Subscribe

Issue No.11 - November (2008 vol.57)

pp: 1443-1453

Bijan Ansari , University of Waterloo, Waterloo

M. Anwar Hasan , University of Waterloo, Waterloo

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2008.133

ABSTRACT

A high performance architecture of elliptic curve scalar multiplication based on the Montgomery ladder method over finite field GF(2m) is proposed. A pseudo-pipelined word serial finite field multiplier with word size w, suitable for the scalar multiplication is also developed. Implemented in hardware, this system performs a scalar multiplication in approximately 6⌈m/w⌉(m−1) clock cycles and the gate delay in the critical path is equal to TAND + ⌈log2(w/k)⌉TXOR, where TAND and TXOR are delays due to two-input AND and XOR gates respectively and 1 ≤ k ≪ w is used to shorten the critical path.

INDEX TERMS

Elliptic curves, finite fields, scalar multiplication

CITATION

Bijan Ansari, M. Anwar Hasan, "High-Performance Architecture of Elliptic Curve Scalar Multiplication",

*IEEE Transactions on Computers*, vol.57, no. 11, pp. 1443-1453, November 2008, doi:10.1109/TC.2008.133REFERENCES

- [1] P.K. Mishra, “Pipelined Computation of Scalar Multiplication in Elliptic Curve Cryptosystems,”
IEEE Trans. Computers, vol. 55, no. 8, pp. 1000-1010, Aug. 2006.- [3] N. Gura, S.C. Shantz, H. Eberle, S. Gupta, V. Gupta, D. Finchelstein, E. Goupy, and D. Stebila, “An End-to-End Systems Approach to Elliptic Curve Cryptography,”
Proc. Fourth Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '02), B.S. Kaliski, Ç.K. Koç, and C. Paar, eds., pp. 349-365, 2002.- [4] T. Izu and T. Takagi, “Fast Elliptic Curve Multiplications with SIMD Operations,”
Proc. Fourth Int'l Conf. Information and Comm. Security (ICICS '02), R.H. Deng, S. Qing, F. Bao, and J. Zhou, eds., pp. 217-230, 2002.- [7] K. Sakiyama, L. Batina, B. Preneel, and I. Verbauwhede, “Superscalar Coprocessor for High-Speed Curve-Based Cryptography,”
Proc. Eighth Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '06), L. Goubin and M. Matsui, eds., pp. 415-429, 2006.- [10] G. Orlando and C. Paar, “A High Performance Reconfigurable Elliptic Curve Processor for ${\rm GF}(2^{m})$ ,”
Proc. Second Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '00), Ç.K. Koç and C. Paar, eds., pp. 41-56, 2000.- [11] C. Grabbe, M. Bednara, J. von zur Gathen, J. Shokrollahi, and J. Teich, “A High Performance VLIW Processor for Finite Field Arithmetic,”
Proc. 17th Int'l Parallel and Distributed Processing Symp. (IPDPS '03), pp. 189-194, 2003.- [12] M. Ernst, M. Jung, F. Madlener, S. Huss, and R. Blümel, “A Reconfigurable System on Chip Implementation for Elliptic Curve Cryptography over ${\rm GF}(2^{n})$ ,”
Proc. Fourth Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '02), B.S. Kaliski, Ç.K. Koç, and C. Paar, eds., pp. 381-399, 2002.- [15] M. Bednara, M. Daldrup, J. von zur Gathen, J. Shokrollahi, and J. Teich, “Reconfigurable Implementation of Elliptic Curve Crypto Algorithms,”
Proc. 16th Int'l Parallel and Distributed Processing Symp. (IPDPS), 2002.- [16] D. Hankerson, A. Menezes, and S. Vanstone,
Guide to Elliptic Curves Cryptography. Springer, 2003.- [17] I. Blake, G. Seroussi, and N. Smart,
Elliptic Curves in Cryptography. Cambridge Univ. Press, 2002.- [19] J. López and R. Dahab, “Fast Multiplication on Elliptic Curves over ${\rm GF}(2^{m})$ without Precomputation,”
Proc. First Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '99), Ç.K. Koç and C. Paar, eds., pp. 316-327, 1999.- [21] B. Ansari and M.A. Hasan, “High Performance Architecture of Elliptic Curve Scalar Multiplication,” technical report, Univ. of Waterloo, http://www.cacr.math.uwaterloo.ca/techreports/ 2006cacr2006-01.pdf, Jan. 2006.
- [23] D. Catalano, R. Cramer, I. Damgard, G.D. Crescenzo, D. Pointcheval, and T. Takagi,
Contemporary Cryptology. Birkhäuser Basel, 2005.- [25] A. Weimerskirch and C. Paar,
Generalizations of the Karatsuba Algorithm for Efficient Implementations, Cryptology ePrint Archive, Report 2006/224, http:/eprint.iacr.org/, 2006. |