Generating random factored ideals in number fields

Abstract

We present a randomized polynomial-time algorithm to generate an ideal and its factorization uniformly at random in a given number field. We do this by generating a random integer and its factorization according to the distribution of norms of ideals at most N N in the given number field. Using this randomly generated norm, we can produce a random factored ideal in the ring of algebraic integers uniformly at random among ideals with norm up to N N , in randomized polynomial time. We also present a variant of this algorithm for generating random factored ideals in function fields.