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(ω
ⁱ
)/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.