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Prime Numbers and Computer Methods for Factorization pp 173–225Cite as

Birkhäuser

Modern Factorization Methods

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Part of the book series: Progress in Mathematics ((PM,volume 126))

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.

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Bibliography

  1. J. M. Pollard, “Theorems on Factorization and Primality Testing,” Proc. Cambr. Philos. Soc. 76 (1974) pp. 521–528.

    Article  MathSciNet  MATH  Google Scholar 

  2. H. C. Williams. “A p +1 Method of Factoring,” Math. Comp. 39 (1982) pp. 225–234.

    MathSciNet  MATH  Google Scholar 

  3. J. M. Pollard, “A Monte Carlo Method for Factorization,” Nordisk Tidskrift för Informationsbehandling (BIT) 15 (1975) pp. 331–334.

    MathSciNet  MATH  Google Scholar 

  4. Richard P. Brent, “An Improved Monte Carlo Factorization Algorithm,” Nordisk Tidskrift för Informationsbehandling (BIT) 20 (1980) pp. 176–184.

    MathSciNet  MATH  Google Scholar 

  5. Richard P. Brent and J. M. Pollard, “Factorization of the Eighth Fermat Number,” Math. Comp. 36 (1981) pp. 627–630.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. F. Gauss, Disquisitiones Aritmeticae, Yale University Press, New Haven, 1966, Art. 329–332.

    Google Scholar 

  7. Duncan A. Buell, Binary Quadratic Forms. Classical Theory and Modern Computations, Springer-Verlag, New York, 1989.

    MATH  Google Scholar 

  8. Daniel Shanks, “Class Number, A Theory of Factorization, and Genera,” Amer. Math. Soc. Proc. Symposia in Pure Math. 20 (1971) pp. 415–440.

    MathSciNet  Google Scholar 

  9. Louis Monier, Algorithmes de Factorisations d’Entiers, IRIA, Paris, 1980, pp. 3.13–3.24.

    Google Scholar 

  10. 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.

    Google Scholar 

  11. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993.

    MATH  Google Scholar 

  12. Michael A. Morrison and John Brillhart, “A Method of Factoring and the Factorization of F 7,” Math. Comp. 29 (1975) pp. 183–205.

    MathSciNet  MATH  Google Scholar 

  13. D. H. Lehmer and R. E. Powers, “On Factoring Large Numbers,” Bull. Am. Math. Soc. 37 (1931) pp. 770–776.

    Article  MathSciNet  Google Scholar 

  14. Maurice Kraïtchik, Théorie des Nombres, Tome II, Gauthiers-Villars, Paris, 1926, pp. 195–208.

    MATH  Google Scholar 

  15. 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.

    Google Scholar 

  16. Carl Pomerance and Samuel S. Wagstaff, Jr., “Implementation of the Continued Fraction Integer Factoring Algorithm,” Congr. Num. 37 (1983) pp. 99–118.

    MathSciNet  Google Scholar 

  17. Thorkil Naur, Integer Factorization, DAIMI report, University of Aarhus, 1982.

    Google Scholar 

  18. Thorkil Naur, “New Integer Factorizations,” Math. Comp. 41 (1983) pp. 687–695.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. A. Mullin, “Recursive Function Theory,” Bull Am. Math. Soc. 69 (1963) p. 737.

    Article  Google Scholar 

  20. Carl Pomerance, “The Quadratic Sieve Factoring Algorithm,” in Advances of Cryptology, Proc. of Eurocrypt 84, Lecture Notes in Computer Sc. 209, Springer-Verlag, Berlin, 1985 pp. 169–182.

    Google Scholar 

  21. James A. Davis and Diane B. Holdridge, “Factorization Using the Quadratic Sieve Algorithm,” SANDIA report, SAND83-1346, SANDIA National Laboratories, Livermore, 1983.

    Google Scholar 

  22. R. D. Silverman, “The Multiple Polynomial Quadratic Sieve,” Math. Comp. 48 (1987) pp. 329–339.

    Article  MathSciNet  MATH  Google Scholar 

  23. D. Wiedemann, “Solving Sparse Linear Equations over Finite Fields,” IEEE Trans. Inform. Theory 32 (1986) pp. 54–62.

    Article  MathSciNet  MATH  Google Scholar 

  24. Brigitte Vallée, “Generation of Elements with Small Modular Squares and Provably Fast Integer Factoring Algorithms, Math. Comp. 56 (1991) pp. 823–849.

    Article  MathSciNet  MATH  Google Scholar 

  25. C. P. Schnorr and H. W. Lenstra, Jr., “A Monte Carlo Factoring Algorithm with Linear Storage,” Math. Comp. 43 (1984) pp. 289–311.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. W. Lenstra, Jr. “Factoring Integers with Elliptic Curves,” Ann. of Math. (2) 126 (1987) pp. 649–673.

    Article  MathSciNet  MATH  Google Scholar 

  27. Peter Montgomery, “Speeding the Pollard and Elliptic Curve Methods of Factorization,” Math. Comp. 48 (1987) pp. 243–264.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. O. L. Atkin and F. Morain, “Finding Suitable Curves for the Elliptic Curve Method of Factorization,” Math. Comp. 60 (1993) pp. 399–405.

    Article  MathSciNet  MATH  Google Scholar 

  29. Robert D. Silverman and Samuel S. Wagstaff, Jr. “A Practical Analysis of the Elliptic Curve Factoring Algorithm, Math. Comp. 61 (1993) pp. 445–462.

    Article  MathSciNet  MATH  Google Scholar 

  30. 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.

    Chapter  Google Scholar 

  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.

    Chapter  Google Scholar 

  32. 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.

    Article  MathSciNet  MATH  Google Scholar 

  33. 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.

    Chapter  Google Scholar 

  34. 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.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Hans Riesel

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  1. Hans Riesel
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© 1994 Springer Science+Business Media New York

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Riesel, H. (1994). Modern Factorization Methods. In: Prime Numbers and Computer Methods for Factorization. Progress in Mathematics, vol 126. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0251-6_6

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  • DOI: https://doi.org/10.1007/978-1-4612-0251-6_6

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6681-5

  • Online ISBN: 978-1-4612-0251-6

  • eBook Packages: Springer Book Archive

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