We present an algorithm that computes the product of two $n$-bit integers in $O(n \log n)$ bit operations, thus confirming a conjecture of Schönhage and Strassen from 1971. Our complexity analysis takes place in the multitape Turing machine model, with integers encoded in the usual binary representation. Central to the new algorithm is a novel “Gaussian resampling” technique that enables us to reduce the integer multiplication problem to a collection of multidimensional discrete Fourier transforms over the complex numbers, whose dimensions are all powers of two. These transforms may then be evaluated rapidly by means of Nussbaumer’s fast polynomial transforms.

我们提出了一种算法，该算法计算$ O（n \ log n）$位操作中两个$ n $位整数的乘积，从而确认了1971年以来Schönhage和Strassen的猜想。我们的复杂度分析发生在多带Turing机中模型，以通常的二进制表示形式编码整数。新算法的核心是一种新颖的“高斯重采样”技术，使我们能够将整数乘法问题简化为复数的多维离散傅里叶变换的集合，这些维的维数均为2的幂。然后可以借助Nussbaumer的快速多项式变换来快速评估这些变换。