Arithmetic in the Finite Fields Using Optimal Normal and Polynomial Bases in Combination

Abstract

In this chapter the idea of using optimal normal bases (o.n.b.) of second and third types in combination with polynomial basis of field F(q n ) is detailed using a new modification of o.n.b. called reduced optimal normal basis –1, β 1, …, β n − 1 corresponding to a permutated o.n.b. β 1, …, β n − 1 Operations of multiplication, rising to power q i , rising to arbitrary power and inversion in reduced o.n.b. in combination with polynomial basis as well as converting operations between these bases in the fields of characteristic three has been described, estimated and expanded to the fields of characteristic two. This allows get efficient implementations of cryptographic protocols using operation of Tate pairing on supersingular elliptic curve.