Asymptotically fast factorization of integers

Abstract

The paper describes a "probabilistic algorithm" for finding a factor of any large composite integer n (the required input is the integer n together with an auxiliary sequence of random numbers). It is proved that the expected number of operations which will be required is O ( exp ⁡ { β ( ln ⁡ n ln ⁡ ln ⁡ n ) 1 / 2 } ) O(\exp \{ \beta {(\ln n\ln \ln n)^{1/2}}\} ) for some constant β > 0 \beta > 0 . Asymptotically, this algorithm is much faster than any previously analyzed algorithm for factoring integers; earlier algorithms have all required O ( n α ) O({n^\alpha }) operations where α > 1 / 5 \alpha > 1/5 .