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 $s=1$.

翻译：2016年，Cramer、Ducas、Peikert和Regev提出了一种高效算法，用于在$q$为素数幂的$q$次分圆域中恢复主理想的短生成元。本文中，我们借助狄利克雷$L$函数在$s=1$处的负二次矩，改进了他们在广义黎曼假设下、且$q$为素数时对对数分圆单位格对偶基的分析。作为推论，我们在此特殊情形下得到了该算法成功概率的更好下界。为了证明主要结果，我们还分析了狄利克雷$L$函数在$s=1$处负$2k$次矩的渐近行为。