Efficient pth root computations in finite fields of characteristic p

Abstract

We present a method for computing pth roots using a polynomial basis over finite fields ${\mathbb F_q}$ of odd characteristic p, p ≥ 5, by taking advantage of a binomial reduction polynomial. For a finite field extension ${\mathbb F_{q^m}}$ of ${\mathbb F_q}$ our method requires p − 1 scalar multiplications of elements in ${\mathbb F_{q^m}}$ by elements in ${\mathbb F_q}$. In addition, our method requires at most ${(p-1)\lceil m/p \rceil}$ 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 ^1/p, defined as the Hamming weight of z ^1/p or ${{\rm wt}\left(z^{1/p} \right)}$, is directly related to the computational cost of the pth root computation. Using trinomials in characteristic 3, Ahmadi et al. (Discrete Appl Math 155:260–270, 2007) give ${{\rm wt}\left(z^{1/3} \right)}$ is greater than 1 in nearly all cases. Using a binomial reduction polynomial over odd characteristic p, p ≥ 5, we find ${{\rm wt}\left(z^{1/p}\right) = 1}$ always.