Integer multiplication is at least as hard as matrix transposition
David Harvey, Joris van der Hoeven
TL;DR
The paper establishes a principled reduction from matrix transposition to integer multiplication on multitape Turing machines, using a Bluestein–Kronecker FFT framework with fixed-point arithmetic to preserve data integrity. By encoding matrix data as complex-fixed-point vectors and alternating forward/inverse DFT steps with careful data ordering, the authors show that strong lower bounds for transposition would imply corresponding lower bounds for multiplication, and provide precise quantitative relations under various regimes of the log-iterated complexity. Key contributions include a main transposition-to-multiplication reduction, a decomposition approach that breaks transpositions into subproblems, and recursive/ell-parameter refinements that link growth rates of $\text{T}(m)$ and $\text{M}(m)$ across multiple scales. The work highlights a novel pathway for proving superlinear lower bounds on multiplication by focusing on transposition and related data-movement problems, and discusses extensions to other permutations, coefficient sizes, and computational models. Overall, it builds a bridge between data rearrangement and arithmetic complexity with potential impact on our understanding of fundamental limits of computation.
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 $Ω(n^2 \log n)$ steps on a Turing machine, then our reduction implies that multiplying $n$-bit integers requires $Ω(n \log n)$ steps. In other words, if matrix transposition is as hard as expected, then integer multiplication is also as hard as expected.