Integer multiplication below n log n

We give a deterministic algorithm that multiplies two n-bit integers in $O(n(\lg n{)}^{1-\kappa })$ worst-case time, with $\kappa =2^{-182}$, on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the $n\log n$ optimality conjecture of Schönhage and Strassen in this model.