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.

翻译：二元域扩张是许多应用的基础，例如多元公钥密码学、基于编码的密码学以及纠错码。其实现需要数论和代数几何基础，并必须利用高效基。计算能力的持续提升以及新型（量子）计算机的设计，增加了对系统安全性的威胁，并迫使采用具有极高多项式次数或扩张次数的日益苛刻的加密标准。出于密码学目的或有限域算术的其他常见实现，有必要探索具有不同基的多种实现方案。与某些基不同，多项式基和高斯正规基已有详尽文献记载并被广泛使用。本文探讨 $\mathbb{F}_{2^n}$ 在 $\mathbb{F}_2$ 上的其他基形式，以展示在不同范围内运算的高效实现。为此，我们利用 Couveignes 和 Lercier 引入、随后由 Ezome 和 Sall 扩展的快速计算和椭圆周期结果。这为我们建立了在二元域上进行高效计算的新表格。