A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix $A$ of dimension $n$. The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by $O(n^3 (\log n + \log ||A||)^2(\log n)^2)$ bit operations, where $||A||= \max_{ij} |A_{ij}|$ denotes the largest entry of $A$ in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time $(n^3 \log ||A||)^{1+o(1)}$ bit operations, where the exponent $"+o(1)"$ captures additional factors $c_1 (\log n)^{c_2} (\log \log ||A||)^{c_3}$ for positive real constants $c_1,c_2,c_3$.

翻译：提出了一种拉斯维加斯随机算法，用于计算维度为$n$的非奇异整数矩阵$A$的埃尔米特标准型。该算法采用二次整数乘法与三次矩阵乘法，其运行时间在比特操作意义下受$O(n^3 (\log n + \log ||A||)^2(\log n)^2)$限制，其中$||A||= \max_{ij} |A_{ij}|$表示$A$中元素的绝对值最大值。还给出该算法的一个变体，采用伪线性整数乘法，其运行时间在比特操作意义下为$(n^3 \log ||A||)^{1+o(1)}$，其中指数部分"+o(1)"涵盖正实常数$c_1,c_2,c_3$所对应的额外因子$c_1 (\log n)^{c_2} (\log \log ||A||)^{c_3}$。