Improved Stage 2 to P ± 1 Factoring Algorithms

  • Peter L. Montgomery
  • Alexander Kruppa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5011)


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.


Integer factorization convolution discrete Fourier transform number theoretic transform P–1 P+1 multipoint polynomial evaluation reciprocal polynomials 


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  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  2. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bernstein, D.J., Sorenson, J.P.: Modular exponentiation via the explicit Chinese remainder theorem. Math. Comp. 76, 443–454 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Crandall, R., Fagin, B.: Discrete weighted transforms and large-integer arithmetic. Math. Comp. 62, 305–324 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Granlund, T.: GNU MP: The GNU Multiple Precision Arithmetic Library,
  6. 6.
    Montgomery, P.L.: Modular multiplication without trial division. Math. Comp. 44, 519–521 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comp. 48, 243–264 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Montgomery, P.L., Silverman, R.D.: An FFT extension to the P − 1 factoring algorithm. Math. Comp. 54, 839–854 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Montgomery, P.L.: An FFT Extension to the Elliptic Curve Method of Factorization. UCLA dissertation (1992),
  10. 10.
    Nussbaumer, H.J.: Fast Fourier Transform and convolution algorithms, 2nd edn. Springer, Heidelberg (1982)Google Scholar
  11. 11.
    Pollard, J.M.: Theorems on factorization and primality testing. Proc. Cambridge Philosophical Society 76, 521–528 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Williams, H.C.: A p + 1 method of factoring. Math. Comp. 39, 225–234 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Peter L. Montgomery
    • 1
  • Alexander Kruppa
    • 2
  1. 1.Microsoft Research, One Microsoft WayRedmondUSA
  2. 2.LORIA, Campus ScientifiqueVandœuvre-lès-Nancy CedexFrance

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