# Naccache–Stern cryptosystem

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 $(\mathbb{Z}/n\mathbb{Z})^*$ 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 $u = \prod_{i=1}^{k/2} p_i$ and $v = \prod_{k/2+1}^k p_i$.
• Set $\sigma = uv = \prod_{i=1}^k p_i$
• 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 $\mathbb{Z}/\sigma\mathbb{Z}$.

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

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 $\sigma$.

Given a ciphertext c, to decrypt, we calculate

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

where $m_i \equiv m \mod p_i$.

• Since pi is chosen to be small, mi can be recovered by exhaustive search, i.e. by comparing $c_i$ to $g^{j\phi(n)/p_i}$ 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.

## References

Naccache, David; Stern, Jacques (1998). "A New Public Key Cryptosystem Based on Higher Residues". "Proceedings of the 5th ACM Conference on Computer and Communications Security". CCS '98. ACM. pp. 59–66. doi:10.1145/288090.288106. ISBN 1-58113-007-4.