Some parallel algorithms for integer factorisation

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

Volumn 1685 LNCS, Issue , 1999, Pages 1-22

  Brent, Richard P ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

PUBLIC KEY CRYPTOGRAPHY;

COMPOSITE NUMBERS; CURRENT CAPABILITY; MOORE'S LAW; PARALLEL MACHINE; PRACTICAL IMPORTANCE; PRIME FACTORS; PUBLIC KEY CRYPTOSYSTEMS;

FACTORIZATION;

EID: 84878646831     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/3-540-48311-x_1     Document Type: Conference Paper

References (92)

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. A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp. 61 (1993), 29-68. Programs available from ftp://ftp.inria.fr/INRIA/ ecpp.V3.4.1.tar.Z.
3. H. Boender and H. J. J. te Riele, Factoring integers with large prime variations of the quadratic sieve, Experimental Mathematics, 5 (1996), 257-273.
4. R. P. Brent, A Fortran multiple-precision arithmetic package, ACM Transactions on Mathematical Software 4 (1978), 57-70.
5. R. P. Brent, An improved Monte Carlo factorisation algorithm, BIT 20 (1980), 176-184.
6. R. P. Brent, Some integer factorisation algorithms using elliptic curves, Australian Computer Science Communications 8 (1986), 149-163. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/rpb102.dvi.gz.
7. R. P. Brent, Parallel algorithms for integer factorisation, in Number Theory and Cryptography (edited by J. H. Loxton), London Mathematical Society Lecture Note Series 154, Cambridge University Press, 1990, 26-37.
8. R. P. Brent, Vector and parallel algorithms for integer factorisation, Proceedings Third Australian Supercomputer Conference University of Melbourne, December 1990, 12 pp. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/ Richard.Brent/rpb122.dvi.gz.
9. R. P. Brent, Large factors found by ECM, Oxford University Computing Laboratory, May 1999. ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/ Richard.Brent/champs.txt.
10. R. P. Brent, Factorization of the tenth Fermat number, Math. Comp. 68 (1999), 429-451.
11. Preliminary version available as Factorization of the tenth and eleventh Fermat numbers, Technical Report TR-CS-96-02, CSL, ANU, Feb. 1996, 25pp. ftp://ftp.comlab.ox.ac.uk:/pub/Documents/techpapers/Richard.Brent/rpb161tr.dvi. gz.
12. R. P. Brent and J. M. Pollard, Factorisation of the eighth Fermat number, Math. Comp. 36 (1981), 627-630.
13. 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.
14. D. A. Buell, Factoring: algorithms, computations, and computers, J. Supercomputing 1 (1987), 191-216.
15. C. Caldwell, The Dubner PC Cruncher - a microcomputer coprocessor card for doing integer arithmetic, review in J. Rec. Math. 25 (1), 1993.
16. T. R. Caron and R. D. Silverman, Parallel implementation of the quadratic sieve, J. Supercomputing 1 (1988), 273-290.
17. 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.
18. 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.
19. 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.
20. H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, Berlin, 1993.
21. H. Cohen and H. W. Lenstra, Jr., Primality testing and Jacobi sums, Math. Comp. 42 (1984), 297-330.
22. S. Contini, The factorization of RSA-140, RSA Laboratories Bulletin 10, 8 (March 1999). Available from http://www.rsa/com/rsalabs/html/bulletins.html.
23. 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. (Pubitemid 126146241)
24. R. E. Crandall, Parallelization of Pollard-rho factorization, preprint, 23 April 1999.
25. R. Crandall and B. Fagin, Discrete weighted transforms and large-integer arithmetic, Math. Comp. 62 (1994), 305-324.
26. D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Roy. Soc. London, Ser. A 400 (1985), 97-117. (Pubitemid 15563359)
27. D. Deutsch, Quantum computational networks, Proc. Roy. Soc. London, Ser. A 425 (1989), 73-90.
28. W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory 22 (1976), 472-492.
29. B. Dixon and A. K. Lenstra, Massively parallel elliptic curve factoring, Proc. Eurocrypt '92, LNCS 658, Springer-Verlag, Berlin, 1993, 183-193.
30. B. Dodson and A. K. Lenstra, NFS with four large primes: an explosive experiment, Proc. Crypto'95, LNCS 963, Springer-Verlag, Berlin, 1995, 372-385.
31. T. El Gamal, A public-key cryptosystem and a signature scheme based on discrete logarithms, Advances in Cryptology: Proc. CRYPTO'84, Springer-Verlag, Berlin, 1985, 10-18.
32. T. El Gamal, A public-key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. on Information Theory 31 (1985), 469-472.
33. M. Elkenbracht-Huizing, An implementation of the number field sieve, Experimental Mathematics, 5 (1996), 231-253.
34. M. Elkenbracht-Huizing, Factoring integers with the number field sieve, Doctor's thesis, Leiden University, 1997.
35. M. Elkenbracht-Huizing, A multiple polynomial general number field sieve Algorithmic Number Theory - ANTS III, LNCS 1443, Springer-Verlag, Berlin, 1998, 99-114.
36. K. F. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, 1982.
37. D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison Wesley, third edition, 1997.
38. N. Koblitz, A Course in Number Theory and Cryptography, Springer-Verlag, New York, 1994.
39. S. Lang, Elliptic Curves - Diophantine Analysis, Springer-Verlag, Berlin, 1978.
40. R. S. Lehman, Factoring large integers, Math. Comp. 28 (1974), 637-646.
41. 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.
42. 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.
43. 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.
