Generating subgroups of ray class groups with small prime ideals

TL;DR

This paper provides explicit bounds, assuming the Riemann Hypothesis, on prime ideals generating subgroups of ray class groups, with applications to cryptology, algebraic number theory, and elliptic curve isogenies.

Contribution

First explicit bounds for prime ideals generating subgroups of ray class groups under the Riemann Hypothesis, improving previous asymptotic estimates.

Findings

Prime ideals generating subgroups of ray class groups are bounded by 16(i log m)^2 under RH.

Bounds are applicable to cryptology, class groups, and isogeny graphs of Jacobians.

Results improve understanding of generators in algebraic number theory and cryptographic contexts.

Abstract

Explicit bounds are given on the norms of prime ideals generating arbitrary subgroups of ray class groups of number fields, assuming the Extended Riemann Hypothesis. These are the first explicit bounds for this problem, and are significantly better than previously known asymptotic bounds. Applied to the integers, they express that any subgroup of index $i$ of the multiplicative group of integers modulo $m$ is generated by prime numbers smaller than $16(i\log m)^2$, subject to the Riemann Hypothesis. Two particular consequences relate to mathematical cryptology. Applied to cyclotomic fields, they provide explicit bounds on generators of the relative class group, needed in some previous work on the shortest vector problem on ideal lattices. Applied to Jacobians of hyperelliptic curves, they allow one to derive bounds on the degrees of isogenies required to make their horizontal isogeny graphs connected. Such isogeny graphs are used to study the discrete logarithm problem on said Jacobians.