Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method

  • Kenji Koyama
  • Yukio Tsuruoka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 740)


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.


Elliptic Curve Elliptic Curf Binary Representation Window Method Binary Method 
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.


  1. 1.
    Bos, J. and Coster, M: “Addition chain heuristics” Proc. of CRYPTO’89 (1989).Google Scholar
  2. 2.
    Brickell, E. F.: “A fast modular multiplication algorithm with application to two key cryptography” Proc. of CRYPTO’82 (1982).Google Scholar
  3. 3.
    Brickell, E.F., Gordon, D.M., McCurley, K.S., and Wilson, D.: “Fast exponentiation with precomputation” Proc. of EUROCRYPT’92 (1992).Google Scholar
  4. 4.
    Diffie, W. and Hellman, M.E.: “New directions in cryptography”, IEEE Transactions on Information Theory, Vol. 22, No. 6, (1976), pp. 644–654.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Downey, P. Leony, B. and Sethi, R: “Computing sequences with addition chains”, Siam J. Comput. 3 (1981) pp. 638–696.CrossRefGoogle Scholar
  6. 6.
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Goldwasser, S. and Killian, J.: “Almost all primes can be quickly certified”, Proc. 18th STOC. Berkeley, (1986), pp. 316–329.Google Scholar
  8. 8.
    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.CrossRefzbMATHGoogle Scholar
  9. 9.
    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).Google Scholar
  10. 10.
    Knuth, D.E.: “Seminumerical algorithm (arithmetic)” The Art of Computer Programming Vol.2, Addison Wesley, (1969).Google Scholar
  11. 11.
    Koblitz, N.: A course in number theory and cryptography, Berlin: Springer-Verlag, (1987).zbMATHGoogle Scholar
  12. 12.
    Lenstra, H. W. Jr.: “Factoring integers with elliptic curves”, Ann. of Math. 126 (1987), pp. 649–673.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Montgomery, P.L.: “Speeding the Pollard and elliptic curve methods of factorization”, Math. Comp. 48, (1987), pp. 243–264.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    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.zbMATHMathSciNetGoogle Scholar
  15. 15.
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Yacobi, Y.: “Exponentiating faster with addition chains” Proc. of EUROCRYPT’90 (1990).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Kenji Koyama
    • 1
  • Yukio Tsuruoka
    • 1
  1. 1.NTT Communication Science LaboratoriesSeikacho, KyotoJapan

Personalised recommendations