Note: this is not to be confused with the Naccache–Stern knapsack cryptosystem.

The Naccache–Stern cryptosystem is a homomorphic public-key cryptosystem whose security rests on the higher residuosity problem. The Naccache–Stern cryptosystem was discovered by David Naccache and Jacques Stern in 1998.

## Scheme Definition

Like many public key cryptosystems, this scheme works in the group ${\displaystyle (\mathbb{Z}/n\mathbb{Z})^*}$ where n is a product of two large primes. This scheme is homomorphic and hence malleable.

### Key Generation

• Pick a family of k small distinct primes p1,...,pk.
• Divide the set in half and set ${\displaystyle u = \prod_{i=1}^{k/2} p_i}$ and ${\displaystyle v = \prod_{k/2+1}^k p_i}$.
• Set ${\displaystyle \sigma = uv = \prod_{i=1}^k p_i}$
• Choose large primes a and b such that both p = 2au+1 and q=2bv+1 are prime.
• Set n=pq.
• Choose a random g mod n such that g has order φ(n)/4.

The public key is the numbers σ,n,g and the private key is the pair p,q.

When k=1 this is essentially the Benaloh cryptosystem.

### Message Encryption

This system allows encryption of a message m in the group ${\displaystyle \mathbb{Z}/\sigma\mathbb{Z}}$.

• Pick a random ${\displaystyle x \in \mathbb{Z}/n\mathbb{Z}}$.
• Calculate ${\displaystyle E(m) = x^\sigma g^m \mod n}$

Then E(m) is an encryption of the message m.

### Message Decryption

To decrypt, we first find m mod pi for each i, and then we apply the Chinese remainder theorem to calculate m mod n.

Given a ciphertext c, to decrypt, we calculate

• ${\displaystyle c_i \equiv c^{\phi(n)/p_i} \mod n}$. Thus
${\displaystyle \begin{matrix} c^{\phi(n)/p_i} &\equiv& x^{\sigma \phi(n)/p_i} g^{m\phi(n)/p_i} \mod n\\ &\equiv& g^{(m_i + y_ip_i)\phi(n)/p_i} \mod n \\ &\equiv& g^{m_i\phi(n)/p_i} \mod n \end{matrix}}$

where ${\displaystyle m_i \equiv m \mod p_i}$.

• Since pi is chosen to be small, mi can be recovered be exhaustive search, i.e. by comparing ${\displaystyle c_i}$ to ${\displaystyle g^{j\phi(n)/p_i}}$ for j from 1 to pi-1.
• Once mi is known for each i, m can be recovered by a direct application of the Chinese remainder theorem.

## Security

The semantic security of the Naccache–Stern cryptosystem rests on an extension of the quadratic residuosity problem known as the higher residuosity problem.