The Two-Modular Fourier Transform of Binary Functions

TL;DR

This paper introduces the two-modular Fourier transform (TMFT) for binary functions over n-bit vectors, providing an efficient computation method and analyzing its properties and complexity.

Contribution

The paper presents the first solution for computing the Fourier transform of binary functions over n-bit vectors using the TMFT and its fast algorithm.

Findings

Defined the two-modular Fourier transform (TMFT) for binary functions.

Established properties and the convolution theorem for TMFT.

Analyzed the complexity of the fast TMFT and its inverse.

Abstract

In this paper, we provide a solution to the open problem of computing the Fourier transform of a binary function defined over $n$-bit vectors taking $m$-bit vector values. In particular, we introduce the two-modular Fourier transform (TMFT) of a binary function $f:G\rightarrow {\cal R}$, where $G = (\mathbb{F}_2^n,+)$ is the group of $n$ bit vectors with bitwise modulo two addition $+$, and ${\cal R}$ is a finite commutative ring of characteristic $2$. Using the specific group structure of $G$ and a sequence of nested subgroups of $G$, we define the fast TMFT and its inverse. Since the image ${\cal R}$ of the binary functions is a ring, we can define the convolution between two functions $f:G\rightarrow {\cal R}$. We then provide the TMFT properties, including the convolution theorem, which can be used to efficiently compute convolutions. Finally, we derive the complexity of the fast TMFT and the inverse fast TMFT.