Abstract
The art of decomposing large integers into prime factors has advanced considerably during the last 25 years. It is the advent of high-speed computers that has rekindled interest in this field. This development has followed several lines. In one of these, already existing theoretical methods and known algorithms have been carefully analyzed and perfected. As an example of this work we mention Michael Morrison’s and John Brillhart’s analysis of an old factorization method, the continued fraction algorithm, going back to ideas introduced already by Legendre and developed further by Maurice Kraïtchik, D. H. Lehmer and R. E. Powers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
J. M. Pollard, “Theorems on Factorization and Primality Testing,” Proc. Cambr. Philos. Soc. 76 (1974) pp. 521–528.
H. C. Williams, “A p+1 Method of Factoring,” Math. Comp. 39 (1982) pp. 225–234.
J. M. Pollard, “A Monte Carlo Method for Factorization,” Nordisk Tidskriftför Infor-mationsbehandling (BIT) 15 (1975) pp. 331–334.
Richard P. Brent, “An Improved Monte Carlo Factorization Algorithm,” Nordisk Tidskriftför Informationsbehandling (BIT) 20 (1980) pp. 176–184.
Richard P. Brent and J. M. Pollard, “Factorization of the Eighth Fermat Number,” Math. Comp. 36 (1981) pp. 627–630.
C. F. Gauss, Disquisitiones Aritmeticae, Yale University Press, New Haven, 1966, Art. 329–332.
Duncan A. Buell, Binary Quadratic Forms. Classical Theory and Modern Computations, Springer-Verlag, New York, 1989.
Daniel Shanks, “Class Number, A Theory of Factorization, and Genera,” Amer. Math. Soc. Proc. Symposia in Pure Math. 20 (1971) pp. 415–440.
Louis Monier, Algorithmes de Factorisations d’Entiers, IRIA, Paris, 1980, pp. 3.13– 3.24.
R. J. Schoof, “Quadratic Fields and Factorization,” in H. W. Lenstra, Jr. and R. Tijdeman (eds.), Computational Methods in Number Theory, Part II, Mathematisch Centrum, Amsterdam, 1982, pp. 235–286.
Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993.
Michael A. Morrison and John Brillhart, “A Method of Factoring and the Factorization of F 7,” Math. Comp. 29 (1975) pp. 183 2013; 205.
D. H. Lehmer and R. E. Powers, “On Factoring Large Numbers,” Bull. Am. Math. Soc. 37 (1931) pp. 770–776.
Maurice Kraïtchik, Théorie des Nombres, Tome II, Gauthiers-Villars, Paris, 1926, pp. 195–208.
Carl Pomerance, “Analysis and Comparison of Some Integer Factoring Algorithms,” in H. W. Lenstra, Jr. and R. Tijdeman (eds.), Computational Methods in Number Theory, Part I, Mathematisch Centrum Tract 154, Amsterdam, 1982, pp. 89–139.
Carl Pomerance and Samuel S. Wagstaff, Jr., “Implementation of the Continued Fraction Integer Factoring Algorithm,” Congr. Num. 37 (1983) pp. 99–118.
Thorkil Naur, Integer Factorization, DAIMI report, University of Aarhus, 1982.
Thorkil Naur, “New Integer Factorizations;” Math. Comp. 41 (1983) pp. 687–695.
A. A. Mullin, “Recursive Function Theory,” Bull. Am. Math. Soc. 69 (1963) p. 737.
Carl Pomerance, “The Quadratic Sieve Factoring Algorithm,” in Advances of Cryp-tology, Proc. of Eurocrypt 84, Lecture Notes in Computer Sc. 209, Springer-Verlag, Berlin, 1985 pp. 169–182.
James A. Davis and Diane B. Holdridge, “Factorization Using the Quadratic Sieve Algorithm,” SANDIA report, SAND83–1346, SANDIA National Laboratories, Livermore, 1983.
R. D. Silverman, “The Multiple Polynomial Quadratic Sieve,” Math. Comp. 48 (1987) pp. 329–339.
D. Wiedemann, “Solving Sparse Linear Equations over Finite Fields,” IEEE Trans. Inform. Theory 32 (1986) pp. 54–62.
Brigitte Vallée, “Generation of Elements with Small Modular Squares and Provably Fast Integer Factoring Algorithms, Math. Comp. 56 (1991) pp. 823–849.
C. P. Schnorr and H. W. Lenstra, Jr., “A Monte Carlo Factoring Algorithm with Linear Storage,” Math. Comp. 43 (1984) pp. 289–311.
H. W. Lenstra, Jr. “Factoring Integers with Elliptic Curves,” Ann. of Math. (2) 126 (1987) pp. 649–673.
Peter Montgomery, “Speeding the Pollard and Elliptic Curve Methods of Factorization,” Math. Comp. 48 (1987) pp. 243–264.
A. O. L. Atkin and F. Morain, “Finding Suitable Curves for the Elliptic Curve Method of Factorization,” Math. Comp. 60 (1993) pp. 399–405.
Robert D. Silverman and Samuel S. Wagstaff, Jr. “A Practical Analysis of the Elliptic Curve Factoring Algorithm, Math. Comp. 61 (1993) pp. 445–462.
J. M. Pollard, “Factoring with Cubic Integers,” in A. K. Lenstra and H. W. Lenstra, Jr. (eds.), The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, New York, 1993, pp. 4–10.
31. J. P. Buhler, H. W. Lenstra, and C. Pomerance, “Factoring Integers with the Number Field Sieve,” in A. K. Lenstra and H. W. Lenstra, Jr. (eds.), The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, New York, 1993, pp. 50–94.
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) pp. 319–349.
Daniel J. Bernstein and A. K. Lenstra, “A General Number Field Sieve Implementation,” in A. K. Lenstra and H. W. Lenstra, Jr. (eds.), The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, New York, 1993, pp. 103– 126.
J. Buchmann, J. Loho and J. Zayer, “An Implementation of the General Number Field Sieve,” in Douglas R. Stinson (ed.), Advances in Cryptology CRYPTO ’93, Lecture Notes in Computer Sc. 773, Springer-Verlag, New York, 1994, pp. 159–165.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Riesel, H. (2011). Modern Factorization Methods. In: Prime Numbers and Computer Methods for Factorization. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8298-9_6
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8298-9_6
Published:
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-8297-2
Online ISBN: 978-0-8176-8298-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)