Integer multiplication is at least as hard as matrix transposition

David Harvey; Joris van der Hoeven

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arXiv · 2503.22848

Integer multiplication is at least as hard as matrix transposition

David Harvey·Joris van der Hoeven

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Abstract

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.

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David Harvey, Joris van der Hoeven. 2025-03-28. Integer multiplication is at least as hard as matrix transposition. https://arxiv.org/abs/2503.22848

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