The Polynomial Transform

TL;DR

The paper introduces the Polynomial Transform, a new finite field DFT variant that enables efficient multiplication of large numbers and disproves the Network Coding Conjecture.

Contribution

It presents a novel polynomial-based DFT with linear complexity over finite fields and demonstrates its implications for computational complexity and network coding.

Findings

Transform operates in O(n) time over finite fields.

Enables multiplication of n-bit numbers in near-linear time.

Disproves the Network Coding Conjecture.

Abstract

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.