On efficient normal bases over binary fields
Mohamadou Sall, M. Anwar Hasan
TL;DR
This work targets efficient arithmetic in large binary field extensions $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$, aiming to go beyond classical bases by exploiting bases tied to FFT/NTT-like algorithms. Building on results by Couveignes–Lercier and Ezome–Sall, the authors develop normal bases from Gauss periods and from one-dimensional algebraic groups, notably elliptic normal bases, to widen the set of cases where fast multiplication is possible, and they provide new tables and practical guidance for embedding-degree choices. They also explore sub-normal constructions via Artin–Schreier–Witt and Kummer theory to handle instances where standard Gaussian normal bases are unavailable, with quantified costs that favor sparse multiplication and vector-matrix operations. The findings expand the toolbox for efficient binary-field arithmetic in cryptographic and coding contexts, offering concrete tables, Magma code support, and architecture-aware strategies for selecting embedding degrees and representations.
Abstract
Binary field extensions are fundamental to many applications, such as multivariate public key cryptography, code-based cryptography, and error-correcting codes. Their implementation requires a foundation in number theory and algebraic geometry and necessitates the utilization of efficient bases. The continuous increase in the power of computation, and the design of new (quantum) computers increase the threat to the security of systems and impose increasingly demanding encryption standards with huge polynomial or extension degrees. For cryptographic purposes or other common implementations of finite fields arithmetic, it is essential to explore a wide range of implementations with diverse bases. Unlike some bases, polynomial and Gaussian normal bases are well-documented and widely employed. In this paper, we explore other forms of bases of $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$ to demonstrate efficient implementation of operations within different ranges. To achieve this, we leverage results on fast computations and elliptic periods introduced by Couveignes and Lercier, and subsequently expanded upon by Ezome and Sall. This leads to the establishment of new tables for efficient computation over binary fields.