Provably sublinear point multiplication on Koblitz curves and its hardware implementation

IEEE Transactions on Computers

Volumn 57, Issue 11, 2008, Pages 1469-1481

  Dimitrov, Vassil S ^a   Järvinen, Kimmo U ^b   Jacobson Jr , Micheal J ^a   Chan, Wai Fong ^a   Huang, Zhun ^c  

^a UNIVERSITY OF CALGARY   (Canada)

^b AALTO UNIVERSITY   (Finland)

^c VIA Technologies ^*  (China)

Author keywords

Elliptic curve cryptography; Field programmable gate arrays; Koblitz curves; Multiple base expansions; Parallel processing; Sublinearity

Indexed keywords

BOOLEAN FUNCTIONS; CRYPTOGRAPHY; HARDWARE;

ELLIPTIC CURVE CRYPTOGRAPHY; FIELD-PROGRAMMABLE GATE ARRAYS; KOBLITZ CURVES; MULTIPLE-BASE EXPANSIONS; PARALLEL PROCESSING; SUBLINEARITY;

FIELD PROGRAMMABLE GATE ARRAYS (FPGA);

EID: 54049083879     PISSN: 00189340     EISSN: None     Source Type: Journal    
DOI: 10.1109/TC.2008.65     Document Type: Article

References (38)

