Naccache–Stern cryptosystem

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 p**i 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 p**i is chosen to be small, m**i can be recovered by exhaustive search, i.e. by comparing c_i to g^{j\phi(n)/p_i} for j from 1 to p**i-1.
• Once m**i 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

| book-title = Proceedings of the 5th ACM Conference on Computer and Communications Security | doi-access = free