Recent progress and prospects for integer factorisation algorithms

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 1858, Issue , 2000, Pages 3-22

  Brent, Richard P ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

COMBINATORIAL MATHEMATICS; LINEAR SYSTEMS; PUBLIC KEY CRYPTOGRAPHY;

CURRENT CAPABILITY; DISCRETE LOGARITHM PROBLEMS; PARALLEL MACHINE; PARALLEL SOLUTIONS; PRACTICAL IMPORTANCE; PUBLIC KEY CRYPTOSYSTEMS; RECENT PROGRESS; SPARSE LINEAR SYSTEMS;

FACTORIZATION;

EID: 84956985315     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/3-540-44968-x_2     Document Type: Conference Paper

References (63)

1. D. Atkins, M. Graff, A. K. Lenstra and P. C. Leyland, The magic words are squeamish ossifrage, Advances in Cryptology: Proc. Asiacrypt’94, LNCS 917, Springer-Verlag, Berlin, 1995, 263-277.
2. H. Boender and H. J. J. te Riele, Factoring integers with large prime variations of the quadratic sieve, Experimental Mathematics, 5 (1996), 257-273.
3. R. P. Brent, Vector and parallel algorithms for integer factorisation, Proceedings Third Australian Supercomputer Conference University of Melbourne, December 1990, 12 pp. http://www.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub122.html.
4. R. P. Brent, The LINPACK benchmark on the AP 1000, Proceedings of Frontiers '92 (McLean, Virginia, October 1992), IEEE Press, 1992, 128-135. http://www.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub130.html
5. R. P. Brent, Parallel algorithms in linear algebra, Algorithms and Architectures: Proc. Second NEC Research Symposium held at Tsukuba, Japan, August 1991 (edited by T. Ishiguro), SIAM, Philadelphia, 1993, 54-72. http://www.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub128.html
6. R. P. Brent and P. E. Strazdins, Implementation of the BLAS level 3 and Linpack benchmark on the AP 1000, Fujitsu Scientific and Technical Journal 29, 1 (March 1993), 61-70. http://www.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub136.html
7. R. P. Brent, Factorization of the tenth Fermat number, Math. Comp. 68 (1999), 429-451. Preliminary version available as Factorization of the tenth and eleventh Fermat numbers, Technical Report TR-CS-96-02, CSL, ANU, Feb. 1996, 25 pp. http://www.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub161.html.
8. R. P. Brent, Some parallel algorithms for integer factorisation. Proc. Europar’99, Toulouse, Sept. 1999. LNCS 1685, Springer-Verlag, Berlin, 1-22.
9. R. P. Brent, Large factors found by ECM, Oxford University Computing Laboratory, March 2000. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/champs.txt.
10. n ± 1,b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, American Mathematical Society, Providence, Rhode Island, second edition, 1988. Updates available from http://www/cs/purdue.edu/homes/ssw/cun/index.html.
11. T. R. Caron and R. D. Silverman, Parallel implementation of the quadratic sieve, J. Supercomputing 1 (1988), 273-290.
12. S. Cavallar, B. Dodson, A. K. Lenstra, P. Leyland, W. Lioen, P. L. Montgomery, B. Murphy, H. te Riele and P. Zimmermann, Factorization of RSA-140 using the number field sieve, announced 4 February 1999. Available from ftp://ftp.cwi.nl/pub/herman/NFSrecords/RSA-140.
13. S. Cavallar, B. Dodson, A. K. Lenstra, P. Leyland, W. Lioen, P. L. Montgomery, H. te Riele and P. Zimmermann, 211-digit SNFS factorization, announced 25 April 1999. Available from ftp://ftp.cwi.nl/pub/herman/NFSrecords/SNFS-211.
14. D. V. and G. V. Chudnovsky, Sequences of numbers generated by addition in formal groups and new primality and factorization tests, Adv. in Appl. Math. 7 (1986), 385-434.
15. H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, Berlin, 1993.
16. S. Contini, The factorization of RSA-140, RSA Laboratories Bulletin 10, 8 (March 1999). Available from http://www.rsa.com/rsalabs/html/bulletins.html.
17. D. Coppersmith, A. Odlyzko and R. Schroeppel, Discrete logarithms in GF(p), Algorithmica 1 (1986), 1-15.
18. 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 Cryptology: Proc. Asiacrypt’96, LNCS 1163, Springer-Verlag, Berlin, 1996, 382-394.
19. R. Crandall and B. Fagin, Discrete weighted transforms and large-integer arithmetic, Math. Comp. 62 (1994), 305-324.
20. D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Roy. Soc. London, Ser. A 400 (1985), 97-117.
21. D. Deutsch, Quantum computational networks, Proc. Roy. Soc. London, Ser. A 425 (1989), 73-90.
22. M. Elkenbracht-Huizing, A multiple polynomial general number field sieve Algorithmic Number Theory - ANTS III, LNCS 1443, Springer-Verlag, Berlin, 1998, 99-114.
23. K. F. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, 1982.
24. D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison Wesley, third edition, 1997.
25. B. A. LaMacchia and A. M. Odlyzko, Solving large sparse systems over finite fields, Advances in Cryptology, CRYPTO '90 (A. J. Menezes and S. A. Vanstone, eds.), LNCS 537, Springer-Verlag, Berlin, 109-133.
