Fast square-free decomposition of integers using class groups

Let n = a 2 b , where b is square-free. In this paper we present an algorithm based on class groups of binary quadratic forms that finds the square-free decomposition of n , i.e. a and b , in heuristic expected time: O ~ ( L b [ 1 / 2 , 1 ] ln ( n ) + L b [ 1 / 2 , 1 / 2 ] ln ( n ) 2 ) . If a , b are both primes of roughly the same cryptographic size, then our method is currently the fastest known method to factor n . This has applications in cryptography, since some cryptosystems rely on the hardness of factoring integers of this form.