.) First International Workshop on Cryptographic Hardware and Embedded Systems (CHES 99). Lecture Notes in Computer Science, vol. 1717, pp. 316-327. Springer (1999).
36. m). In: Roy, B. K., Okamoto, E. (eds.) 1st International Conference in Cryptology in India (INDOCRYPT 2000). Lecture Notes in Computer Science, vol. 1977, pp. 203-212. Springer (2000).
37. Lutz, J., Hasan, M. A.: High performance FPGA based elliptic curve cryptographic co-processor. In: International Conference on Information Technology: Coding and Computing (ITCC'04), vol. 2, pp. 486-492. IEEE Computer Society (2004).
38. Montgomery P. L.: Five, six, and seven-term Karatsuba-like formulae. IEEE Trans. Comput. 54(3), 362-369 (2005).
39. National Institute of Standards and Technology (NIST).: Recommended Elliptic Curves for Federal Government Use. NIST Special Publication. http://csrc. nist. gov/csrc/fedstandards. html. Accessed July 1999.
40. Schroeppel, R.: Elliptic curves: Twice as fast! Presentation at the CRYPTO 2000 [4] Rump Session (2000).
41. Solinas J. A.: Efficient arithmetic on Koblitz curves. Des. Codes Cryptogr. 19(2-3), 195-249 (2000).
42. 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: International Workshop on Cryptographic Hardware and Embedded Systems (CHES 2011). Lecture Notes in Computer Science, vol. 6917, Springer, New York (2011).
43. Wall, D. W.: Limits of instruction-level parallelism. In: 4th International Conference on Architectural Support for Programming Languages and Operating System (ASPLOS 91), pp. 176-188. ACM, New York (1991).
44. Wulf W. A., McKee S. A.: Hitting the memory wall: implications of the obvious. SIGARCH Comput. Architect. News 23(1), 20-24 (1995).