Working in the multitape Turing model, we show how to reduce the problem of matrix transposition to the problem of integer multiplication. If transposing an $n \times n$ binary matrix requires $\Omega(n^2 \log n)$ steps on a Turing machine, then our reduction implies that multiplying $n$-bit integers requires $\Omega(n \log n)$ steps. In other words, if matrix transposition is as hard as expected, then integer multiplication is also as hard as expected.

翻译：在多带图灵机模型中，我们展示了如何将矩阵转置问题归约到整数乘法问题。若在二进制$n \times n$矩阵的转置操作需要图灵机执行$\Omega(n^2 \log n)$步，则我们的归约表明$n$位整数乘法需要$\Omega(n \log n)$步。换言之，若矩阵转置符合预期的计算难度，则整数乘法亦将符合预期的计算难度。