Blum–Goldwasser Public Key Encryption System

Definition

The Blum–Goldwasser public key encryption system combines the general construction of Goldwasser–Micali [1] with the concrete Blum–Blum–Shub pseudorandom bit generator [2] to obtain an efficient semantically secure public key encryption whose security is based on the difficulty of factoring Blum integers.

Theory

The system makes use of modular arithmetic and works as follows:

Key Generation. Given a security parameter $\tau \in \mathbb{Z}$ as input, generate two random $\tau$-bit primes p,  q where p = q = 3 mod 4. Set $N = {\it { pq}} \in Z$. The public key is N and private key is (p,  q).

Encryption. To encrypt a message $m = {m}_{1}\ldots {m}_{\ell} \in \{0,\ 1{\}}^{\ell}$:

• 1. Pick a random x in the group ${\mathbb{Z}}_{N}^{{_\ast}}$ and set ${x}_{1} = {x}^{2} \in {\mathbb{Z}}_{N}^{{_\ast}}$.

• 2. For $i = 1,\ \ldots,\ \ell$:

  • (a) View ${x}_{i}$ as an integer in [0,  N − 1] and let ${b}_{i} \in \{0,\ 1\}$ be the least significant bit of ${x}_{i}$.

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