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.

Scheme Definition

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

Pick a family of k small distinct primes p₁,...,p_k.
Divide the set in half and set $u=\prod _{{i=1}}^{{k/2}}p_{i}$ and $v=\prod _{{k/2+1}}^{k}p_{i}$ .
Set $\sigma =uv=\prod _{{i=1}}^{k}p_{i}$
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.

This system allows encryption of a message m in the group ${\mathbb {Z}}/\sigma {\mathbb {Z}}$ .

Then E(m) is an encryption of the message m.

To decrypt, we first find m mod p_i for each i, and then we apply the Chinese remainder theorem to calculate m mod $\sigma$ .

Given a ciphertext c, to decrypt, we calculate

{\begin{matrix}c^{{\phi (n)/p_{i}}}&\equiv &x^{{\sigma \phi (n)/p_{i}}}g^{{m\phi (n)/p_{i}}}\mod n\\&\equiv &g^{{(m_{i}+y_{i}p_{i})\phi (n)/p_{i}}}\mod n\\&\equiv &g^{{m_{i}\phi (n)/p_{i}}}\mod n\end{matrix}}

where $m_{i}\equiv m\mod p_{i}$ .

Since p_i is chosen to be small, m_i can be recovered by exhaustive search, i.e. by comparing $c_{i}$ to $g^{{j\phi (n)/p_{i}}}$ for j from 1 to p_i-1.
Once m_i is known for each i, m can be recovered by a direct application of the Chinese remainder theorem.

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

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.

Algorithms

Integer Factorization	Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Paillier Rabin RSA Okamoto–Uchiyama Schmidt–Samoa

Discrete logarithm	Cramer–Shoup DH DSA ECDH ECDSA EdDSA EKE ElGamal signature scheme MQV Schnorr SPEKE SRP STS

Others	AE CEILIDH EPOC HFE IES Lamport McEliece Merkle–Hellman Naccache–Stern Naccache–Stern knapsack cryptosystem NTRUEncrypt NTRUSign Three-pass protocol XTR

Theory

Standardization

Topics

Cryptography

History of cryptography Cryptanalysis Outline of cryptography

Symmetric-key algorithm Block cipher Stream cipher Public-key cryptography Cryptographic hash function Message authentication code Random numbers Steganography

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