On algebraic graph theory and non-bijective multivariate maps in cryptography

Special family of non-bijective multivariate maps $F_n$ of ${Z_m}^n$ into itself is constructed for $n = 2, 3, \dots$ and composite $m$. The map $F_n$ is injective on $\Omega_n=\{{\rm x}|x_1+x_2 + \dots x_n \in {Z_m}^* \}$ and solution of the equation $F_n({\rm x})={\rm b}, {\rm x}\in \Omega_n$ can be reduced to the solution of equation $z^r=\alpha$, $z \in {Z_m}^*$, $(r, \phi(m))=1$. The “hidden RSA cryptosystem” is proposed.
Similar construction is suggested for the case $\Omega_n={{Z_m}^*}^n$.