1. N. Koblitz, "Elliptic Curve Cryptosystems," Math. Computation, vol. 48, pp. 203-209, 1987.
2. V. Miller, "Use of Elliptic Curves in Cryptography," Advances in Cryptology - CRYPTO "85, pp. 417-426, 1986.
3. N. Koblitz, "CM-Curves with Good Cryptographic Properties," Advances in Cryptology - CRYPTO '91, pp. 279-287, 1992.
4. Digital Signature Standard (DSS), Fed. Information Processing Standard, FIPS PUB 186-2, Nat'l Inst. of Standards and Technology (NIST) Computer Security FIPS PUB 186-2, Jan. 2000.
5. J. Solinas, "Efficient Arithmetic on Koblitz Curves," Designs, Codes and Cryptography, vol. 19, pp. 195-249, 2000.
6. R. Avanzi, C. Heuberger, and H. Prodinger, "Minimality of the Hamming Weight of the τ-NAF for Koblitz Curves and Improved Combination with Point Halving," Selected Areas in Cryptography - SAC '05, pp. 332-344, 2005.
7. V. Dimitrov, G. Jullien, and W. Miller, "An Algorithm for Modular Exponentiation," Information Processing Letters, vol. 66, no. 3, pp. 155-159, 1998.
8. M. Ciet and F. Sica, "An Analysis of Double Base Number Systems and a Sublinear Scalar Multiplication Algorithm," Progress in Cryptology - Mycrypt '05, pp. 171-182, 2005.
9. V. Dimitrov, L. Imbert, and P. Mishra, "Efficient and Secure Elliptic Curve Point Multiplication Using Double-Base Chains," Advances in Cryptology - ASIACRYPT '05, pp. 59-78, 2005.
10. V.S. Dimitrov, K.U. Järvinen, M.J. Jacobson Jr., W.F. Chan, and Z. Huang, "FPGA Implementation of Point Multiplication on Koblitz Curves Using Kleinian Integers," Cryptographic Hardware and Embedded Systems - CHES '06, pp. 445-459, 2006.
11. R. Avanzi and F. Sica, "Scalar Multiplication on Koblitz Curves Using Double Bases," Progress in Cryptology - VIETCRYPT '06, pp. 131-146, 2006.
12. F. Sica, Scalar Multiplication on Koblitz Curves Using Double Bases Univ. of Calgary, invited talk, Apr. 2006.
13. J. Conway and D. Smith, On Quaternions and Octonions. AK Peters, 2003.
14. R. Tijdeman, "On Integers with Many Small Prime Factors," Composition Math., vol. 26, no. 3, pp. 319-330, 1973.
15. A. Baker, "Linear Forms in the Logarithms of Algebraic Numbers IV," Math., vol. 15, pp. 204-216, 1968.
16. M. Mignotte and M. Waldshmidt, "Linear Forms in Two Logarithms and Schneider's Method III," Annales de la Faculté des Sciences de Toulouse, pp. 43-75, 1990.
17. R. Tijdeman, personal communication, 2006.
18. n)," Selected Areas in Cryptography - SAC '98, pp. 201-212, 1998.
19. C. Doche and T. Lange, "Arithmetic of Elliptic Curves," Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman and Hall/CRC, H. Cohen and G. Frey, eds., chapter 13, pp. 267-302, 2006.
20. n Using Projective Coordinates," Information Processing Letters, vol. 76, no. 3, pp. 101-103, 2000.
21. n)," IEEE Trans. Computers, vol. 51, no. 8, pp. 972-975, Aug. 2002.
22. m) Using Normal Bases," Information and Computation, vol. 78, no. 3, pp. 171-177, Sept. 1988.
23. Stratix II Device Handbook, Altera, http://www.altera.com/ literature/hb/stx2/stratix2_handbook.pdf, May 2007.
24. Stratix II EP2S180 DSP Development Board - Reference Manual, Altera, http://www.altera.com/literature/manual/ mnl_SII_DSP_RM_11Aug06.pdf, Aug. 2006.
25. m)," IEEE Trans. Computers, vol. 34, no. 8, pp. 709-717, Aug. 1985.
26. B. Ansari and M.A. Hasan, "High Performance Architecture of Elliptic Curve Scalar Multiplication," Technical Report CACR 2006-1, Univ. of Waterloo, 2006.
27. n) in Optimal Normal Basis on a Reconfigurable Computer," Proc. Int'l Conf. Field Programmable Logic and Application (FPL '04), pp. 1098-1100, 2004.
28. M. Bednara, M. Daldrup, J. von zur Gathen, J. Shokrollahi, and J. Teich, "Reconfigurable Implementation of Elliptic Curve Crypto Algorithms," Proc. Reconfigurable Architectures Workshop, Int'l Parallel and Distributed Processing Symp. (IPDPS '02), pp. 157-164, Apr. 2002.
29. m) Elliptic Curve Cryptography Processors," IEEE Trans. Very Large Scale Integration Systems, vol. 14, no. 6, pp. 659-662, June 2006.
30. R. Cheung, N. Telle, W. Luk, and P. Cheung, "Customizable Elliptic Curve Cryptosystem," IEEE Trans. Very Large Scale Integration Systems, vol. 13, pp. 1048-1059, Sept. 2005.
31. m)," Technical Report SMLI TR-2003-123, Sun Microsystems, May 2003.
32. K. Järvinen, J. Forsten, and J. Skyttä, "FPGA Design of Self-Certified Signature Verification on Koblitz Curves," Cryptographic Hardware and Embedded Systems - CHES '07, pp. 256-271, 2007.
33. P. Leong and K. Leung, "A Microcoded Elliptic Curve Processor Using FPGA Technology," IEEE Trans. Very Large Scale Integration Systems vol. 10, no. 5, pp. 550-559, Oct. 2002.
34. J. Lutz and A. Hasan, "High Performance FPGA Based Elliptic Curve Cryptographic Co-Processor," Proc. Int'l Conf. Information Technology: Coding and Computing (ITCC '04), vol. 2, pp. 486-492, Apr. 2004.
35. m) on an FPGA," Cryptographic Hardware and Embedded Systems - CHES '00, pp. 25-40, 2000.
36. m)," Cryptographic Hardware and Embedded Systems - CHES '00, pp. 41-56, 2000.
37. m" Microprocessors and Microsystems, vol. 28, nos. 5-6, pp. 329-339, Aug. 2004.
38. C. Shu, K. Gaj, and T. El-Ghazawi, "Low Latency Elliptic Curve Cryptography Accelerators for NIST Curves over Binary Fields," Proc. IEEE Int'l Conf. Field-Programmable Technology (FPT '05), pp. 309-310, Dec. 2005.