44. A. K. Lenstra and M. S. Manasse, Factoring by electronic mail, Proc. Eurocrypt '89, LNCS 434, Springer-Verlag, Berlin, 1990, 355-371.
45. A. K. Lenstra and M. S. Manasse, Factoring with two large primes, Math. Comp. 63 (1994), 785-798.
46. H. W. Lenstra, Jr., Factoring integers with elliptic curves, Annals of Mathematics (2) 126 (1987), 649-673.
47. K. S. McCurley, The discrete logarithm problem, in Cryptography and Computational Number Theory, C. Pomerance, ed., Proc. Symp. Appl. Math., Amer. Math. Soc., 1990.
48. A. Menezes, Elliptic Curve Public Key Cryptosystems, Kluwer Academic Publishers, Boston, 1993.
49. A. Menezes, Elliptic curve cryptosystems, CryptoBytes 1, 2 (1995), 1-4. Available from http://www.rsa.com/rsalabs/pubs/cryptobytes.
50. P. L. Montgomery, Modular multiplication without trial division, Math. Comp. 44 (1985), 519-521.
51. P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorisation, Math. Comp. 48 (1987), 243-264.
52. P. L. Montgomery, An FFT extension of the elliptic curve method of factorization, Ph. D. dissertation, Mathematics, University of California at Los Angeles, 1992. ftp://ftp.cwi.nl/pub/pmontgom/ucladissertation.psl.Z.
53. 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.
54. P. L. Montgomery, Square roots of products of algebraic numbers, Mathematics of Computation 1943-1993, Proc. Symp. Appl. Math. 48 (1994), 567-571.
55. 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.
56. F. Morain, Courbes elliptiques et tests de primalité, Ph. D. thesis, Univ. Claude Bernard - Lyon I, France, 1990. ftp://ftp.inria.fr/INRIA/ publication/Theses/TU-0144.tar.Z.
57. 7, Math. Comp. 29 (1975), 183-205.
58. R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.
59. B. A. Murphy, Modelling the yield of number field sieve polynomials, Algorithmic Number Theory - ANTS III, LNCS 1443, Springer-Verlag, Berlin, 1998, 137-150.
60. B. A. Murphy, Polynomial selection for the number field sieve integer factorisation algorithm, Ph. D. thesis, Australian National University, 1999.
61. B. A. Murphy and R. P. Brent, On quadratic polynomials for the number field sieve, Australian Computer Science Communications 20 (1998), 199-213.
62. 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.
63. A. M. Odlyzko, The future of integer factorization, CryptoBytes 1, 2 (1995), 5-12. Available from http://www.rsa.com/rsalabs/pubs/cryptobytes.
64. P. C. van Oorschot and M. J. Wiener, Parallel collision search with application to hash functions and discrete logarithms, Proc 2nd ACM Conference on Computer and Communications Security, ACM, New York, 1994, 210-218.
65. P. C. van Oorschot and M. J. Wiener, Parallel collision search with cryptanalytic applications, J. Cryptology 12 (1999), 1-28.
66. J. M. Pollard, Theorems in factorisation and primality testing, Proc. Cambridge Philos. Soc. 76 (1974), 521-528.
67. J. M. Pollard, A Monte Carlo method for factorisation, BIT 15 (1975), 331-334.
68. C. Pomerance, The quadratic sieve factoring algorithm, Advances in Cryptology, Proc. Eurocrypt '84, LNCS 209, Springer-Verlag, Berlin, 1985, 169-182.
69. C. Pomerance, The number field sieve, Proceedings of Symposia in Applied Mathematics 48, Amer. Math. Soc., Providence, Rhode Island, 1994, 465-480.
70. C. Pomerance, A tale of two sieves, Notices Amer. Math. Soc. 43 (1996), 1473-1485.
71. 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.
72. 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/.
73. M. O. Rabin, Probabilistic algorithms for testing primality, J. Number Theory 12 (1980), 128-138.
74. 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.
75. H. Riesel, Prime numbers and computer methods for factorization, 2nd edition, Birkhäuser, Boston, 1994.
76. R. L. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM 21 (1978), 120-126. (Pubitemid 8591219)
77. RSA Laboratories, Information on the RSA challenge, http://www.rsa/com/ rsalabs/html/challenges.html.
78. B. Schneier, Applied Cryptography, second edition, John Wiley and Sons, 1996.
79. A. Shamir, Factoring large numbers with the TWINKLE device (extended abstract), preprint, 1999. Announced at Eurocrypt'99.
80. 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
81. P. W. Shor, Polynomial time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Computing 26 (1997), 1484-1509. (Pubitemid 127613100)
82. J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986.
83. R. D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), 329-339.
84. R. D. Silverman and S. S. Wagstaff, Jr., A practical analysis of the elliptic curve factoring algorithm, Math. Comp. 61 (1993), 445-462.
85. I. N. Stewart and D. O. Tall, Algebraic Number Theory, second edition, Chapman and Hall, 1987.
86. D. Stinson, Cryptography - Theory and Practice, CRC Press, Boca Raton, 1995.
87. A. M. Turing, On computable numbers, with an application to the Entschei-dungsproblem, Proc. London Math. Soc. (2) 42 (1936), 230-265. Errata ibid 43 (1937), 544-546.
88. U. Vazirani, Introduction to special section on quantum computation, SIAM J. Computing 26 (1997), 1409-1410.
89. D. Weber, Computing discrete logarithms with the number field sieve, Algorithmic Number Theory - ANTS II, LNCS 1122, Springer-Verlag, Berlin, 1996, 99-114.
90. D. Weber, On the computation of discrete logarithms in finite prime fields, Ph. D. thesis, Universität des Saarlandes, 1997.
91. D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54-62.
92. J. Zayer, Faktorisieren mit dem Number Field Sieve, Ph. D. thesis, Universität des Saarlandes, 1995.