二进制多项式模约简的对数反馈深度

中文摘要 AI 辅助

多项式模约简是二进制有限域算术和重复 Frobenius 幂运算的核心。稀疏自顶向下折叠使用较少的移位/异或操作，但靠近首项的抽头会形成长反馈链。我们将此递推关系表述为幂零移位算子的求逆，并通过特征二 Frobenius 幂对其逆进行因式分解。由此产生的 Frobenius 因式分解约简（FFR）适用于每个首一二进制模数，无需具体化倒数或稠密约简矩阵，且其移位可以在线生成，无需持久的模数特定调度。对于次数 $m$、非首项支撑大小 $s$ 和最近抽头距离 $\Delta_{\min}$，FFR 具有精确反馈深度 $\lceil\log_2(m/\Delta_{\min})\rceil$ 和调度工作量 $O(ms(1+\log(m/s)))$。在次数高达 $131072$ 的 1096 个支撑上的可移植 C 评估识别出不同的 FFR、López--Dahab 和基于 gf2x 的 Barrett 区域。在四个认证的不可约模数上，FFR 使完整的 Rabin 不可约性测试比 NTL 快 $1.35$--$8.04$ 倍，比匹配的 Barrett 实现快 $1.60$--$6.81$ 倍。

英文摘要

Polynomial modular reduction is central to binary finite-field arithmetic and repeated Frobenius powering. Top-down shift/XOR folding can use few operations when the nonleading support is sparse, but a tap near the leading term creates a long feedback chain. We formulate this recurrence as inversion of a nilpotent shift operator and factor its inverse by characteristic-two Frobenius powers. The resulting Frobenius-factorized reduction (FFR) applies to every monic binary modulus without materializing a reciprocal or dense reduction matrix, and its shifts can be generated online without a persistent modulus-specific schedule. For degree $m$ and nonempty nonleading support of size $s$, with nearest-tap distance $Δ_{\min}$, FFR uses exactly $\lceil\log_2(m/Δ_{\min})\rceil$ sequential feedback stages and has scheduled work $O(ms(1+\log(m/s)))$. A portable-C evaluation on 1,096 supports through degree $131072$ identifies distinct FFR, López--Dahab, and gf2x-backed Barrett regions. On four certified irreducible moduli, FFR makes complete Rabin irreducibility testing $1.35$--$8.04$ times faster than NTL and $1.60$--$6.81$ times faster than the matched Barrett implementation.

发表机构

• Sichuan University（四川大学）
• Tsinghua University（清华大学）

机构由 AI 辅助整理，请以论文原文为准。