Refined analysis and improvements on some factoring algorithms

  • C. P. Schnorr
Session 1: A. Paz, Chairman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 115)


By combining the principles of known factoring algorithms we obtain some improved algorithms which by heuristic arguments all have a time bound O(exp √c ln n ln ln n) for various constants c≥3. In particular, Miller's method of solving index equations and Shanks's method of computing ambiguous quadratic forms with determinant −n can be modified in this way. We show how to speed up the factorization of n by using preprocessed lists of those numbers in [−u,u] and [n−u,n+u],O<<u<<n which only have small prime factors. These lists can be uniformly used for the factorization of all numbers in [n−u,n+u]. Given these lists, factorization takes O(exp[2 (ln n)1/3 (ln ln n)2/3]) steps. We slightly improve Dixon's rigorous analysis of his Monte Carlo factoring algorithm. We prove that this algorithm with probability 1/2 detects a proper factor of every composite n within o(exp √6ln n ln ln n) steps.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • C. P. Schnorr
    • 1
  1. 1.Fachbereich MathematikUniversität FrankfurtGermany

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