The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a smooth number, then the divide-and-conquer approach leads to the fastest known FFT algorithms. Depending on the type of group that the set of evaluation points forms, these algorithms are classified as multiplicative (Math of Comp. 1965) and additive (FOCS 2014) FFT algorithms. In this work, we provide a unified framework for FFT algorithms that include both multiplicative and additive FFT algorithms as special cases, and beyond: our framework also works when $q+1$ is smooth, while all known results require $q$ or $q-1$ to be smooth. For the new case where $q+1$ is smooth (this new case was not considered before in literature as far as we know), we show that if $n$ is a divisor of $q+1$ that is $B$-smooth for a real $B>0$, then our FFT needs $O(Bn\log n)$ arithmetic operations in $\mathbb{F}_q$. Our unified framework is a natural consequence of introducing the algebraic function fields into the study of FFT.

翻译：在有限域 $\mathbb{F}_q$ 上的快速傅里叶变换（FFT）计算给定次数小于 $n$ 的多项式在 $\mathbb{F}_q$ 中一组特定选择的 $n$ 个不同求值点处的求值。若 $q$ 或 $q-1$ 为光滑数，则分治策略可导出已知最快的 FFT 算法。根据求值点集合所构成的群类型，这些算法被分类为乘法型（Math. of Comp. 1965）和加法型（FOCS 2014）FFT 算法。本文提出了一个统一的 FFT 算法框架，将乘法型与加法型 FFT 算法作为特例囊括其中，并进一步拓展：该框架在 $q+1$ 为光滑数时同样适用，而所有已知结果要求 $q$ 或 $q-1$ 为光滑数。针对 $q+1$ 为光滑数这一新情形（据我们所知，该情形此前未被文献考虑过），我们证明：若 $n$ 是 $q+1$ 的约数且对实数 $B>0$ 为 $B$-光滑，则我们的 FFT 在 $\mathbb{F}_q$ 中仅需 $O(Bn\log n)$ 次算术运算。该统一框架是将代数函数域引入 FFT 研究的自然结果。