Let
$p=3n+1$
be a prime with
$n\in \mathbb {N}=\{0,1,2,\ldots \}$
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
$a_1 \bmod p,\, a_2 \bmod p,\ldots , a_n \bmod p$
is a permutation of the sequence
$g^3 \bmod p,\,g^6 \bmod p,\ldots , g^{3n} \bmod p$
. We determine the sign of this permutation.