Mathematical formalisation of encryption & decryption

• $\mathcal{A}$, the alphabet, is a finite set.
• $\mathcal{M}\subseteq {\mathcal{A}}^{\ast }$ is the message space. $M\in \mathcal{M}$ is a plaintext (message).
• $\mathcal{C}$ is the ciphertext space, whose alphabet may differ from $\mathcal{M}$.
• $\mathcal{K}$ denotes the key space of keys.
• Each $e\in \mathcal{K}$ determines a bijective function from $\mathcal{M}$ to $\mathcal{C}$, denoted by $E_e$. $E_e$ is the encryption function (or transformation). Note: we write $E_e(P)=C$ or equivalently $E(e,P)=C$.
• For each $d\in \mathcal{K}$, $D_d$ denotes a bijection from $\mathcal{C}$ to $\mathcal{M}$. $D_d$ is the decryption function.
• Applying $E_e$ is called encryption, applying $D_d$ is called decryption.