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Computing a square root for the number field sieve

  • Jean-Marc Couveignes
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1554)

Abstract

The number field sieve is a method proposed by Lenstra, Lenstra, Manasse and Pollard for integer factorization (this volume, pp. 11–42). A heuristic analysis indicates that this method is asymptotically faster than any other existing one. It has had spectacular successes in factoring numbers of a special form. New technical difficulties arise when the method is adapted for general numbers (this volume, pp. 50–94). Among these is the need for computing the square root of a huge algebraic integer given as a product of hundreds of thousands of small ones. We present a method for computing such a square root that avoids excessively large numbers. It works only if the degree of the number field that is used is odd. The method is based on a careful use of the Chinese remainder theorem.

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References

  1. 1.
    L.M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC) (1991), 64–71.Google Scholar
  2. 2.
    D.J. Bernstein, A.K. Lenstra, A general number field sieve implementation, this volume, pp. 103–126.Google Scholar
  3. 3.
    J.P. Buhler, H.W. Lenstra, Jr., Carl Pomerance, Factoring integers with the number field sieve, this volume, pp. 50–94.Google Scholar
  4. 4.
    D.E. Knuth, The art of computer programming, volume 2, second edition, Addison-Wesley, Reading, Mass., 1981.zbMATHGoogle Scholar
  5. 5.
    E. Landau, Sur quelques théorèmes de M. Petrovic relatifs aux zéros des fonctions analytiques, Bull. Soc. Math. France 33 (1905), 251–261.MathSciNetzbMATHGoogle Scholar
  6. 6.
    S. Lang, Algebraic number theory, Addison-Wesley, Reading, Massachusetts, 1970.zbMATHGoogle Scholar
  7. 7.
    A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse, J.M. Pollard, The number field sieve, this volume, pp. 11–42. Extended abstract: Proc. 22nd Annual ACM Symp. on Theory of Computing (STOC) (1990), 564–572.Google Scholar
  8. 8.
    M. Mignotte, Mathématiques pour le calcul formel, Presses Universitaires de France, Paris, 1989.zbMATHGoogle Scholar
  9. 9.
    P.L. Montgomery, R.D. Silverman, An FFT extension to the P — 1 factoring algorithm, Math. Comp. 54 (1990), 839–854.MathSciNetzbMATHGoogle Scholar
  10. 10.
    B.L. van der Waerden, Algebra, seventh edition, Springer-Verlag, Berlin, 1966.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Jean-Marc Couveignes
    • 1
    • 2
  1. 1.U. M. R. d'Algorithmique Arithmétique de BordeauxUniversité de Bordeaux IFrance
  2. 2.Groupe de recherche en complexité et cryptographie, L. I. E. N. S., URA 1327 du CNRSD. M. I., Ecole Normale SupérieureParis Cedex 05France

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