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. By definition, an element of $\mathbb{Z}[G_m]$, where $G_m$ denotes the Galois group of the field, is called short whenever it writes as ${\sum }_{\sigma \in G_m}{\epsilon }_{\sigma }\sigma$ with all ${\epsilon }_{\sigma }\in \{0,1\}$. One ingredient for building such a basis consists in picking wisely generators ${\alpha }_m(b)$ in a large family of short elements. 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.

Keywords: Cyclotomic fields, Stickelberger ideal, short basis, relative class number. 20 pages