Superlinear lower bounds for matrix transposition in the Turing model

Establish a non-trivial (superlinear) lower bound for the worst-case time complexity of transposing an n × n binary matrix on a multitape Turing machine. Specifically, prove that for every binary transposition machine T operating on input size m = n^2 bits, the running time Tcost_T(m) grows faster than linear in m (for example, show Tcost_T(m) = ω(m) or stronger).

Background

The paper studies reductions from matrix transposition to integer multiplication in the multitape Turing model, showing that lower bounds for transposition imply lower bounds for multiplication. Despite folklore algorithms achieving O(m log m) for transposition, virtually no superlinear lower bounds are known in the full Turing model, unlike restricted models. Establishing any superlinear lower bound for transposition would immediately strengthen lower-bound consequences for multiplication via the reductions developed in the paper.