One-time pad

Definition

Let $\mathcal{M} = \{0,1\}^n$, $\mathcal{K} = \{0,1\}^n$, $\mathcal{C} = \{0,1\}^n$, $$\begin{align} \operatorname{Enc}_\mathbf{K}(m) = m \oplus k \\ \operatorname{Dec}_\mathbf{K}(m) = c \oplus k \end{align}$$ where $\oplus$ denotes bitwise XOR

Proposition

1. A one-time pad (OTP) is a Perfectly secret Encryption scheme (correctness). $$\operatorname{Dec}_\mathbf{K}(\operatorname{Enc}_\mathbf{K}(m)) = (m \oplus k) \oplus k = m \oplus (k \oplus k) = m \oplus 0^n = m$$
2. OTP is perfectly secure by previous theorem suffices to show that OTP is Perfectly indistinguishable.

#incomplete