On efficient normal bases over binary fields

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.