Naccache–Stern knapsack cryptosystem

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

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. 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 $x \in \{0,1\}^n$ such that

$c \equiv \prod_{i=0}^n v_i^{x_i} \mod p$

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 $\prod_{i=0}^np_i < p$.
• Pick a secret integer s < p-1, such that gcd(p-1,s) = 1.
• Set $v_i = \sqrt[s]{p_i} \mod p$.

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

$c = \prod_{i=0}^n v_i^{m_i} \mod p$

where mi is the ith bit of the message m.

Decryption

To decrypt a message c, calculate

$m = \sum_{i=0}^n \frac{2^i}{p_i-1} \times \left( \gcd(p_i,c^s \mod p) -1 \right)$

This works because the fraction

$\frac{ \gcd(p_i,c^s \mod p) - 1 }{p_i - 1}$

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