In this paper we extend a normal basis of a finite field over its base field to a new basis which permits both computationally inexpensive exponentiation and multiplication. We focus primarily on extensions of the finite field F2. These bases are motivated by Artin–Schreier theory and we conclude that they are particularly useful in Artin–Schreier extensions; that is, extensions F2n of F2 with n a power of two.

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Finite Fields and their Applications
School of Mathematics and Statistics

Thomson, D, & Weir, C. (Colin). (2018). Artin–Schreier extensions of normal bases. Finite Fields and their Applications, 53, 267–286. doi:10.1016/j.ffa.2018.06.010