Generating subgroups of ray class groups with small prime ideals

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.

Mildly Short Vectors in Cyclotomic Ideal Lattices in Quantum Polynomial Time

“…This restriction on the norms seems reasonable considering that it has been proven that prime ideals of norm poly(m) that fall in Cl − K are sufficient to generate Cl − K , assuming GRH and Assumption 2 (see [JW15, Corollary 6.5]). Explicitly, is has been proved in [Wes18b] that prime ideals of norm at most (2.71h + K log ∆ K + 4.13) 2 are sufficient to generate Cl − K .…”