Rigorous Computation of Fundamental Units in Algebraic Number Fields

We present an algorithm that unconditionally computes a representation of the unit group of a number field of discriminant ${\mathrm{\Delta }}_K$, given a full-rank subgroup as input, in asymptotically fewer bit operations than the baby-step giant-step algorithm. If the input is assumed to represent the full unit group, for example, under the assumption of the Generalized Riemann Hypothesis, then our algorithm can unconditionally certify its correctness in expected time $O({\mathrm{\Delta }}_K^{n\mathrm{/}(4n+2)+ϵ})=O({\mathrm{\Delta }}_K^{1\mathrm{/}4-1\mathrm{/}(8n+4)+ϵ})$ where $n$ is the unit rank.