Public Key Encryption

Public Key Enchamism (PKE) is a quadruple of algorithms ${\mathcal {K}}=\{{\text{Setup}},{\text{KeyGen}},{\text{Encaps}},{\text{Decaps}}\}$ and a corresponding key space $K$, a ciphertext space ${\text{C}}$ with

1. The parameter generation ${\text{Setup}}$ algorithm takes input $1^{k}$, where $k\in \mathbb {N}$ is the security parameter and return global parameter $\pi$.
2. The key generation ${\text{KeyGen}}$ algorithm takes input param $\pi$ and returns a public and secret-key pair (${\text{pk}},{\text{sk}}$).
3. The encryption ${\text{Enc}}$ algorithm takes $1^{k},\pi ,pk$ and a message $M\in (0,1)^{*}$ and returns a ciphertext $c\in C$.
4. The deterministic decryption ${\text{Dec}}(sk,c)$ algorithm takes $1^{k},\pi ,sk,c$ and returns a message $M\cup \{\perp \}$, with $\perp$ denoting as a failure.