Puneet Varma (Editor)

Naccache–Stern knapsack cryptosystem

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

Note: this is not to be confused with the Naccache–Stern cryptosystem based on the higher residuosity problem.

Contents

The Naccache–Stern Knapsack Cryptosystem is an atypical public-key cryptosystem developed by David Naccache and Jacques Stern in 1997. This cryptosystem is deterministic, and hence is not semantically secure. While unbroken to date, this system also lacks provable security.

System Overview

This system is based on a type of knapsack problem. Specifically, the underlying problem is this: given integers c,n,p and v0,...,vn, find a vector x ∈ { 0 , 1 } n such that

c ≡ ∏ i = 0 n v i x i mod p

The idea here is that when the vi are relatively prime and much smaller than the modulus p this problem can be solved easily. It is this observation which allows decryption.

Key Generation

To generate a public/private key pair

  • Pick a large prime modulus p.
  • Pick a positive integer n and for i from 0 to n, set pi to be the ith prime, starting with p0 = 2 and such that ∏ i = 0 n p i < p .
  • Pick a secret integer s < p-1, such that gcd(p-1,s) = 1.
  • Set v i = p i s mod p .

    • The public key is then p,n and v0,...,vn. The private key is s.

    Encryption

    To encrypt an n-bit long message m, calculate

    c = ∏ i = 0 n v i m i mod p

    where mi is the ith bit of the message m.

    Decryption

    To decrypt a message c, calculate

    m = ∑ i = 0 n 2 i p i − 1 × ( gcd ( p i , c s mod p ) − 1 )

    This works because the fraction

    gcd ( p i , c s mod p ) − 1 p i − 1

    is 0 or 1 depending on whether pi divides cs mod p.

    References

    Naccache–Stern knapsack cryptosystem Wikipedia
    (Text) CC BY-SA