Cryptography is the fundamental of secure communication. A public key cryptosystem is necessary to realize secure and efficient communication on public network. To construct a public key cryptosystem, it is essential to find a one-way trap-door function. In this paper, we propose a new one-way trap-door function, and present corresponding encryptosystem and entity authentication scheme. Meanwhile, it is proven that our algorithm and schemes are feasible, secure and efficient. Compared with recent algorithm such as RSA or ElGamal systems, our algorithm is more secure under the condition of the nearly similar efficiency.