Consider an encryption scheme $(Gen,Enc,Dec)$ where $Gen$ is the key generation algorithm, $Enc$ is the encryption algorithm, where $c \leftarrow Enc_{k}(m) $ is taken to mean that the message $m$ in some message space $M$ encrypted with a key $k$ generated from some keyspace $K$ using $Gen$ produces the ciphertext $c$ which belongs to some cipherspace $C$, and where $Dec_{k}(c) = m$ is the decryption algorithm which is always deterministic.
In Shannon perfect secrecy, it is assumed that $|K| = |M| = |C|$. Does this imply that $Enc$ is deterministic?
i.e. is it impossible that there exist two different ciphertexts $c_{1},c_{2} \in C$ that are both produced by encrypting the same (message, key) pair? It seems once you restrict the three spaces to have equal cardinality, the encryption algorithm must be deterministic, but I cannot seem to prove this formally.