An efficient generation of geometric sequences defined with trace and legendre symbol

Let p and m be an odd characteristic and extension degree, respectively. Then, let ! be a primitive element in F*_pm, a geometric sequence S = {s_i}, s_i =(Tr(ωⁱ)/p) for i = 0, 1, 2, ... has been proposed together with its some properties, where Tr (·) is the trace function over F_p and (a/p) is Legendre symbol for a in F_p. This paper shows an efficient method for generating the sequence when the base field F_pm is defined by an irreducible binomial over F_p with small odd characteristic p.