Advertisement

Factoring

  • Arjen K. Lenstra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 857)

Abstract

A brief survey of general purpose integer factoring algorithms and their implementations.

Keywords

Single Instruction Multiple Data Quadratic Residue Fermat Number Star Configuration Relation Collection 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. M. Adleman, Factoring numbers using singular integers, proc 23rd Annual ACM Symposium on Theory of Computing (STOC) (1991) 64–71Google Scholar
  2. 2.
    W. R. Alford, C. Pomerance, Implementing the self initializing quadratic sieve on a distributed network, manuscript, 1994Google Scholar
  3. 3.
    D. Atkins, M. Graff, A. K. Lenstra, P. C. Leyland, THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE (in preparation)Google Scholar
  4. 4.
    D. J. Bernstein, A. K. Lenstra, A general number field sieve implementation, 103–126 in: [26]Google Scholar
  5. 5.
    R. P. Brent, Factorization of the eleventh Fermat number (preliminary report), Abstracts Amer. Math. Soc. 10 (1989) 73Google Scholar
  6. 6.
    R. P. Brent, Parallel algorithms for integer factorisation, pp. 26–37 in: J. H. Loxton (ed.), Number theory and cryptography, London Math. Soc. Lecture Note Series 154, Cambridge University Press, Cambridge, 1990Google Scholar
  7. 7.
    R. P. Brent, J. M. Pollard, Factorization of the eighth Fermat number, Math. Comp. 36 (1981) 627–630Google Scholar
  8. 8.
    J. Buchmann, J. Loho, J. Zayer, An implementation of the general number field sieve, Advances in Cryptology, Crypto '93, Lecture Notes in Comput. Sci. 773 (1994) 159–165.Google Scholar
  9. 9.
    T. R. Caron, R. D. Silverman, Parallel implementation of the quadratic sieve, The Journal of Supercomputing 1 (1988) 273–290CrossRefGoogle Scholar
  10. 10.
    S. Coppersmith, Solving linear equations over GF(2): block Lanczos algorithm, Linear algebras and its applications 192 (1993) 33–60CrossRefGoogle Scholar
  11. 11.
    S. Coppersmith, Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm, Math. Comp. 62 (1994) 333–350Google Scholar
  12. 12.
    J. A. Davis, D. B. Holdridge, Factorization using the quadratic sieve algorithm, Tech. Report SAND 83-1346, Sandia National Laboratories, Albuquerque, NM, 1983Google Scholar
  13. 13.
    T. Denny, B. Dodson, A. K. Lenstra, M. S. Manasse, On the factorization of RSA-120, Advances in Cryptology, Crypto '93, Lecture Notes in Comput. Sci. 773 (1994) 166–174Google Scholar
  14. 14.
    A. Díaz, M. Hitz, E. Kaltofen, A. Lobo, T. Valente, Process scheduling in DCS and the large sparse linear systems challenge, J. Symbolic Computation (submitted)Google Scholar
  15. 15.
    B. Dixon, A. K. Lenstra, Factoring integers using SIMD sieves, Advances in Cryptology, Eurocrypt '93, Lecture Notes in Comput. Sci. 765 (1994) 28–39Google Scholar
  16. 16.
    J. D. Dixon, Asymptotically fast factorization of integers, Math. Comp. 36 (1981) 255–260Google Scholar
  17. 17.
    B. Dodson, A. K. Lenstra, NFS with four large primes: an explosive experiment (in preparation)Google Scholar
  18. 18.
    M. Gardner, Mathematical games, A new kind of cipher that would take millions of years to break, Scientific American, August 1977, 120–124Google Scholar
  19. 19.
    J. L. Gerver, Factoring large numbers with a quadratic sieve, Math. Comp. 36 (1983) 287–294Google Scholar
  20. 20.
    R. Golliver, A. K. Lenstra, K. S. McCurley, Lattice sieving and trial division, Algorithmic number theory symposium, proceedings, Cornell, 1994 (to appear)Google Scholar
  21. 21.
    R. K. Guy, How to factor a number, Proc. Fifth Manitoba Conf. Numer. Math., Congressus Numerantium 16 (1976) 49–89Google Scholar
  22. 22.
    G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 5th ed., 1979Google Scholar
  23. 23.
    E. Kaltofen, Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems, Math. Comp. (to appear)Google Scholar
  24. 24.
