Analysis of two knapsack public key cryptosystems

Two knapsack-based public key cryptosystems were proposed recently, in which the entries of the secret knapsack sequences are composed of products of random integers and the knapsack problems have nonbinary solutions. These features make the cryptosystems to be secure against low density attacks. In this study, the authors present lattice-based complete private key recovery attacks on these two schemes. The authors attacks firstly find short vectors related to the public key and with some orthogonality to the secret knapsack sequences and then recover some components of the private key by computing common factors of the entries of the short vectors. Especially, the authors attack can both completely recover the unique key for these two schemes. The attacks are of practical complexities and verified by experiments.