Summary: Extensive studies have been made of the public key cryptosystems based on multivariate polynomials. However most of the proposed public key cryptosystems of rate 1.0 based on multivariate polynomials, are proved not secure. In this paper, we propose several types of new constructions of public key cryptosystems based on two classes of randomly generated simultaneous equations, namely, a class based on bijective transformation and another class based on random transformation. One of the features of the proposed cryptosystems is that the sets of random simultaneous equations significantly improve the utilization factor of the transformation. We show an example of the proposed cryptosystem whose size of the ciphertext is only 100 bits.