Optimization Strategies for Hardware-Based Cofactorization

  • Daniel Loebenberger
  • Jens Putzka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5867)


We use the specific structure of the inputs to the cofactorization step in the general number field sieve (GNFS) in order to optimize the runtime for the cofactorization step on a hardware cluster. An optimal distribution of bitlength-specific ECM modules is proposed and compared to existing ones. With our optimizations we obtain a speedup between 17% and 33% of the cofactorization step of the GNFS when compared to the runtime of an unoptimized cluster.


General Number Field Sieve (GNFS) Elliptic Curve Method (ECM) hardware cluster cofactorization step 


  1. 1.
    Bellman, R.: Dynamic Programming. Princeton University Text (1957)Google Scholar
  2. 2.
    Cohen, H.: A course in computational algebraic number theory. Springer, Berlin (1997)Google Scholar
  3. 3.
    Franke, J., Kleinjung, T.: RSA 640 (2005),
  4. 4.
    Franke, J., Kleinjung, T.: Continued Fractions and Lattice Sieving (Unpublished) (2006),
  5. 5.
    von zur Gathen, J., Güneysu, T., Kargl, A., Loebenberger, D., Paar, C., Putzka, J.: Faktorisierung großer Zahlen: Hardware für Elliptische Kurven Faktorisierung. Technical report, HGI Bochum, b-it Bonn & Siemens AG München (2007)Google Scholar
  6. 6.
    Kleinjung, T.: Cofactorisation Strategies for the Number Field Sieve and an Estimate for the Sieving Step for Factoring 1024-bit Integers (Unpublished) (2004),
  7. 7.
    Kleinjung, T.: On Polynomial Selection for the General Number Field Sieve. Mathematics of Computation 75(256), 2037–2047 (2006), zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kumar, S., Paar, C., Pelzl, J., Pfeiffer, G., Schimmler, M.: Breaking ciphers with COPACOBANA –A Cost-Optimized Parallel Code Breaker. In: Goubin, L., Matsui, M. (eds.) CHES 2006. LNCS, vol. 4249, pp. 101–118. Springer, Heidelberg (2006), CrossRefGoogle Scholar
  9. 9.
    Lenstra, A.K., Lenstra Jr., H.W. (eds.): The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Berlin (1993)zbMATHGoogle Scholar
  10. 10.
    Lenstra Jr., H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)Google Scholar
  11. 11.
    Pollard, J.M.: Theorems on factorization and primality testing. Proceedings of the Cambridge Philosophical Society 76, 521–528 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    RSA Laboratories. The RSA Challenge Numbers (2007) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Daniel Loebenberger
    • 1
  • Jens Putzka
    • 2
  1. 1.b-itBonn
  2. 2.MPI für MathematikBonn

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