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

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. 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 where.

Security

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