There are choice collusion attack

random collusion attack and linear collusion attack in CPK.The user’s private key of this algorithm was generated by multiplying the basic private key with the inverse of the assistant private key.The basic private key and the assistant private key was the result of combining the elements of the basic private key matrix and the assistant private key matrix separately.The relationship among user’s private keys is nonlinear.By analyzing

this algorithm can defend from the choice collusion attack and random collusion attack that existed in CPK.The size of assistant key matrix can be chose according to need;the security of the system can be controlled.