Public key cryptosystems are constructed by embedding a trapdoor into a one-way function. So, the one-wayness and the trapdoorness are vital to public key cryptography. In this paper, we propose a novel public key cryptographic primitive called preimage selective trapdoor function. This scenario allows to use exponentially many preimage to hide a plaintext even if the underlying function is not one-way. The compact knapsack problem is used to construct a probabilistic public key cryptosystem, the underlying encryption function of which is proven to be preimage selective trapdoor one-way functions under some linearization attack models. The constructive method can guarantee the noninjectivity of the underlying encryption function and the unique decipherability for ciphertexts simultaneously. It is heuristically argued that the security of the proposal cannot be compromised by a polynomial-time adversary even if the compact knapsack is easy to solve. We failed to provide any provable security results about the proposal; however, heuristic illustrations show that the proposal is secure against some known attacks including brute force attacks, linearization attacks, and key-recovery attacks. The proposal turns out to have acceptable key sizes and performs efficiently and hence is practical
