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Integer Factoring

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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.

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References

  1. L. M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC), New Orleans, (May 6–8, 1991) pp. 64–71.

  2. W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math, Vol. 140 (1994) pp. 703–722.

    Google Scholar 

  3. W. R. Alford, A. Granville, and C. Pomerance, On the difficulty of finding reliable witnesses, ANTS'94, Lecture Notes in Comput. Sci., 877 (1994) pp. 1–16.

  4. D. Atkins, M. Graff, A. K. Lenstra, and P. C. Leyland, The magic words are squeamish ossifrage, Advances in Cryptology, Asiacrypt'94, Lecture Notes in Comput. Sci., 917 (1995) pp. 265–277.

  5. D. J. Bernstein, The multiple-lattice number field sieve, Chapter 3 of Ph.D. thesis; ftp://koobera.math.uic.edu/pub/papers/mlnfs.dvi.

  6. W. Bosma and A. K. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (W. Bosma and A. van der Poorten, eds.), Kluwer Academic Publishers, Dordrecht, Boston, London (1995) pp. 119–136.

    Google Scholar 

  7. R. P. Brent, Factorization of the tenth and eleventh Fermat Numbers, manuscript (1996).

  8. R. P. Brent and J. M. Pollard, Factorization of the eighth Fermat number, Math. Comp., Vol. 36 (1981) pp. 627–630.

    Google Scholar 

  9. J. Buchmann, J. Loho, and J. Zayer, An implementation of the general number field sieve, Advances in Cryptology, Crypto '93, Lecture Notes in Comput. Sci, 773 (1994) pp. 159–165.

  10. E. R. Canfield, P. Erdös, and C. Pomerance, On a problem of Oppenheim concerning "Factorisatio Numerorum," J. Number Theory, Vol. 17 (1983) pp. 1–28.

    Google Scholar 

  11. H. Cohen, A course in computational number theory, Graduate Texts in Mathematics, Vol. 138, Springer-Verlag, Berlin (1993).

    Google Scholar 

  12. S. Contini and A. K. Lenstra, Implementations of blocked Lanczos and Wiedemann algorithms, manuscript.

  13. J. Cowie, B. Dodson, R. M. Elkenbracht-Huizing, A. K. Lenstra, P. L. Montgomery, and J. Zayer, A World Wide Number Field Sieve factoring record: on to 512 bits, Advances in Cryptography, Asiacrypt '96, Lecture Notes in Computer Science, 1163 (1996) pp. 382–394.

  14. J. A. Davis and D. B. Holdridge, Factorization using the quadratic sieve algorithm, Tech. Report SAND 83–1346, Sandia National Laboratories, Albuquerque, NM (1983).

    Google Scholar 

  15. N. G. de Bruijn, On the number of positive integers ≤ x and free of prime factors > y, II, Indag. Math., Vol. 38 (1966) pp. 239–247.

    Google Scholar 

  16. M. Deleglise and J. Rivat, Computing π(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp., Vol. 65 (1996) pp. 235–245.

    Google Scholar 

  17. T. Denny, B. Dodson, A. K. Lenstra, and M. S. Manasse, On the factorization of RSA-120, Advances in Cryptology, Crypto '93, Lecture Notes in Comput. Sci., 773 (1994) pp. 166–174.

  18. B. Dodson and A. K. Lenstra, NFS with four large primes: an explosive experiment, Advances in Cryptology, Crypto '95, Lecture Notes in Comput. Sci., 963 (1995) pp. 372–385.

  19. R. M. Elkenbracht-Huizing, An implementation of the number field sieve, Technical Report NM-R9511, Centrum voor Wiskunde en Informatica, Amsterdam, 1995; to appear in Experimental Mathematics.

  20. R. M. Elkenbracht-Huizing, A multiple polynomial general number field sieve, Preproceedings ANTS II (H. Cohen, ed.), Universit´e de Bordeaux (1996) pp. 101–116.

  21. M. Gardner, Mathematical games, A new kind of cipher that would take millions of years to break, Scientific American (August 1977) pp. 120–124.

  22. J. Gillogly, A. K. Lenstra, P. C. Leyland, and A. Muffett, An unnoticed factoring attack on a PGP key, presented at Crypto '95 rump session.

  23. R. Golliver, A. K. Lenstra, and K. McCurley, Lattice sieving and trial division, ANTS'94, Lecture Notes in Comput. Sci., 877 (1994) pp. 18–27.

  24. G. H. Hardy and W. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, Oxford (1979).

    Google Scholar 

  25. D. E. Knuth, Art of computer programming, volume 2, Seminumerical Algorithms, 2nd ed., Addison-Wesley, Reading, Massachusetts (1981).

    Google Scholar 

  26. M. Kraïtchik, Theorie de Nombres, II, Gauthiers-Villars, Paris (1926) pp. 195–208.

    Google Scholar 

  27. J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, Computing π(x): The Meissel-Lehmer Method, Math. Comp., Vol. 44 (1985) pp. 537–560.

