A Monte Carlo factoring algorithm with linear storage

Abstract

We present an algorithm which will factor an integer n quite efficiently if the class number h ( − n ) h( - n) is free of large prime divisors. The running time T ( n ) T(n) (number of compositions in the class group) satisfies prob ⁡ [ T ( m ) ⩽ n 1 / 2 r ] ≳ ( r − 2 ) − ( r − 2 ) \operatorname {prob}[T(m) \leqslant {n^{1/2r}}] \gtrsim {(r - 2)^{ - (r - 2)}} for random m ∈ [ n / 2 , n ] m \in [n/2,n] and r ⩾ 2 r \geqslant 2 . So far it is unpredictable which numbers will be factored fast. Running the algorithm on all discriminants - ns with s ⩽ r r s \leqslant {r^r} and r = ln ⁡ n / ln ⁡ ln ⁡ n r = \sqrt {\ln n/\ln \ln n} , every composite integer n will be factored in o ( exp ⁡ ln ⁡ n ln ⁡ ln ⁡ n ) o(\exp \sqrt {\ln n\ln \ln n} ) bit operations. The method requires an amount of storage space which is proportional to the length of the input n. In our analysis we assume a lower bound on the frequency of class numbers h ( − m ) h( - m) , m ⩽ n m \leqslant n , which are free of large prime divisors.