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
[edit]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
[edit]- Pick a family of k small distinct primes p1,...,pk.
- Divide the set in half and set and .
- Set
- 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.
Message Encryption
[edit]This system allows encryption of a message m in the group .
- Pick a random .
- Calculate
Then E(m) is an encryption of the message m.
Message Decryption
[edit]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
where .
- 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.
Security
[edit]The semantic security of the Naccache–Stern cryptosystem rests on an extension of the quadratic residuosity problem known as the higher residuosity problem.
References
[edit]Naccache, David; Stern, Jacques (1998). "A New Public Key Cryptosystem Based on Higher Residues". Proceedings of the 5th ACM Conference on Computer and Communications Security. CCS '98. ACM. pp. 59–66. doi:10.1145/288090.288106. ISBN 1-58113-007-4.