椭圆曲线快速傅里叶变换（ECFFT）第一部分：有限域上的快速多项式算法
Elliptic Curve Fast Fourier Transform (ECFFT) Part I: Fast Polynomial Algorithms over all Finite Fields
摘要：
对于光滑有限域$F_q$（即，当$q-1$分解成小素数时），快速傅立叶变换（FFT）导致用于许多基本多项式运算（例如乘法、除法、插值和多点求值）的最快已知代数算法。然而，在没有光滑阶单位根的域上的相同操作遭受渐近减速。Schonhage和Strassen的经典算法在光滑情况的基础上引入了一个乘性的减速因子$\log \log n$。Harvey、van der Hoeven和Lecerf最近的显著结果将这种乘法开销显著减少到了$\exp(\log^* (n))$。我们介绍了一种新的方法来快速算法的多项式运算在所有大型有限域。其关键思想是用一组与一个精心选择的椭圆曲线群（集合$L$本身不是一个群）适当相关的点$L \subset F$来替换单位根群。这种方法的主要优点是椭圆曲线群可以是Hasse-Weil区间$[q+1 \pm 2\sqrt{q}]$中的任何大小，因此可以具有大的光滑阶的子群，类似FFT的分治算法可以利用这些子群。将其与阶必须整除$q-1$的乘法子群进行比较。对于多项式表示的评价子集的$L$，我们表明，乘法，除法，度计算，插值，评价和里德-所罗门编码（也称为低程度的扩展）与固定的评价点都可以计算与算术电路的大小类似的是实现与经典的FFT时，字段大小是特殊的。对于几个问题，这产生了渐近最小的已知算术电路，即使在多项式的标准单项表示。
英文摘要：
For smooth finite fields $F_q$ (i.e., when $q-1$ factors into small primes) the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division, interpolation and multi-point evaluation. However, the same operations over fields with no smooth order root of unity suffer from an asymptotic slowdown. The classical algorithm of Schonhage and Strassen incurred a multiplicative slowdown factor of $\log \log n$ on top of the smooth case. Recent remarkable results of Harvey, van der Hoeven and Lecerf dramatically reduced this multiplicative overhead to $\exp(\log^* (n))$. We introduce a new approach to fast algorithms for polynomial operations over all large finite fields. The key idea is to replace the group of roots of unity with a set of points $L \subset F$ suitably related to a well-chosen elliptic curve group (the set $L$ itself is not a group). The key advantage of this approach is that elliptic curve groups can be of any size in the Hasse-Weil interval $[q+1 \pm 2\sqrt{q}]$ and thus can have subgroups of large, smooth order, which an FFT-like divide and conquer algorithm can exploit. Compare this with multiplicative subgroups over whose order must divide $q-1$. For polynomials represented by their evaluation over subsets of $L$, we show that multiplication, division, degree-computation, interpolation, evaluation and Reed-Solomon encoding (also known as low-degree extension) with fixed evaluation points can all be computed with arithmetic circuits of size similar to what is achievable with the classical FFTs when the field size is special. For several problems, this yields the asymptotically smallest known arithmetic circuits even in the standard monomial representation of polynomials.
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