Composite Cyclotomic Fourier Transforms with Reduced Complexities

TL;DR

This paper introduces composite cyclotomic Fourier transforms (CCFTs) that significantly reduce computational complexities for large-length DFTs over finite fields, with efficient algorithms and practical hardware advantages.

Contribution

It proposes CCFTs that lower complexities for moderate to long DFT lengths and introduces efficient cyclic convolution algorithms, including the first 2047- and 4095-point DFTs.

Findings

CCFTs outperform previous FFTs in complexity for large lengths

Efficient 11-point cyclic convolution algorithms are developed

First known 2047- and 4095-point DFTs over GF(2^{11})

Abstract

Discrete Fourier transforms~(DFTs) over finite fields have widespread applications in digital communication and storage systems. Hence, reducing the computational complexities of DFTs is of great significance. Recently proposed cyclotomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexities. Unfortunately, there are two issues with CFFTs: (1) they rely on efficient short cyclic convolution algorithms, which has not been investigated thoroughly yet, and (2) they have very high additive complexities when directly implemented. In this paper, we address both issues. One of the main contributions of this paper is efficient bilinear 11-point cyclic convolution algorithms, which allow us to construct CFFTs over GF$(2^{11})$. The other main contribution of this paper is that we propose composite cyclotomic Fourier transforms (CCFTs). In comparison to previously proposed fast Fourier transforms, our CCFTs achieve lower overall complexities for moderate to long lengths, and the improvement significantly increases as the length grows. Our 2047-point and 4095-point CCFTs are also first efficient DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also advantageous for hardware implementations due to their regular and modular structure.