Generating random factored ideals in number fields

Charles, Zachary

doi:10.1090/mcom/3283

Generating random factored ideals in number fields
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by Zachary Charles;

Math. Comp. 87 (2018), 2047-2056

DOI: https://doi.org/10.1090/mcom/3283

Published electronically: October 17, 2017

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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 $N$ 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 $N$, in randomized polynomial time. We also present a variant of this algorithm for generating random factored ideals in function fields.

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