    B. A. LaMacchia, A. M. Odlyzko, Computation of discrete logarithms in prime fields, Designs, Codes and Cryptography 1 (1991) 47–62Google Scholar
  25. 25.
    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, Elsevier, Amsterdam, 1990Google Scholar
  26. 26.
    A. K. Lenstra, H. W. Lenstra, Jr. (eds), The development of the number field sieve, Lecture Notes in Math. 1554, Springer-Verlag, Berlin, 1993Google Scholar
  27. 27.
    A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The number field sieve, 11–42 in: [26]Google Scholar
  28. 28.
    A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993) 319–349Google Scholar
  29. 29.
    A. K. Lenstra, M. S. Manasse, Factoring by electronic mail, Advances in Cryptology, Eurocrypt '89, Lecture Notes in Comput. Sci. 434 (1990) 355–371Google Scholar
  30. 30.
    A. K. Lenstra, M. S. Manasse, Factoring with two large primes, Advances in Cryptology, Eurocrypt '90, Lecture Notes in Comput. Sci., 473 (1990) 72–82; Math. Comp. (to appear)Google Scholar
  31. 31.
    H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987) 649–673Google Scholar
  32. 32.
    H. W. Lenstra, Jr., C. Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc. 5 (1992) 483–516Google Scholar
  33. 33.
    H. W. Lenstra, Jr., R. Tijdeman (eds), Computational methods in number theory, Math. Centre Tracts 154/155, Mathematisch Centrum, Amsterdam, 1984Google Scholar
  34. 34.
    E. Messmer, Bellcore leads team effort to crack RSA encryption code, Network World, May 2, 1994Google Scholar
  35. 35.
    P. L. Montgomery, Record number field sieve factorizations, announcement on NmbrThry@VM1.NODAK.EDU, July 12, 1994Google Scholar
  36. 36.
    P. L. Montgomery, A block Lanczos algorithm for finding dependencies over GF(2), Draft manuscript, June 17, 1994Google Scholar
  37. 37.
    M. A. Morrison, J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp. 29 (1975) 183–205Google Scholar
  38. 38.
    R. Peralta, A quadratic sieve on the n-dimensional hypercube, Advances in Cryptology, Crypto '92, Lecture Notes in Comput. Sci. 740 (1993) 324–332Google Scholar
  39. 39.
    J. M. Pollard, Theorems on factorization and primality testing, Proc. Cambr. Philos. Soc 76 (1974) 521–528Google Scholar
  40. 40.
    J. M. Pollard, A Monte Carlo method for factorization, BIT 15 (1975) 331–334CrossRefGoogle Scholar
  41. 41.
    J. M. Pollard, Factoring with cubic integers, 4–10 in [26]Google Scholar
  42. 42.
    C. Pomerance, Analysis and comparison of some integer factoring algorithms, pp. 89–139 in: [33]Google Scholar
  43. 43.
    C. Pomerance, J. W. Smith, Reduction of huge, sparse matrices over finite fields via created catastrophes, Experiment. Math. 1 (1992) 89–94MathSciNetGoogle Scholar
  44. 44.
    C. Pomerance, J. W. Smith, R. Tuler, A pipe-line architecture for factoring large integers with the quadratic sieve algorithm, SIAM J. Comput. 17 (1988) 387–403CrossRefGoogle Scholar
  45. 45.
    R. L. Rivest, letter to Martin Gardner, 1977Google Scholar
  46. 46.
    R. L. Rivest, A. Shamir, L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 (1978) 120–126CrossRefGoogle Scholar
  47. 47.
    R. J. Schoof, Quadratic fields and factorization, pp 235–286 in: [33]Google Scholar
  48. 48.
    R. C. Schroeppel, personal communication, May 1994Google Scholar
  49. 49.
    P. W. Shor, Algorithms for quantum computation: Discrete log and factoring, DIMACS Technical Report 94-37; to appear in Proc. 35th Symposium on Foundations of Computer Science, 1994Google Scholar
  50. 50.
    R. D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987) 329–339Google Scholar
  51. 51.
    J. W. Smith, S. S. Wagstaff, Jr., An extended precision operand computer, Proc. 21st Southeast Region ACM Conf. (1983) 209–216Google Scholar
  52. 52.
    D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986) 54–62CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Arjen K. Lenstra
    • 1
  1. 1.MorristownUSA

Personalised recommendations