Efficient p th root computations in finite fields of characteristic p

We present a method for computing pth roots using a polynomial basis over finite fields Fq of odd characteristic p, p ≥ 5, by taking advantage of a binomial reduction polynomial. For a finite field extension Fqm of Fq our method requires p− 1 scalar multiplication of elements in Fqm by elements in Fq. In addition, our method requires at most (p − 1)dm/pe additions in the extension field. In certain cases, these additions are not required. If z is a root of the irreducible reduction polynomial, then the number of terms in the polynomial basis expansion of z, defined as the Hamming weight of z or wt ( z ) , is directly related to the computational cost of the pth root computation. Using trinomials in characteristic 3, Ahmadi et al. [1] give wt ( z ) is greater than 1 in nearly all cases. Using a binomial reduction polynomial over odd characteristic p, p ≥ 5, we find wt ( z ) = 1 always.