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.
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Communicated by P. Wild.
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Panario, D., Thomson, D. Efficient pth root computations in finite fields of characteristic p . Des. Codes Cryptogr. 50, 351–358 (2009). https://doi.org/10.1007/s10623-008-9236-0
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DOI: https://doi.org/10.1007/s10623-008-9236-0