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Naccache–Stern cryptosystem

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Title: Naccache–Stern cryptosystem  
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Language: English
Subject: Naccache–Stern knapsack cryptosystem, Homomorphic encryption, Computational hardness assumption, Benaloh cryptosystem, CEILIDH
Publisher: World Heritage Encyclopedia

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 1
    • Key Generation 1.1
    • Message Encryption 1.2
    • Message Decryption 1.3
  • Security 2
  • References 3

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.


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


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