复杂度降低的复合分圆Fourier变换
Composite Cyclotomic Fourier Transforms with Reduced Complexities
摘要：
有限域上的离散傅里叶变换在数字通信和存储系统中有着广泛的应用。因此，降低DFT的计算复杂性具有重要意义。最近提出的分圆快速傅里叶变换（CFFT）是有前途的，由于其低乘法复杂度。不幸的是，CFFT存在两个问题：（1）它们依赖于有效的短循环卷积算法，这还没有被彻底研究，以及（2）它们在直接实现时具有非常高的加性复杂度。在本文中，我们解决这两个问题。本文的主要贡献之一是高效的双线性11点循环卷积算法，它允许我们在GF $(2^{11})$上构造CFFT。本文的另一个主要贡献是，我们提出了复合分圆傅立叶变换（CCFTs）。与以前提出的快速傅立叶变换相比，我们的CCFT实现了较低的整体复杂度，为中等到长的长度，并改善显着增加的长度增长。据我们所知，我们的2047点和4095点CCFT也是第一个如此长度的高效DFT。最后，由于其规则和模块化结构，我们的CCFT也有利于硬件实现。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
