Naccache–Stern cryptosystem

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.

Like many public key cryptosystems, this scheme works in the group ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{*}}$ where n is a product of two large primes. This scheme is homomorphic and hence malleable.

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 ${\displaystyle \mathbb {Z} /\sigma \mathbb {Z} }$.

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