Block Encryption and Pairs

Definition 1

An l -bit block encryption is a function $e: P_{l} \times {\cal K}_{m} \rightarrow C_{l}$, such that for each m -bit key $k\in {\cal K}_{m}$ and block $p \in P_{l}$, $e_{\tiny k}(p)$ has an invertible mapping. The inverse mapping is block decryption, denoted $e^{-1}_{\tiny k}{(c)}$. 

Definition 2

A plaintext-ciphertext pair is an ordered 2-tuple $\{\langle p, c \rangle~ \vert~ \exists k\mbox{ s.t. }c= e_{\tiny k}(p)\wedge p= e^{-1}_{\tiny k}{(c)}\}.$

Definition 3

A plaintext-ciphertext pair under key, k , is an ordered 3-tuple $\{\langle p, c, k \rangle~ \vert~ c= e_{\tiny k}(p)\wedge p= e^{-1}_{\tiny k}{(c)}\}.$

We will use the term pair to refer to a plaintext-ciphertext pair.