The number field sieve (original) (raw)
The second author was supported by NSF under Grant No. DMS-9002939 and by NSA/MSP under Grant No. MDA90-H-4043.
References
- L.M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC), New Orleans, May 6–8, 1991, 64–71.
Google Scholar - D. J. Bernstein, A. K. Lenstra, A general number field sieve implementation, this volume, pp. 103–126.
Google Scholar - J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, S. S. Wagstaff, Jr., Factorizations of b n ± 1, b=2, 3, 5, 6, 7, 10, 11, 12 up to high powers, second edition, Contemp. Math. 22, Amer. Math. Soc., Providence, 1988.
MATH Google Scholar - J. Buchmann, Complexity of algorithms in algebraic number theory, in: R. A. Mollin (ed.), Proceedings of the first conference of the Canadian Number Theory Association, De Gruyter, Berlin, 1990, 37–53.
Google Scholar - J. Buchmann, H. W. Lenstra, Jr., Approximating rings of integers in number fields, in preparation.
Google Scholar - J. Buchmann, V. Shoup, Constructing nonresidues in finite fields and the extended Riemann hypothesis, in preparation. Extended abstract: Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC), New Orleans, May 6–8, 1991, 72–79.
Google Scholar - J. P. Buhler, H. W. Lenstra, Jr., C. Pomerance, Factoring integers with the number field sieve, this volume, pp. 50–94.
Google Scholar - H. Cohen, A course in computational algebraic number theory, Springer-Verlag, to appear.
Google Scholar - H. Cohen, H. W. Lenstra, Jr., Heuristics on class groups, pp. 33–62 in: H. Jager (ed.), Number theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer-Verlag, Heidelberg.
Google Scholar - D. Coppersmith, Fast evaluations of logarithms in fields of characteristic 2, IEEE Trans. Inform. Theory 30 (1984), 587–594.
Article MathSciNet MATH Google Scholar - D. Coppersmith, Modifications to the number field sieve, J. Cryptology, to appear; IBM Research Report RC 16264, 1990.
Google Scholar - D. Coppersmith, Solving linear equations over GF(2): block Lanczos algorithm, Linear Algebra Appl., to appear; IBM Research Report RC 16997, 1991.
Google Scholar - D. Coppersmith, Solving linear equations over GF(2) II: block Wiedemann algorithm, Math. Comp., to appear; IBM Research Report RC 17293, 1991.
Google Scholar - D. Coppersmith, A. M. Odlyzko, R. Schroeppel, Discrete logarithms in GF(p), Algorithmica 1 (1986), 1–15.
Article MathSciNet MATH Google Scholar - J.-M. Couveignes, Computing a square root for the number field sieve, this volume, pp. 95–102.
Google Scholar - J. D. Dixon, Asymptotically fast factorization of integers, Math. Comp. 36 (1981), 255–260.
Article MathSciNet MATH Google Scholar - T. ElGamal, A subexponential-time algorithm for computing discrete logarithms over GF(p 2), IEEE Trans. Inform. Theory 31 (1985), 473–481.
Article MathSciNet MATH Google Scholar - D. M. Gordon, Discrete logarithms in GF(p) using the number field sieve, SIAM J. Discrete Math. 6 (1993), 124–138.
Article MathSciNet MATH Google Scholar - D. M. Gordon, K. S. McCurley, Massively parallel computation of discrete logarithms, Advances in cryptology, Crypto '92, to appear.
Google Scholar - J. L. Hafner, K. S. McCurley, Asymptotically fast triangularization of matrices over rings, SIAM J. Comput. 20 (1991), 1068–1083.
Article MathSciNet MATH Google Scholar - D. E. Knuth, The art of computer programming, volume 2, Seminumerical algorithms, second edition, Addison-Wesley, Reading, Massachusetts, 1981.
MATH Google Scholar - B. A. LaMacchia, A. M. Odlyzko, Solving large sparse systems over finite fields, Advances in cryptology, Crypto '90, Lecture Notes in Comput. Sci. 537 (1991), 99–129.
MATH Google Scholar - B. A. LaMacchia, A. M. Odlyzko, Computation of discrete logarithms in prime fields, Designs, Codes and Cryptography 1 (1991), 47–62.
Article MathSciNet MATH Google Scholar - S. Lang, Algebra, third edition, Addison-Wesley, Reading, Massachusetts, 1993.
MATH Google Scholar - 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, 1990.
Google Scholar - A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), to appear.
Google Scholar - A. K. Lenstra, M. S. Manasse, Factoring by electronic mail, Advances in cryptology, Eurocrypt '89, Lecture Notes in Comput. Sci. 434 (1990), 355–371.
Article MathSciNet Google Scholar - A. K. Lenstra, M. S. Manasse, Factoring with two large primes, Math. Comp., to appear.
Google Scholar - H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.
Article MathSciNet MATH Google Scholar - H. W. Lenstra, Jr., Algorithms in algebraic number theory, Bull. Amer. Math. Soc. 26 (1992), 211–244.
Article MathSciNet MATH Google Scholar - H. W. Lenstra, Jr., C. Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc. 5 (1992), 483–516.
Article MathSciNet MATH Google Scholar - M. A. Morrison, J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp. 29 (1975), 183–205.
MathSciNet MATH Google Scholar - M. Pohst, H. Zassenhaus, Algorithmic algebraic number theory, Cambridge University Press, Cambridge, 1989.
Book MATH Google Scholar - C. Pomerance, Analysis and comparison of some integer factoring algorithms, pp. 89–139 in: H. W. Lenstra, Jr., R. Tijdeman (eds), Computational methods in number theory, Math. Centre Tracts 154/155, Mathematisch Centrum, Amsterdam, 1983.
Google Scholar - C. Pomerance, Fast, rigorous factorization and discrete logarithm algorithms, in: D. S. Johnson, T. Nishizeki, A. Nozaki, H. S. Wilf (eds), Discrete algorithms and complexity, Academic Press, Orlando, 1987, 119–143.
Google Scholar - C. Pomerance (ed.), Cryptology and computational number theory, Proc. Sympos. Appl. Math. 42, Amer. Math. Soc., Providence, 1990.
MATH Google Scholar - C. Pomerance, J. W. Smith, Reduction of huge, sparse matrices over finite fields via created catastrophes, Experiment. Math. 1 (1992), 89–94.
Article MathSciNet MATH Google Scholar - C. P. Schnorr, Refined analysis and improvements on some factoring algorithms, J. Algorithms 3 (1982), 101–127.
Article MathSciNet MATH Google Scholar - O. Schirokauer, On pro-finite groups and on discrete logarithms, Ph.D. thesis, University of California, Berkeley, 68 pages, May 1992.
Google Scholar - I. N. Stewart, D. O. Tall, Algebraic number theory, second edition, Chapman and Hall, London, 1987.
MATH Google Scholar - B. Vallée, Generation of elements with small modular squares and provably fast integer factoring algorithms, Math. Comp. 56 (1991), 823–849.
Article MathSciNet MATH Google Scholar - D. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54–62.
Article MathSciNet MATH Google Scholar - H. G. Zimmer, Computational problems, methods and results in algebraic number theory, Lecture Notes in Math. 262, Springer-Verlag, Berlin, 1972.
MATH Google Scholar