  We compute the $C_p$-equivariant dual Steenrod algebras associated to the
constant Mackey functors $\underline{\mathbb{F}}_p$ and
$\underline{\mathbb{Z}}_{(p)}$, as $\underline{\mathbb{Z}}_{(p)}$-modules. The
$C_p$-spectrum $\underline{\mathbb{F}}_p \otimes \underline{\mathbb{F}}_p$ is
not a direct sum of $RO(C_p)$-graded suspensions of $\underline{\mathbb{F}}_p$
when $p$ is odd, in contrast with the classical and $C_2$-equivariant dual
Steenrod algebras.
