Efficient pth root computations in finite fields of characteristic p

• Authors: D. Panario;D. Thomson
• Venue: Designs, Codes and Cryptography
• Year: 2009

We present a method for computing pth roots using a polynomial basis over finite fields $${\mathbb F_q}$$ of odd characteristic p, p 驴 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 驴 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 驴 5, we find $${{\rm wt}\left(z^{1/p}\right) = 1}$$ always.