Paper 2025/2303

Suwako: A Logarithmic-Depth Modular Reduction for Arbitrary Trinomials over $\mathbb{F}_{2^m}$ without Pre-computation

Abstract

Modular reduction over binary extension fields $\mathbb{F}_{2^m}$ is a fundamental operation in cryptographic implementations, including GCM and Elliptic Curve Cryptography. Traditional reduction algorithms (e.g., linear LFSR-based methods) are highly sensitive to the algebraic structure of the defining polynomial. This sensitivity is especially acute for trinomials $P(x) = x^m + x^t + 1$, where cryptographic standards have historically mandated the use of ``friendly'' polynomials (with small $t$) to avoid the linear performance degradation associated with ``random'' or ``unfriendly'' parameters. In this paper, we challenge this constraint by introducing Suwako, a novel reduction algorithm. By exploiting the self-similar algebraic structure of the reduction map, Suwako transforms the reduction process from a serial iterative chain (dependent on the degree gap $\Delta = m-t$) into a logarithmic-depth binary-doubling structure. We theoretically prove that Suwako achieves $O(\log m)$ folding depth for arbitrary trinomials, regardless of the position of the middle term $t$. Furthermore, unlike window-based or Montgomery/Barrett reduction methods, Suwako requires no pre-computation, making it optimal for dynamic environments.