  Let $p=3n+1$ be a prime with $n\in\mathbb{N}=\{0,1,\cdots\}$, and let
$g\in\mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots<a_n<p$ be
all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the
sequence $$a_1\ {\rm mod}\ p,\ a_2\ {\rm mod}\ p,\cdots, a_n\ {\rm mod}\ p$$ is
a permutation $s_p(g)$ of the sequence $$g^3\ {\rm mod}\ p,\ g^6\ {\rm mod}\
p,\cdots, g^{3n}\ {\rm mod}\ p.$$ In this paper, we shall determine the sign of
this permutation.
