World Library  
Flag as Inappropriate
Email this Article

Naccache–Stern cryptosystem

Article Id: WHEBN0006149634
Reproduction Date:

Title: Naccache–Stern cryptosystem  
Author: World Heritage Encyclopedia
Language: English
Subject: Naccache–Stern knapsack cryptosystem, Homomorphic encryption, Computational hardness assumption, Benaloh cryptosystem, CEILIDH
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

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.

Contents

  • Scheme Definition 1
    • Key Generation 1.1
    • Message Encryption 1.2
    • Message Decryption 1.3
  • Security 2
  • References 3

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.

Key Generation

  • Pick a family of k small distinct primes p1,...,pk.
  • 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.

Message Encryption

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

  • Pick a random x \in \mathbb{Z}/n\mathbb{Z}.
  • Calculate E(m) = x^\sigma g^m \mod n

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

Message Decryption

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

Given a ciphertext c, to decrypt, we calculate

  • c_i \equiv c^{\phi(n)/p_i} \mod n. Thus
\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_ip_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 pi is chosen to be small, mi can be recovered by exhaustive search, i.e. by comparing c_i to g^{j\phi(n)/p_i} 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

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

References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.