  We explore a new form of DFT, which we call the Polynomial Transform. It
functions over finite fields, and a size $n$ transform takes $O(n)$ operations.
In the multitape Turing machine model, it allows us to multiply two $n$ bit
numbers in time $n(k^{\log^*{n}} + \log{p})$, where $k$ is a constant and
$\log^*{n}$ is the iterated logarithm. One important consequence is that the
Network Coding Conjecture is false.
