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Issue No.05 - May (2012 vol.61)
pp: 732-737
Jithra Adikari , University of Waterloo, Waterloo
Vassil S. Dimitrov , University of Calgary, Calgary
Kimmo U. Järvinen , Aalto University of Technology, Aalto
ABSTRACT
Scalar multiplication in elliptic curve cryptography is the most computational intensive operation. Efficiency of this operation can be significantly improved in hardware implementations by using Frobenius endomorphisms which require integer to \tau-adic nonadjacent form conversion. Because conversion is one of the limiting factors in some of Koblitz curve-based cryptosystems, it has become an interesting problem. In this paper, we propose two algorithms and a novel hardware architecture to double the speed of integer to \tau-adic nonadjacent form conversion.
INDEX TERMS
Elliptic curve cryptography, Koblitz curves, integer to \tauNAF conversion, field programmable gate array, application specific integrated circuit.
CITATION
Jithra Adikari, Vassil S. Dimitrov, Kimmo U. Järvinen, "A Fast Hardware Architecture for Integer to \tauNAF Conversion for Koblitz Curves", IEEE Transactions on Computers, vol.61, no. 5, pp. 732-737, May 2012, doi:10.1109/TC.2011.87
REFERENCES
[1] N. Koblitz, “Elliptic Curve Cryptosystems,” Math. Computation, vol. 48, pp. 203-209, Jan. 1987.
[2] V.S. Miller, “Use of Elliptic Curves in Cryptography,” Proc. Advances in Cryptology—(CRYPTO '85), pp. 417-426, 1986.
[3] V.S. Dimitrov, L. Imbert, and P.K. Mishra, “The Double-Base Number System and Its Application to Elliptic Curve Cryptography,” Math. Computation, vol. 77, no. 262, pp. 1075-1104, 2008.
[4] C. Doche and L. Habsieger, “A Tree-Based Approach for Computing Double-Base Chains,” Proc. 13th Australasian Conf. Information Security and Privacy (ACISP '08), pp. 433-446, 2008.
[5] J. Adikari, V.S. Dimitrov, and L. Imbert, “Hybrid Binary-Ternary Joint Form and Its Application in Elliptic Curve Cryptography,” Proc. IEEE 19th Symp. Computer Arithmetic (ARITH '09), pp. 76-83, June 2009.
[6] J. Adikari, V.S. Dimitrov, and L. Imbert, “Hybrid Binary-Ternary Number System for Elliptic Curve Cryptosystems,” IEEE Trans. Computers, vol. 60, no. 2, pp. 254-265, Feb. 2011.
[7] N. Koblitz, “CM-Curves with Good Cryptographic Properties,” Proc. 11th Ann. Int'l Cryptology Conf. Advances in Cryptology—(CRYPTO '91), pp. 279-287, 1992.
[8] W. Meier and O. Staffelbach, “Efficient Multiplication on Certain Nonsupersingular Elliptic Curves,” Proc. 12th Ann. Int'l Cryptology Conf. Advances in Cryptology—(CRYPTO '92), pp. 333-344, 1993.
[9] J.A. Solinas, “Efficient Arithmetic on Koblitz Curves,” Design, Codes and Cryptography, vol. 19, pp. 195-249, Mar. 2000.
[10] S. Okada, N. Torii, K. Itoh, and M. Takenaka, “Implementation of Elliptic Curve Cryptographic Coprocessor Over ${GF}(2^m)$ on an FPGA,” Proc. Second Int'l Workshop Cryptographic Hardware and Embedded Systems—(CHES '00), pp. 25-40, 2000.
[11] J. Lutz and A. Hasan, “High Performance FPGA Based Elliptic Curve Cryptographic Co-Processor,” Proc. Int'l Conf. Information Technology: Coding and Computing, vol. 2, pp. 486-492, 2004.
[12] K.U. Järvinen and J.O. Skyttä, “High-Speed Elliptic Curve Cryptography Accelerator for Koblitz Curves,” Proc. Ann. Int'l Symp. Field-Programmable Custom Computing Machines (FCCM '08), pp. 109-118, 2008.
[13] K. Järvinen and J. Skyttä, “Fast Point Multiplication on Koblitz Curves: Parallelization Method and Implementations,” Microprocessors and Microsystems, vol. 33, pp. 106-116, Mar. 2009.
[14] K. Järvinen, J. Forsten, and J. Skyttä, “Efficient Circuitry for Computing $\tau$ -Adic Non-Adjacent Form,” Proc. IEEE 13th Int'l Conf. Electronics, Circuits and Systems (ICECS '06), pp. 232-235, Dec. 2006.
[15] B.B. Brumley and K.U. Järvinen, “Koblitz Curves and Integer Equivalents of Frobenius Expansions,” Proc. Int'l Conf. Selected Areas in Cryptography—(SAC '07), pp. 126-137, 2007.
[16] V.S. Dimitrov, K.U. Järvinen, M.J. Jacobson, W.F. Chan, and Z. Huang, “FPGA Implementation of Point Multiplication on Koblitz Curves Using Kleinian Integers,” Proc. Cryptographic Hardware and Embedded Systems—(CHES '06), pp. 445-459, Oct. 2006.
[17] V.S. Dimitrov, K.U. Järvinen, M.J. Jacobson, W.F. Chan, and Z. Huang, “Provably Sublinear Point Multiplication on Koblitz Curves and Its Hardware Implementation,” IEEE Trans. Computers, vol. 57, no. 11, pp. 1469-1481, Nov. 2008.
[18] B.B. Brumley and K.U. Järvinen, “Conversion Algorithms and Implementations for Koblitz Curve Cryptography,” IEEE Trans. Computers, vol. 59, no. 1, pp. 81-92, Jan. 2010.
[19] National Institute of Standard and Tech nology “FIPS 186-2, Digital Signature Standard,” Fed. Information Processing Standards Publication, 2000.
[20] Certicom Research “SEC 2: Recommended Elliptic Curve Domain Parameters,” Standards for Efficient Cryptography, 2000.
[21] D. Hankerson, A. Menezes, and S. Vanstone, Guide to Elliptic Curve Cryptography. Springer, 2004.
[22] H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, July 2005.
[23] L.C. Washington, Elliptic Curves: Number Theory and Cryptography, first ed. Chapman & Hall/CRC, May 2003.
[24] D.E. Knuth, The Art of Computer Programming Volume 2: Seminumerical Algorithms. Addsion Weslsey Longman Publishing Group, May 1969.
[25] J.H. Conway and D. Smith, On Quaternions and Octonions, first ed. AK Peters, 2003.
15 ms
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