Abstract
The basic operation in elliptic cryptosystems is the computation of a multiple d·P of a point P on the elliptic curve modulo n. We propose a fast and systematic method of reducing the number of operations over elliptic curves. The proposed method is based on pre-computation to generate an adequate addition-subtraction chain for multiplier the d. By increasing the average length of zero runs in a signed binary representation of d, we can speed up the window method. Formulating the time complexity of the proposed method makes clear that the proposed method is faster than other methods. For example, for d with length 512 bits, the proposed method requires 602.6 multiplications on average. Finally, we point out that each addition/subtraction over the elliptic curve using homogeneous coordinates can be done in 3 multiplications if parallel processing is allowed.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bos, J. and Coster, M: “Addition chain heuristics” Proc. of CRYPTO’89 (1989).
Brickell, E. F.: “A fast modular multiplication algorithm with application to two key cryptography” Proc. of CRYPTO’82 (1982).
Brickell, E.F., Gordon, D.M., McCurley, K.S., and Wilson, D.: “Fast exponentiation with precomputation” Proc. of EUROCRYPT’92 (1992).
Diffie, W. and Hellman, M.E.: “New directions in cryptography”, IEEE Transactions on Information Theory, Vol. 22, No. 6, (1976), pp. 644–654.
Downey, P. Leony, B. and Sethi, R: “Computing sequences with addition chains”, Siam J. Comput. 3 (1981) pp. 638–696.
ElGamal, T.: “A public key cryptosystem and a signature scheme based on the discrete logarithm”, IEEE Transactions on Information Theory, Vol. 31, No. 4, (1985), pp. 469–472.
Goldwasser, S. and Killian, J.: “Almost all primes can be quickly certified”, Proc. 18th STOC. Berkeley, (1986), pp. 316–329.
Jedwab, J. and Mitchell, C, J.: “Minimum weight modified signed-digit representations and fast exponentiation”, Electronics Letters Vol. 25, No. 17, (1989), pp. 1171–1172.
Koyama, K. Maurer, U. Okamoto, T and Vanstone, S, A: “New public-key schemes based on elliptic curves over the ring Z n”, Proc. of CRYPTO’91 (1991).
Knuth, D.E.: “Seminumerical algorithm (arithmetic)” The Art of Computer Programming Vol.2, Addison Wesley, (1969).
Koblitz, N.: A course in number theory and cryptography, Berlin: Springer-Verlag, (1987).
Lenstra, H. W. Jr.: “Factoring integers with elliptic curves”, Ann. of Math. 126 (1987), pp. 649–673.
Montgomery, P.L.: “Speeding the Pollard and elliptic curve methods of factorization”, Math. Comp. 48, (1987), pp. 243–264.
Morain, F. and Olivos, J.: “Speeding up the computations on an elliptic curve using addition-subtraction chains” Theoretical Informatics and Applications Vol. 24, No. 6 (1990) pp. 531–544.
Rivest, R.L. Shamir, A. and Adleman, L.: “A method for obtaining digital signatures and public-key cryptosystems”, Communications of the ACM, Vol. 21, No. 2, (1978), pp. 120–126.
Yacobi, Y.: “Exponentiating faster with addition chains” Proc. of EUROCRYPT’90 (1990).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Koyama, K., Tsuruoka, Y. (1993). Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method. In: Brickell, E.F. (eds) Advances in Cryptology — CRYPTO’ 92. CRYPTO 1992. Lecture Notes in Computer Science, vol 740. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48071-4_25
Download citation
DOI: https://doi.org/10.1007/3-540-48071-4_25
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57340-1
Online ISBN: 978-3-540-48071-6
eBook Packages: Springer Book Archive