Maximality of affine group, and hidden graph cryptosystems

We describe a new algebraic-combinatorial method of public key encryption with a certain similarity to the well known Imai–Matsumoto. We use the general idea to treat vertices of a linguistic graph (see [21] and further references) as messages and use the iterative process to walk on such graph as encryption process. To hide such encryption (graph and walk on it) we will use two affine transformation. Like in Imai–Matsumoto encryption the public rule is just a direct polynomial map from the plaintext to the ciphertext.
The knowledge about graph and chosen walk on them (the key) allow to decrypt a ciphertext fast. We hope that the system is secure even in the case when the graph is Public but the walk is hidden. In case of “public” graph we can use same encryption as private key algorithm with the resistance to attacks when adversary knows several pairs:(plaintext, ciphertext).
We shall discuss the general idea of combining affine transformations and chosen polynomial map of ${\rm deg}\ge 2$ in case of prime field $F_p$. As it follows from the maximality of affine group each bijection on ${F_p}^n$ can be obtained by such combining.