Abstract
We present an algorithm which speeds scalar multiplication on a general elliptic curve by an estimated 3.8% to 8.5% over the best known general methods when using affine coordinates. This is achieved by eliminating a field multiplication when we compute 2P +Q from given points P, Q on the curve. We give applications to simultaneous multiple scalar multiplication and to the Elliptic Curve Method of factorization. We show how this improvement together with another idea can speed the computation of the Weil and Tate pairings by up to 7.8%.
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References
Paulo S. L. M. Barreto, Hae Y. Kim, Ben Lynn and Michael Scott, Efficient algorithms for pairing-based cryptosystems, in Advances in Cryptology-Crypto 2002, M. Yung (Ed.), LNCS 2442, Springer-Verlag, 2002, pp. 354–368. 349, 350
I. F. Blake, G. Seroussi, N. P. Smart, Elliptic Curves in Cryptography, LMS 265 Cambridge University Press, 1999. 344, 346
Dan Boneh and Matt Franklin, Identity-based encryption from the Weil pairing, in Advances in Cryptology-Crypto 2001, J. Kilian (Ed.), LNCS 2139, Springer-Verlag, 2001, pp. 213–229. Appendix available at http://crypto.stanford.edu/~dabo/papers/ibe.pdf. 349
Dan Boneh, Ben Lynn, and Hovav Shacham, Short signatures from the Weil pairing, in Advances in Cryptology-Asiacrypt 2001, C. Boyd (Ed.), LNCS 2248, Springer-Verlag, 2001, pp. 514–532. 352
D. M. Gordon, A survey of fast exponentiation methods, J. Algorithms, 27, pp. 129–146, 1998. 345, 347
Antoine Joux, The Weil and Tate Pairings as building blocks for public key cryptosystems (survey), in Algorithmic Number Theory, 5th International Symposium ANTS-V, Sydney, Australia, July 7–12, 2002 Proceedings, Claus Fieker and David R. Kohel (Eds.), LNCS 2369, Springer-Verlag, 2002, pp. 20–32. 349
Donald E. Knuth, The Art of Computer Programming, vol. 2, Seminumerical Algorithms, Addison-Wesley, 3rd edition, 1997. 345
C. K. Koç and E. Savaş, Architectures for Unified Field Inversion with Applications in Elliptic Curve Cryptography, The 9th IEEE International Conference on Electronics, Circuits and Systems, ICECS 2002, Dubrovnik, Croatia, September 15–18, 2002, vol. 3, pp. 1155–1158. 346
Bodo Möller, Algorithms for multi-exponentiation, in Selected Areas in Cryptography 2001, Toronto, Ontario, Serge Vaudenay and Amr M. Youssef(Eds.), LNCS 2259, Springer-Verlag, 2002, pp. 165–180
Peter L. Montgomery, Speeding the Pollard and Elliptic Curve Methods of Factorization, Math. Comp., v. 48(1987), pp. 243–264. 349
Peter L. Montgomery, Evaluating Recurrences of Form Xm+n = f(Xm, Xn, Xm-n) via Lucas Chains. Available at ftp://ftp.cwi.nl:/pub/pmontgom/lucas.ps.gz. 349
Yasuyuki Sakai, Kouichi Sakurai, On the Power of Multidoubling in Speeding up Elliptic Scalar Multiplication, in Selected Areas in Cryptography 2001, Toronto, Ontario, Serge Vaudenay and Amr M. Youssef(Eds.), LNCS 2259, Springer-Verlag, 2002, pp. 268–283. 347
Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106, 1986. 344, 349, 353
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Eisenträger, K., Lauter, K., Montgomery, P.L. (2003). Fast Elliptic Curve Arithmetic and Improved Weil Pairing Evaluation. In: Joye, M. (eds) Topics in Cryptology — CT-RSA 2003. CT-RSA 2003. Lecture Notes in Computer Science, vol 2612. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36563-X_24
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DOI: https://doi.org/10.1007/3-540-36563-X_24
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