# 1 Construction

Assume that all data are encoded using the alphabet $$\Sigma = \{0,\dots,n-1\}$$ (where $$n$$ is the length of the alphabet).

Shift cipher (Caesar cipher). Given $$\ell \in \mathbb{Z}^+$$ which is the fixed length of the plaintext, define the following encryption scheme $$\Pi=(\mathsf{Gen},\mathsf{Enc},\mathsf{Dec})$$:

• Message space $$\mathcal{M} = \{0,\dots,n-1\}^\ell$$.
• Key space $$\mathcal{K} = \{0,\dots,n-1\}$$.
• Ciphertext space $$\mathcal{C} = \{0,\dots,n-1\}^\ell$$.
• $$\mathsf{Gen}$$ outputs a key $$k \in \mathcal{K}$$ according to the uniform distribution.
• $$\mathsf{Enc}$$ takes as input a key $$k \in \mathcal{K}$$ and a message $$m \in \mathcal{M}$$, and outputs the ciphertext $\mathsf{Enc}_k(m_1 \dots m_\ell) = c_1 \dots c_\ell \textrm{, where } c_i := [(m_i+k) \operatorname{mod} n].$
• $$\mathsf{Dec}$$ takes as input a key $$k \in \mathcal{K}$$ and a ciphertext $$c \in \mathcal{C}$$, and outputs the message $\mathsf{Dec}_k(c_1 \dots c_\ell) = m_1 \dots m_\ell \textrm{, where } m_i := [(c_i-k) \operatorname{mod} n].$
Since modulo arithmetic is associative: \begin{align*} (m_i + k) - k &\equiv m_i + (k - k) \\ &\equiv m_i \pmod n \end{align*}

this encryption scheme satisfies the correctness requirement.

# 2 Secrecy and Cryptanalysis

Notice that $$|\mathcal{K}| = n$$, $$|\mathcal{M}| = n^\ell$$. If $$|\mathcal{K}| < |\mathcal{M}|$$, then the scheme cannot be perfectly secret.

Theorem 2.1. Shift cipher is not perfectly secret when $$\ell > 1$$.

Possible cryptanalysis: Frequency analysis.