Extensive studies have been made of the public-key cryptosystems based on multivariate polynomials. However most of the proposed public-key cryptosystems of the rate 1.0 based on multivariate polynomials, are proved not secure, although the size of the public-key assumes very large value. 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 size of the public-key is much shortened compared with the conventional public-key cryptosystem based on multivariate polynomials. We also show that the sets of random simultaneous equations significantly improve the utilization factor of the public-key space. We show an example of the proposed cryptosystem over extension field where the size of the cipher-text takes on the small values between 80 and 160. We see that our proposed system, regardless of the small size of public-key, seems to be apparently secure, in a sense that the utilization factor is sufficiently large compared with the conventional public-key cryptosystems based on multivariate polynomials.
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