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(n³ (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 (n³ log ||A||)^1+o(1) bit operations, where the exponent “+ o(1)” captures additional factors c₁ (log n)^c2 (loglog||A||)^c3 for positive real constants c₁,c₂,c₃.

给出了拉斯维加斯随机算法来计算维度为n的非奇异整数矩阵A的 Hermite 范式。该算法使用二次整数乘法和三次矩阵乘法，运行时间受O(n ³ (log n + log ||A||) ² (log n ) ² ) 位运算限制，其中 || A ||= 最大_ij | A _ij | 表示A的绝对值中最大的条目。给出了使用伪线性整数乘法的算法的变体，其运行时间为(n ³ log || A ||) ^{1+ o (1)}位运算，其中指数“ + o (1)”捕获附加因子c ₁ (log n ) ^c2 (loglog|| A ||) ^c3为正实常数c ₁ ,c ₂ ,c ₃。