Prime Factor Cyclotomic Fourier Transforms with Reduced Complexity over Finite Fields

TL;DR

This paper introduces prime factor cyclotomic Fourier transforms (PFCFTs) that significantly reduce the computational complexity of DFTs over finite fields, especially for long DFTs, enabling efficient hardware implementation.

Contribution

The paper proposes PFCFTs that combine cyclotomic FFTs with the prime factor algorithm to lower additive complexity for long DFTs over finite fields, including very long lengths.

Findings

PFCFTs reduce overall complexity for DFT lengths ≥ 255.

PFCFTs enable efficient 4095-point DFTs, the first of such length in literature.

PFCFTs are advantageous for hardware due to their regular structure.

Abstract

Discrete Fourier transforms~(DFTs) over finite fields have widespread applications in error correction coding. Hence, reducing the computational complexities of DFTs is of great significance, especially for long DFTs as increasingly longer error control codes are chosen for digital communication and storage systems. Since DFTs involve both multiplications and additions over finite fields and multiplications are much more complex than additions, recently proposed cyclotomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexity. Unfortunately, they have very high additive complexity. Techniques such as common subexpression elimination (CSE) can be used to reduce the additive complexity of CFFTs, but their effectiveness for long DFTs is limited by their complexity. In this paper, we propose prime factor cyclotomic Fourier transforms (PFCFTs), which use CFFTs as sub-DFTs via the prime factor algorithm. When the length of DFTs is prime, our PFCFTs reduce to CFFTs. When the length has co-prime factors, since the sub-DFTs have much shorter lengths, this allows us to use CSE to significantly reduce their additive complexity. In comparison to previously proposed fast Fourier transforms, our PFCFTs achieve reduced overall complexity when the length of DFTs is at least 255, and the improvement significantly increases as the length grows. This approach also enables us to propose efficient DFTs with very long length (e.g., 4095-point), first efficient DFTs of such lengths in the literature. Finally, our PFCFTs are also advantageous for hardware implementation due to their regular structure.