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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 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 and .
  • Set
  • 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 .

  • Pick a random .
  • Calculate

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

  • . Thus

where .

  • Since pi is chosen to be small, mi can be recovered be exhaustive search, i.e. by comparing to 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.


Original paper