有限域上简化复杂度的素因子分圆Fourier变换
Prime Factor Cyclotomic Fourier Transforms with Reduced Complexity over Finite Fields
摘要：
有限域上的离散傅里叶变换在纠错编码中有着广泛的应用。因此，降低DFT的计算复杂性具有重要意义，尤其是对于长DFT，因为越来越长的差错控制码被选择用于数字通信和存储系统。由于DFT涉及有限域上的乘法和加法，并且乘法比加法复杂得多，最近提出的分圆快速傅里叶变换（CFFT）由于其低乘法复杂度而很有前途。不幸的是，它们具有非常高的加法复杂性。诸如公共子表达式消除（CSE）之类的技术可以用于降低CFFT的加性复杂度，但是它们对于长DFT的有效性受到它们的复杂度的限制。在本文中，我们提出了素因子分圆傅立叶变换（PFCFTs），它使用CFFT作为子DFT通过素因子算法。当DFT的长度是素数时，我们的PFCFTs减少到CFFTs。当长度具有互质因子时，由于子DFT具有短得多的长度，这允许我们使用CSE来显著降低其加性复杂度。与以前提出的快速傅立叶变换相比，我们的PFCFTs实现了降低的整体复杂性时，DFT的长度至少为255，并改善显着增加的长度增长。这种方法还使我们能够提出具有非常长长度的高效DFT（例如，4095点），这是文献中第一个具有这种长度的高效DFT。最后，我们的PFCFTs也有利于硬件实现，由于其规则的结构。
英文摘要：
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.
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