Recovering short generators via negative moments of Dirichlet $L$-functions

In 2016, Cramer, Ducas, Peikert and, Regev proposed an efficient algorithm for recovering short generators of principal ideals in $q$-th cyclotomic fields with $q$ being a prime power. In this paper, we improve their analysis of the dual basis of the log-cyclotomic-unit lattice under the Generalised Riemann Hypothesis and in the case that $q$ is a prime number by the negative square moment of Dirichlet $L$-functions at $s=1$. As an implication, we obtain a better lower bound on the success probability for the algorithm in this special case. In order to prove our main result, we also give an analysis of the behaviour of negative $2k$-th moments of Dirichlet $L$-functions at $s=1$.