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 \[\mathsf{Dec}_k(c_1 \dots c_\ell) = m_1 \dots m_\ell \textrm{, where } m_i := [(c_i-k) \operatorname{mod} n].\]

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.