Abstract
This paper contains the results collected so far on polynomial composites in terms of many basic algebraic properties. Since it is a polynomial structure, results for monoid domains come in here and there. The second part of the paper contains the results of the relationship between the theory of polynomial composites, the Galois theory and the theory of nilpotents. The third part of this paper shows us some crypto systems. We find generalizations of known ciphers taking into account the infinite alphabet and using simple algebraic methods. We also find two cryptosystems in which the structure of Dedekind rings resides, namely certain elements are equivalent to fractional ideals. Finally, we find the use of polynomial composites and monoid domains in cryptology.
Highlights
Let N = {1, 2, . . . }, N0 = {0, 1, 2, . . . }
By a ring we mean a commutative ring with unity
We will show cryptosystems based on polynomial composites and monoid domains
Summary
In the second section we can find many results about polynomial composites and monoid domains. Basic algebraic properties such as irreducible elements, nilpotents and ideals have been examined. In Theorem 2.23 it turns out that the polynomial composite of the form K + XL[X] (where K ⊂ L be fields) is a Dedekind ring This is a very important class of rings in algebra. Sections four and five are reminder from [12] a generalized RSA cipher and a Diffie-Hellman protocol key exchange Such a reminder is purposeful because we want to draw attention to the replacement of the finite alphabet with the infinite one and the replacement of classical prime numbers with prime ideals. This is a great difficulty in breaking the last cryptosystem
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