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.