    Google Scholar 

  28. B. A. LaMacchia and A. M. Odlyzko, Solving large sparse linear systems over finite fields, Advances in Cryptology, Crypto'90, Lecture Notes in Comput. Sci., 537 (1991) pp. 109–133.

  29. A. K. Lenstra and H. W. Lenstra, Jr., Algorithms in number theory, Chapter 12 in Handbook of Theoretical Computer Science, Volume A, Algorithms and Complexity (J. van Leeuwen, ed.), Elsevier, Amsterdam (1990).

    Google Scholar 

  30. A. K. Lenstra and H. W. Lenstra, Jr., The development of the number field sieve, Lecture Notes in Math., Springer-Verlag, Berlin, 1554 (1993).

    Google Scholar 

  31. A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp., Vol. 61 (1993) pp. 319–349.

    Google Scholar 

  32. A. K. Lenstra and M. S. Manasse, Factoring by electronic mail, Advances in Cryptology, Eurocrypt '89, Lecture Notes in Comput. Sci., 434 (1990) pp. 355–371.

  33. A. K. Lenstra and M. S. Manasse, Factoring with two large primes, Advances in Cryptology, Eurocrypt '90, Lecture Notes in Comput. Sci., 473 (1990) pp. 72–82; Math. Comp., Vol. 63 (1994) pp. 785–798.

  34. H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math, Vol. 126 (1987) pp. 649–673.

    Google Scholar 

  35. H. W. Lenstra, Jr., and C. Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc., Vol. 5 (1992) pp. 483–516.

    Google Scholar 

  36. H. W. Lenstra, Jr. and R. Tijdeman (eds.), Computational methods in number theory, Math. Centre Tracts, Vol. 154/155, Mathematisch Centrum, Amsterdam (1983).

  37. P. C. Leyland, Multiple polynomial quadratic sieve, sans math, ftp://ftp.ox.ac.uk/pub/math/rsa129/ mpqs sans math.Z (1994).

  38. L. Monier, Evaluation and comparison of two efficient probabilistic primality testing algorithms, Theor. Comp. Science, Vol. 11 (1980) pp. 97–108.

    Google Scholar 

  39. P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp., Vol. 48 (1987) pp. 243–264.

    Google Scholar 

  40. P. L. Montgomery, Square roots of products of algebraic numbers, Proceedings of Symposia in Applied Mathematics (Walter Gautschi, ed.), Mathematics of Computation 1943–1993, Vancouver (1993).

  41. P. L. Montgomery, A block Lanczos algorithm for finding dependencies over GF(2), Advances in Cryptology, Eurocrypt'95, Lecture Notes in Comput. Sci., 921 (1995) pp. 106–120.

  42. M. A. Morrison and J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp., Vol. 29 (1975) pp. 183–205.

    Google Scholar 

  43. J. M. Pollard, Theorems on factorization and primality testing, Proc. Cambridge Philos. Soc., Vol. 76 (1974) pp. 521–528.

    Google Scholar 

  44. J. M. Pollard, A Monte Carlo method for factorization, BIT, Vol. 15 (1975) pp. 331–334.

    Google Scholar 

  45. J. M. Pollard, The lattice sieve, Lecture Notes in Math., Springer-Verlag, Berlin, 1554 (1993) pp. 43–49.

    Google Scholar 

  46. C. Pomerance, Analysis and comparison of some integer factoring algorithms, Computational methods in number theory, Math. Centre Tracts, Vol. 154/155, Mathematisch Centrum, Amsterdam (1983) pp. 89–139.

    Google Scholar 

  47. C. Pomerance, Private communication (March 1996).

  48. C. Pomerance and J. W. Smith, Reduction of huge, sparse matrices over finite fields via created catastrophes, Experiment. Math., Vol. 1 (1992) pp. 89–94.

    Google Scholar 

  49. M. O. Rabin, Probabilistic algorithms for primality testing, J. Number Theory, Vol. 12 (1980) pp. 128–138.

    Google Scholar 

  50. H. Riesel, Prime numbers and computer methods for factorization. Progr. Math., Vol. 57, Birkhäuser, Boston (1985).

    Google Scholar 

  51. R. L. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM, Vol. 21 (1978) pp. 120–126.

    Google Scholar 

  52. R. Schoof, Quadratic fields and factorization, Computational methods in number theory, Math. Centre Tracts, Vol. 154/155, Mathematisch Centrum, Amsterdam (1983) pp. 235–286.

    Google Scholar 

  53. D. Shanks, Class number, a theory of factorization, and genera, Proc. Symp. Pure Math., Vol. XX, AMS (1971) pp. 415–440.

    Google Scholar 

  54. R. D. Silverman, The multiple polynomial quadratic sieve, Math. Comp., Vol. 84 (1987) pp. 327–339.

    Google Scholar 

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Lenstra, A.K. Integer Factoring. Designs, Codes and Cryptography 19, 101–128 (2000). https://doi.org/10.1023/A:1008397921377

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