A polynomial composites and monoid domains as algebraic structures and their applications

TL;DR

This paper explores algebraic properties of polynomial composites and monoid domains, investigates their connections with Galois theory and nilpotents, and applies these structures to develop and generalize cryptosystems.

Contribution

It introduces new results on polynomial composites and monoid domains, linking them with Galois theory, nilpotent elements, and cryptography, including generalizations of known ciphers.

Findings

Polynomial composites exhibit specific algebraic properties.

Connections established between polynomial composites, Galois theory, and nilpotent elements.

New cryptosystems based on polynomial composites and monoid domains are proposed.

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 cryptosystems. 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.