Math (commutative algebra, number theory, cryptology, Galois theory)

1. M. Jankowska, L. Matysiak, “A polynomial composites and monoid domains as algebraic structures and their applications”, 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., Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 21:3 (2021), 22
2. L. Matysiak, “Generalized RSA cipher and Diffie-Hellman protocol”, In this paper I am considering several cryptological threads. The problem of the RSA cipher, like the Diffie-Hellman protocol, is the use of finite sets. In this paper, I generalize the RSA cipher and DH protocol for infinite sets using monoids. In monoids we can not find the inverse, which makes it difficult. In the second part of the paper I show the applications in cryptology of polynomial composites and monoid domains. These are less known structures. In this work, I show different ways of encrypting messages based on infinite sets., Journal Applied Mathematics, 39:1-2 (2021), 11
3. P. Jedrezjewicz; M. Marciniak; L. Matysiak; J. Zielinski, “On properties of square-free elements in commutative cancellative monoids”, We discuss various square-free factorizations in monoids in the context of: atomicity, ascending chain condition for principal ideals, decomposition, and a greatest common divisor property. Moreover, we obtain a full characterization of submonoids of factorial monoids in which all square-free elements of a submonoid are square-free in a monoid. We also present a factorial property implying that all atoms of a submonoid are square-free in a monoid, Semigroup Forum, 100 (2020), 21
4. Łukasz Matysiak, Square-Free Factorizations. Some Cryptological and Other Mathematical Aspects, Eliva Press, 2024
5. Jędrzejewicz, P., Marciniak, M., Matysiak, Ł., Zieliński, J., “On properties of square-free elements in commutative cancellative monoids”, Semigroup Forum, 100 (2020), 850-870  
6. Matysiak, Ł., “Generalized RSA cipher and Diffie-Hellman protocol”, Journal of applied mathematics & informatics, 39:1-2 (2021), 93-103

• Kazimierz Wielki University
• Military University of Technology, Warsaw