Constructing Finite Field Extensions with Large Order Elements

Abstract

In this paper, we present an algorithm that, given a fixed prime power q and a positive integer N, finds an integer $n \in [N, 2qN]$ and an element $\alpha \in \mbox{\bf F}_{q^n}$ of order greater than $ 5.8^{n / \log_q n}$, in time polynomial in N. We present another algorithm that finds an integer $n \in [N, N+O(N^{0.77})]$ and an element $\alpha \in \mbox{\bf F}_{q^n}$ of order at least $ 5.8^{\sqrt{n}}$, in time polynomial in N. Our result is inspired by the recent AKS primality testing algorithm [M. Agrawal, N. Kayal, and N. Saxena, Ann. of Math. (2), 160 (2004), pp. 781–793] and the subsequent improvements [P. Berrizbeitia, Math. Comp., 74 (2005), pp. 2043–2059, Q. Cheng, in Proceedings of the 23rd Annual International Cryptology Conference (CRYPTO 2003), D. Boneh, ed., Lecture Notes in Comput. Sci. 2729, Springer-Verlag, Berlin, 2003, pp. 338–348, D. J. Bernstein, Math. Comp., 76 (2007), pp. 389–403].