二元扩域上的加性FFT技术
On the Additive FFT Techniques over Binary Extension Fields
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中文总结 AI 辅助
该研究基于Bailey的四步FFT算法，开发二元扩域仿射子空间多项式求值的加性FFT技术，提出两种Cantor特殊基下的算法，在多数配置中性能优于现有算法，且形式化了部分Cantor特殊基的概念。
中文摘要 AI 辅助
受Bailey于1989年提出的四步FFT算法启发，我们针对二元扩域的仿射子空间上的多项式求值，开发了加性FFT（AFFT）技术。我们的核心见解是，关于子空间的消失多项式的泰勒展开，提供了与Bailey的矩阵公式对应的结构等价物，它将加性FFT（AFFT）分解为与矩阵的列和行相关的独立子AFFT。我们首先提出适用于任意有序基和任意维度拆分的通用基AFFT，为衡量专业化带来的增益提供统一基线。随后我们将该框架专门化为Cantor特殊基，得到两种AFFT算法：第一种支持AFFT维度的任意分解，利用Cantor特殊基结构在泰勒展开阶段无需有限域乘法；第二种采用保留相关子空间多项式二项式形式的分解，恰好需要(1/2)nlog₂n次乘法，以及由m的二进制表示确定的闭式加法计数。我们的实现结果显示，在两个硬件平台测试的42种配置中，该算法有37种比Cantor特殊基上的LCH AFFT更快，这种性能优势源于其完全递归结构，该结构天生具备内存局部性，且无需单独的基转换和求值阶段。最后，在单独的分析中，我们形式化了部分Cantor特殊基的概念，并确定了参数范围，在此范围内von zur Gathen-Gerhard算法和我们的通用基AFFT都比首个Gao-Mateer算法需要更少的加法和乘法。
英文摘要
Motivated by Bailey's four-step FFT algorithm (1989), we develop additive FFT techniques for polynomial evaluation over affine subspaces of binary extension fields. Our key insight is that the Taylor expansion with respect to vanishing polynomials of subspaces provides a structural counterpart to Bailey's matrix formulation. It decomposes an additive FFT (AFFT) into independent sub-AFFTs associated with the columns and rows of a matrix. We first present a general-basis AFFT that applies to any ordered basis and any split of the dimension, providing a unified baseline for measuring the gains from specialization. We then specialize the framework to the Cantor special basis and obtain two AFFT algorithms. The first supports an arbitrary decomposition of the AFFT dimension and exploits the Cantor special basis structure to perform the Taylor expansion stage without finite field multiplications. The second uses a decomposition that preserves the binomial form of the relevant subspace polynomials. It requires exactly $\frac{1}{2}n\log_2 n$ multiplications, together with a closed-form addition count determined by the binary representation of $m$. Our implementation results show that this algorithm is faster than the LCH AFFT over a Cantor special basis in 37 of the 42 configurations tested across two hardware platforms. This performance advantage stems from its fully recursive structure, which provides memory locality by design and avoids separate basis-conversion and evaluation stages. Finally, in a separate analysis, we formalize the notion of partial Cantor special bases and identify parameter regimes in which both the von zur Gathen-Gerhard algorithm and our general-basis AFFT require fewer additions and multiplications than the first Gao-Mateer algorithm.
