A short basis of the Stickelberger ideal of a cyclotomic field

Abstract:

We exhibit an explicit short basis of the Stickelberger ideal of cyclotomic fields of any conductor $m$, i.e., a basis containing only short elements. An element $\sum _{\sigma \in G_m} \varepsilon _{\sigma }\sigma$ of the group ring $\mathbb {Z}[G_{m}]$, where $G_m$ is the Galois group of the field, is said to be short if all of its coefficients $\varepsilon _{\sigma }$ are $0$ or $1$.

As a direct practical consequence, we deduce from this short basis an explicit upper bound on the relative class number that is valid for any conductor. This basis also has several concrete applications, in particular for the cryptanalysis of the Shortest Vector Problem on Ideal Lattices.