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An Implementation of the Elliptic Curve Integer Factorization Method

  • Wieb Bosma
  • Arjen K. Lenstra
Part of the Mathematics and Its Applications book series (MAIA, volume 325)

Abstract

This paper describes the second author’s implementation of the elliptic curve method for the factorization of integers as it is currently available in the computational algebra package Magma, which is under development at the University of Sydney.

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References

  1. [1]
    R. P. Brent, Some integer factorization algorithms using elliptic curves, Research Report CMA-R32–85, The Australian National Univ., Canberra, 1985.Google Scholar
  2. [2]
    J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, S. S. Wagstaff, Jr., Factorizations of bn ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers,second edition, Contemp. Math. 22, Providence: Amer. Math. Soc., 1988.Google Scholar
  3. [3]
    D. V. Chudnovsky, G. V. Chudnovsky, Sequences of numbers generated by addition in formal groups and new primality and factorization tests, IBM Research Report RC 11262, 1985.Google Scholar
  4. [4]
    B. Dixon, A. K. Lenstra, Massively parallel elliptic curve factoring, A.vances in Cryptology, Eurocrypt’92, Lecture Notes in Comput. Sci. 658 (1993), 183–193.CrossRefGoogle Scholar
  5. [5]
    A. K. Lenstra, LIP, a long integer package,available for anonymous ftp from /pub/lenstra on f lash.bellcore. com.Google Scholar
  6. [6]
    A. K. Lenstra, H. W. Lenstra, Jr., Algorithms in number theory, Chapter 12 in: J. van Leeuwen (ed.), Handbook of theoretical computer science, Volume A, Algorithms and complexity, Amsterdam: Elsevier, 1990.Google Scholar
  7. [7]
    A. K. Lenstra, H. W. Lenstra, Jr., (eds.) The development of the number field sieve, Lecture Notes in Math. 1554, Berlin: Springer-Verlag, 1993.Google Scholar
  8. [8]
    A. K. Lenstra, M. S. M.nasse, Factoring by electronic mail, Advances in cryptology, Eurocrypt ‘89, Lecture Notes in Comput. Sci. 434 (1990), 355–371.Google Scholar
  9. [9]
    H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    H. W. Lenstra, Jr., Elliptic curves and number-theoretic algorithms, pp. 99–120 in: A. M. Gleason (ed.), Proceedings of the International Congress of Mathematicians, August 3–11, 1986 (Berkeley, California), Providence: American Mathematical Society, 1987.Google Scholar
  11. [11]
    P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp. 48 (1987) 243–264.MathSciNetCrossRefGoogle Scholar
  12. [12]
    P. L. Montgomery, An FFT extension of the elliptic curve method of factorization, PhD thesis, Los Angeles, 1992.Google Scholar
  13. [13]
    M. S. Paterson, L. J. Stockmeyer, On the number of nonscalar multiplications necessary to evaluate polynomials, SIAM J. Comput. 2 (1973), 60–66.MathSciNetzbMATHGoogle Scholar
  14. [14]
    RSA Data Security Corporation Inc., sci.crypt, May 18, 1991; information available by sending electronic mail to challenge-rsa-list@rsa. com.Google Scholar
  15. [15]
    J. H. Silverman, The arithmetic of elliptic curves, New York: Springer-Verlag, 1986.zbMATHCrossRefGoogle Scholar
  16. [16]
    H. Suyama, Informal preliminary report, October 1985.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Wieb Bosma
    • 1
  • Arjen K. Lenstra
    • 2
  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  2. 2.BellcoreRoom MRE-2Q334MorristownUSA

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