Finite Semigroups and The RSA-Cryptosystem

Abstract

A closer look at the RSA-cryptosystem reveals that its main feature are permutation-polynomials x^c(c>1) over the multiplicative semigroup Z_m of integers modulo m. Thus it is quite natural to see whether there are other finite semigroups S for which permutation polynomials esists. It is quite clear from Z_m that one has to put certain restrictions on S to guarantee the existence of permutation-polynomials x^c. This problem is closely related to that of the ideal generalization of the Euler-Fermat theorem studied recently by [Ecker, 1980 and [Schwarz, 1981]]. Section 2. of this paper gives for reasons of completeness a description of the structure of finite semigroups from [Hewitt and Zuckerman, 1960] and [Lyapin, 1974]. In section 3. the Euler-Fermat theorem in S is treated as in [Ecker, 1980]. In section 4. polynomial-functions x^c over S are considered and necessary and sufficient conditions are given for the existence of permutation-polynomials x^c. Besides that fixed points or alternatively solutions of x^c=x are treated. In section 5. we look at S=Z_m from the point of view underlying sections 2.–4. For this approach although with quite different motivation see also [Hewitt and Zuckerman, 1960] and [Schwarz, 1981]. In section 6. three examples of finite semigroups are studiedj that might serve as a basis for an extended RSA-cryptosystem. We show that the most promising of those semigroups is the multiplicative semigroup of matrices over Z_m.