[Paper Review] On properties of composites and monoid domains

Łukasz Matysiak|arXiv (Cornell University)|Jun 25, 2020

Advanced Differential Equations and Dynamical Systems12 references4 citations

TL;DR

This paper provides a comprehensive algebraic characterization of polynomial composites and monoid domains in commutative algebra, establishing foundational properties such as invertible and irreducible elements, ideals, and graded structures. It further demonstrates their application in cryptology by constructing multilayered encryption schemes using coefficient-wise polynomial composition and monoid domains over finite fields, leveraging non-invertible monoid structures to enhance security.

ABSTRACT

In this paper I consider all possible properties from commutative algebra for polynomial composites and monoid domains. The aim is full characterization of these structures. I start with the examination of group, ring, modules properties, graded, but also the study of invertible elements, irreducible elements, ideals, etc. in these structures. In the second part of the work I give examples of the use of composites and monoid domains in cryptology. Each such polynomial is the sum of the products of the variable and the coefficient. And what if subsequent coefficient sets are appropriate cryptographic systems? Similarly, monoid domains can be a very good tool between encrypting and decrypting messages.

Motivation & Objective

• To fully characterize algebraic properties—such as invertible elements, irreducible elements, ideals, and module structures—of polynomial composites and monoid domains in commutative algebra.
• To generalize the concept of composites beyond standard forms, introducing new types $T_n$ and $T_n'$, and studying their ring-theoretic behavior.
• To investigate the structural role of monoid domains $F[M]$ where $M$ is a submonoid of $\mathbb{Q}_+$, particularly in relation to graded rings and modules.
• To explore the use of these algebraic structures in cryptology, proposing a novel multilayered encryption framework based on coefficient-wise polynomial composition.
• To establish a bridge between abstract algebra and applied cryptography by demonstrating how composites and monoid domains can serve as secure, non-invertible carriers for encrypted messages.

Proposed method

• Define composites as $T = A + XB[X]$ for rings $A \subset B$, and generalize to $T_n = A_0 + A_1X + \cdots + A_{n-1}X^{n-1} + X^nB[X]$ with varying coefficient rings.
• Introduce $I(B,A) = \{f \in B[X] \mid f(A) \subseteq A\}$ as a ring-theoretic construction to study endomorphism-like behavior.
• Construct monoid domains $F[M]$ for a field $F$ and submonoid $M \subset \mathbb{Q}_+$, using formal Laurent-like polynomials with exponents in $M$.
• Apply the algebraic framework to cryptology by modeling encryption as polynomial multiplication: $fg$ encodes message blocks via coefficient-wise operations.
• Use inverse mappings such as $(A_iB_j)^{-1}$ to decrypt pairs of letters, leveraging known cipher types (e.g., Caesar shifts) for reconstruction.
• Model alphabets as $\mathbb{N}_0$-indexed cycles, and map characters to monoid domain elements via $\varphi(m_i) = a_iX^{m_i}$, enabling infinite-alphabet-like behavior with finite fields.

Experimental results

Research questions

• RQ1 What are the complete algebraic properties—such as invertible elements, irreducibility, and ideal structure—of generalized composites $T_n$ and $T_n'$?
• RQ2 How do monoid domains $F[M]$ with $M \subset \mathbb{Q}_+$ behave as rings, and what is their structure as graded modules and algebras?
• RQ3 Can composites and monoid domains be systematically used to construct secure, multilayered encryption schemes in cryptology?
• RQ4 What is the role of non-invertible monoid structures in enhancing cryptographic security compared to standard finite-field ciphers?
• RQ5 How can polynomial composition in $T = A + XB[X]$ be used to encode and decode messages block-wise using multiple cipher types applied per coefficient?

Key findings

• The structure $T_n' = A_0 + A_1X + \cdots + A_{n-1}X^{n-1} + X^nB[X]$ with non-nesting coefficient rings fails to be closed under multiplication, so $(T_n', \cdot)$ is not a semigroup.
• In composites of the form $D + XK[X]$ with $D$ a domain and $K$ a field, every nonzero prime ideal is maximal, indicating a strong ideal-theoretic constraint.
• Irreducible elements in composites are characterized by their constant term being irreducible in $A$ and higher-degree coefficients satisfying specific coprimality conditions.
• The composite $I(B,A)$ of polynomials preserving $A$ under evaluation forms a subring of $B[X]$, providing a new class of polynomial constructions.
• Monoid domains $F[M]$ with $M \subset \mathbb{Q}_+$ allow for infinite-like character encoding via $\mathbb{N}_0$-indexed cycles, enabling non-cyclic, extended-alphabet ciphers.
• A working example shows that a degree-3 composite $fg$ with coefficient ciphers (e.g., Caesar shifts forward/backward) can encrypt an 8-letter message into 16 letters, which Bob decrypts block by block using inverse ciphers, successfully recovering the original message.