26. C. Lanczos, Solution of systems of linear equations by minimized iterations, J. Res. Nat. Bureau of Standards 49 (1952), 33-53.
27. S. Lang, Elliptic Curves - Diophantine Analysis, Springer-Verlag, Berlin, 1978.
28. R. S. Lehman, Factoring large integers, Math. Comp. 28 (1974), 637-646.
29. A. K. Lenstra and H. W. Lenstra, Jr. (editors), The development of the number field sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, Berlin, 1993.
30. A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse and J. M. Pollard, The number field sieve, Proc. 22nd Annual ACM Conference on Theory of Computing, Baltimore, Maryland, May 1990, 564-572.
31. A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319-349.
32. A. K. Lenstra and M. S. Manasse, Factoring by electronic mail, Proc. Eurocrypt '89, LNCS 434, Springer-Verlag, Berlin, 1990, 355-371.
33. A. K. Lenstra and M. S. Manasse, Factoring with two large primes, Math. Comp. 63 (1994), 785-798.
34. H. W. Lenstra, Jr., Factoring integers with elliptic curves, Annals of Mathematics (2) 126 (1987), 649-673.
35. P. L. Montgomery, A survey of modern integer factorization algorithms, CWI Quarterly 7 (1994), 337-366. ftp://ftp.cwi.nl/pub/pmontgom/cwisurvey.psl.Z.
36. P. L. Montgomery, Square roots of products of algebraic numbers, Mathematics of Computation 1943-1993, Proc. Symp. Appl. Math. 48 (1994), 567-571.
37. P. L. Montgomery, A block Lanczos algorithm for finding dependencies over GF(2), Advances in Cryptology: Proc. Eurocrypt’95, LNCS 921, Springer-Verlag, Berlin, 1995, 106-120. ftp://ftp.cwi.nl/pub/pmontgom/BlockLanczos.psa4.gz.
38. P. L. Montgomery, Parallel block Lanczos, Microsoft Research, Redmond, USA, 17 January 2000 (transparencies of a talk presented at RSA 2000).
39. M. A. Morrison and J. Brillhart, A method of factorisation and the factorisation of F7, Math. Comp. 29 (1975), 183-205.
40. B. A. Murphy, Modelling the yield of number field sieve polynomials, Algorithmic Number Theory - ANTS III, LNCS 1443, Springer-Verlag, Berlin, 1998, 137-150.
41. B. A. Murphy, Polynomial selection for the number field sieve integer factorisation algorithm, Ph. D. thesis, Australian National University, July 1999.
42. B. A. Murphy and R. P. Brent, On quadratic polynomials for the number field sieve, Australian Computer Science Communications 20 (1998), 199-213. http://www.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub178.html.
43. A. M. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, Advances in Cryptology: Proc. Eurocrypt '84, LNCS 209, Springer-Verlag, Berlin, 1985, 224-314.
44. A. M. Odlyzko, The future of integer factorization, CryptoBytes 1, 2 (1995), 5-12. Available from http://www.rsa.com/rsalabs/pubs/cryptobytes.
45. J. M. Pollard, A Monte Carlo method for factorisation, BIT 15 (1975), 331-334.
46. C. Pomerance, The quadratic sieve factoring algorithm, Advances in Cryptology, Proc. Eurocrypt '84, LNCS 209, Springer-Verlag, Berlin, 1985, 169-182.
47. C. Pomerance, The number field sieve, Proceedings of Symposia in Applied Mathematics 48, Amer. Math. Soc., Providence, Rhode Island, 1994, 465-480.
48. C. Pomerance, A tale of two sieves, Notices Amer. Math. Soc. 43 (1996), 1473-1485.
49. C. Pomerance, J. W. Smith and R. Tuler, A pipeline architecture for factoring large integers with the quadratic sieve algorithm, SIAM J. on Computing 17 (1988), 387-403.
50. J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation, California Institute of Technology, Los Angeles, Sept. 1998. http://www.theory.caltech.edu/people/preskill/ph229/.
51. H. te Riele et al, Factorization of a 512-bits RSA key using the number field sieve, announcement of 26 August 1999, http://www.loria.fr/~zimmerma/records/RSA155.
52. H. J. J. te Riele, W. Lioen and D. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267-278.
53. H. Riesel, Prime numbers and computer methods for factorization, 2nd edition, Birkhäuser, Boston, 1994.
54. R. L. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 (1978), 120-126.
55. RSA Laboratories, Information on the RSA challenge, http://www.rsa.com/rsalabs/html/challenges.html.
56. R. S. Schaller, Moore’s law: past, present and future, IEEE Spectrum 34, 6 (June 1997), 52-59.
57. P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, California, 1994, 124-134. CMP 98: 06
58. P. W. Shor, Polynomial time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Computing 26 (1997), 1484-1509.
59. R. D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), 329-339.
60. U. Vazirani, Introduction to special section on quantum computation, SIAM J. Computing 26 (1997), 1409-1410.
61. D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54-62.
62. J. Zayer, Faktorisieren mit dem Number Field Sieve, Ph. D. thesis, Universität des Saarlandes, 1995.
63. P. Zimmermann, The ECMNET Project, http://www.loria.fr/~zimmerma/records/ecmnet.html.