Naccache–Stern 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 formula_1 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 formula_5. Then E(m)" is an encryption of the message "m". Message Decryption. To decrypt, we first find "m" mod "p" for each "i", and then we apply the Chinese remainder theorem to calculate "m" mod formula_8. Given a ciphertext "c", to decrypt, we calculate where formula_11. Security. The semantic security of the Naccache–Stern cryptosystem rests on an extension of the quadratic residuosity problem known as the higher residuosity problem.