  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 $\sum_{\sigma\in G_m}
\varepsilon_{\sigma}\sigma$ with all $\varepsilon_\sigma\in\{0,1\}$. 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.
