Example 1
(Symmetric (or secret key) cryptosystem scheme)
Let $P$ be a set of plaintexts and let $C$ be the set of ciphers. Given a key $k$, an encryption map (or algorithm) is a bijective map
\begin{equation}
e_k : P \to C
\end{equation}
such that $e_k(p)$ is a ciphertext for any $p \in P$. The corresponding decryption map would then be the map
\begin{equation}
d_k : C \to P
\end{equation}
such that
\begin{equation}
d_k(e_k(p))
= p
\end{equation}
for all $p \in P$.
In this kind of a symmetric cryptosystem scheme, both algorithms $e$ and $d$ would be assumed to be known to all parties involved, while the security of the scheme relies on the confidentiality of the $k$, used for both encryption and decryption.