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. While unbroken to date, this system also lacks provable security.

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

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

To generate a public/private key pair

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

To encrypt an n-bit long message m, calculate

where m_i is the ith bit of the message m.

To decrypt a message c, calculate

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