Normal bases from 1-dimensional algebraic groups

Abstract This paper surveys and illustrates a geometric method for constructing normal bases allowing fast finite field arithmetic. Indeed, Couveignes and Lercier presented in February 2008 a process for constructing efficient normal bases of F q n over F q from some degree two functions on an elliptic curve over F q . We apply the Couveignes-Lercier method to all the remaining 1-dimensional algebraic groups. That results in natural normal bases with quasi-linear complexity.