Rings of Multisets and Integer Multinumbers

Abstract

In the paper, we consider a ring structure on the Cartesian product of two sets of integer multisets. In this way, we introduce a ring of integer multinumbers as a quotient of the Cartesian product with respect to a natural equivalence. We examine the properties of this ring and construct some isomorphisms to subrings of polynomials and Dirichlet series with integer coefficients. In addition, we introduce finite rings of multinumbers “modulo (p,q)” and propose an algorithm for construction of invertible elements in these rings that may be applicable in Public-key Cryptography. An analog of the Little Fermat Theorem for integer multinumbers is proved.

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

“…These results are at the intersection of combinatorics and functional analysis. On the other hand, symmetric and supersymmetric polynomials are applicable in cryptography [11] and quantum physics [10,41]. Therefore, we can expect that the obtained relations will be useful for modeling quantum ideal gases and in the information theory.…”

“…Applications of algebraic bases of supersymmetric polynomials to models of ideal gases in quantum physics were proposed in [10]. Supersymmetric polynomials over finite fields and their applications in cryptography were considered in [11]. Supersymmetric polynomials on finite-dimensional vector spaces were studied in [12,13].…”

Applications of Supersymmetric Polynomials in Statistical Quantum Physics

“…for fermions, where x i are defined by (16). In addition, according to [2], the co-ordinates (x 1 , x 2 , .…”

“…In [15][16][17], supersymmetric polynomials and analytic functions of abstract Banach spaces were considered. The supersymmetric polynomials of several variables were studied in [18][19][20].…”

Spectra of algebras of block-symmetric analytic functions of bounded type

“…In the literature, S ∞ -symmetric functions often are called symmetric. Symmetric functions of infinitely many variables are important objects in the nonlinear functional analysis [13,20] and are applicable in different areas of the information theory and statistical physics [11,33,38].…”