Naccache–Stern cryptosystem

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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

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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

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  • 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

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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

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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

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

References

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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.