Note: this is not to be confused with the Naccache–Stern knapsack cryptosystem.
Contents
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
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
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.