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Relating transposition with b-bit versus 1-bit entries

Develop reductions, in either direction, between matrix transposition problems with b-bit entries and those with 1-bit entries in the multitape Turing model, clarifying their time-complexity relationship.

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Background

The authors argue it is natural to expect that the cost of transposing matrices with b-bit entries should scale linearly in b compared to the 1-bit case, but they currently lack reductions connecting these two regimes. Establishing such reductions (e.g., proving T(n1,n2,b) = Ω(b·T(n1,n2,1))) would allow leveraging results for larger coefficients to improve bounds and implications for smaller coefficients within their framework.

References

Unfortunately, we have not yet succeeded in designing reductions between transpositions with $b$-bit and $1$-bit entries, in either direction.

— Integer multiplication is at least as hard as matrix transposition (2503.22848 - Harvey et al., 28 Mar 2025) in Section 7.6 (Small versus large coefficients)