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.
- Pick a family of k small distinct primes p1,...,pk.
- Divide the set in half and set and .
- 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.
This system allows encryption of a message m in the group .
- Pick a random .
Then E(m) is an encryption of the message m.
To decrypt, we first find m mod pi for each i, and then we apply the Chinese remainder theorem to calculate m mod .
Given a ciphertext c, to decrypt, we calculate
- . Thus
- Since pi is chosen to be small, mi can be recovered by 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.