Naccache–Stern knapsack cryptosystem
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 such that
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.To generate a public/private key pair
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
where mi is the ith bit of the message m.Decryption
To decrypt a message c, calculate
This works because the fraction
is 0 or 1 depending on whether pi divides cs mod p.
