Elsevier

Journal of Algorithms

Volume 3, Issue 2, June 1982, Pages 101-127
Journal of Algorithms

Refined analysis and improvements on some factoring algorithms

https://doi.org/10.1016/0196-6774(82)90012-8Get rights and content

Abstract

By combining the principles of known factoring algorithms we obtain some improved algorithms which by heuristic arguments all have a time bound O(expc ln n ln lnn) for various constants c ⩾3. In particular, Miller's method of solving index equations and Shanks' method of computing ambiguous quadratic forms with discriminant −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], 0 ⪡ un 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)13(ln ln n)23] steps. We slightly improve Dixon's rigorous analysis of his Monte Carlo factoring algorithm. We prove that this algorithm with probability sol12 detects a proper factor of every composite n within o(exp6 ln n ln lnn) steps.

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    This work started in summer 1980 during a stay at the Stanford Computer Science Department. Preparation of this report was supported in part by National Science Foundation grant MCS-77-23738, and by the Bundesminister für Forschung und Technologie, Federal Republic of Germany, Grant 083 0108.

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