High order elements in finite fields arising from recursive towers

Abstract

AbstractWe illustrate a general technique to construct towers of fields producing high order elements in $$\mathbb {F}_{q^{2^n}}$$ F q 2 n , for odd q, and in $$\mathbb {F}_{2^{2 \cdot 3^n}}$$ F 2 2 · 3 n , for $$n \ge 1$$ n ≥ 1 . These towers are obtained recursively by $$x_{n}^2 + x_{n} = v(x_{n - 1})$$ x n 2 + x n = v ( x n - 1 ) , for odd q, or $$x_{n}^3 + x_{n} = v(x_{n - 1})$$ x n 3 + x n = v ( x n - 1 ) , for $$q=2$$ q = 2 , where v(x) is a polynomial of small degree over the prime field $$\mathbb {F}_{q}$$ F q and $$x_n$$ x n belongs to the finite field extension $$\mathbb {F}_{q^{2^n}}$$ F q 2 n , for an odd q, or to $$\mathbb {F}_{2^{2\cdot 3^n}}$$ F 2 2 · 3 n . Several examples are provided to show the numerical efficacy of our method. Using the techniques of Burkhart et al. (Des Codes Cryptogr 51(3):301–314, 2009) we prove similar lower bounds on the orders of the groups generated by $$x_n$$ x n , or by the discriminant $$\delta _n$$ δ n of the polynomial. We also provide a general framework which can be used to produce many different examples, with the numerical performance of our best examples being slightly better than in the cases analyzed in Burkhart et al. (2009).