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Improved Stage 2 to P ± 1 Factoring Algorithms

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Algorithmic Number Theory (ANTS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5011))

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Abstract

Some implementations of stage 2 of the P–1 method of factorization use convolutions. We describe a space-efficient implementation, allowing convolution lengths around 223 and stage 2 limit around 1016 while attempting to factor 230-digit numbers on modern PC’s. We describe arithmetic algorithms on reciprocal polynomials. We present adjustments for the P+1 algorithm. We list some new findings.

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References

  1. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)

    MATH  Google Scholar 

  2. Baszenski, G., Tasche, M.: Fast polynomial multiplication and convolutions related to the discrete cosine transform. Linear Algebra and its Applications 252, 1–25 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bernstein, D.J., Sorenson, J.P.: Modular exponentiation via the explicit Chinese remainder theorem. Math. Comp. 76, 443–454 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Crandall, R., Fagin, B.: Discrete weighted transforms and large-integer arithmetic. Math. Comp. 62, 305–324 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Granlund, T.: GNU MP: The GNU Multiple Precision Arithmetic Library, http://gmplib.org/

  6. Montgomery, P.L.: Modular multiplication without trial division. Math. Comp. 44, 519–521 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comp. 48, 243–264 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Montgomery, P.L., Silverman, R.D.: An FFT extension to the P − 1 factoring algorithm. Math. Comp. 54, 839–854 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Montgomery, P.L.: An FFT Extension to the Elliptic Curve Method of Factorization. UCLA dissertation (1992), ftp://ftp.cwi.nl/pub/pmontgom

  10. Nussbaumer, H.J.: Fast Fourier Transform and convolution algorithms, 2nd edn. Springer, Heidelberg (1982)

    Google Scholar 

  11. Pollard, J.M.: Theorems on factorization and primality testing. Proc. Cambridge Philosophical Society 76, 521–528 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  12. Williams, H.C.: A p + 1 method of factoring. Math. Comp. 39, 225–234 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Zimmermann, P., Dodson, B.: 20 years of ECM. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 525–542. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA

    Peter L. Montgomery

  2. LORIA, Campus Scientifique, BP 239, 54506, Vandœuvre-lès-Nancy Cedex, France

    Alexander Kruppa

Authors
  1. Peter L. Montgomery
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  2. Alexander Kruppa
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Alfred J. van der Poorten Andreas Stein

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© 2008 Springer-Verlag Berlin Heidelberg

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Montgomery, P.L., Kruppa, A. (2008). Improved Stage 2 to P ± 1 Factoring Algorithms. In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_12

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  • DOI: https://doi.org/10.1007/978-3-540-79456-1_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79455-4

  • Online ISBN: 978-3-540-79456-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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