Naccache–Stern cryptosystem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

  • 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[edit]

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[edit]

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[edit]

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

References[edit]

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.