Knapsack cryptosystems built on NP-hard instance

We construct three public key knapsack cryptosystems. Standard knapsack cryptosystems hide easy instances of the knapsack problem and have been broken. The systems considered in the article face this problem: They hide a random (possibly hard) instance of the knapsack problem. We provide both complexity results (size of the key, time needed to encypher/decypher...) and experimental results. Security results are given for the second cryptosystem ( the fastest one and the one with the shortest key). Probabilistic polynomial reductions show that finding the private key is as difficult as factorizing a product of two primes. We also consider heuristic attacks. First, the density of the cryptosystem can be chosen arbitrarily close to one, discarding low density attacks. Finally, we consider explicit heuristic attacks based on the LLL algorithm and we prove that with respect to these attacks, the public key is as secure as a random key.