Transitive polynomial transformations of residue class rings

2002 ◽  
Vol 12 (2) ◽  
Author(s):  
M.V. Larin
Keyword(s):  
Residue Class ◽  
Integer Coefficients ◽  
Polynomial Transformations

AbstractWe give a complete description of the polynomials f(x) with integer coefficients such that the period of the recurring sequence u

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

2006 ◽  
Vol 149 (1) ◽  
pp. 31-41 ◽  
Author(s):  
Anca Iuliana Bonciocat ◽  
Alexandru Zaharescu
Keyword(s):  
Integer Coefficients

scholarly journals Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 710
Author(s):  
Modjtaba Ghorbani ◽  
Maryam Jalali-Rad ◽  
Matthias Dehmer
Keyword(s):  
Integer Coefficients ◽  
Integer Polynomial ◽  
Counting Polynomial

Suppose ai indicates the number of orbits of size i in graph G. A new counting polynomial, namely an orbit polynomial, is defined as OG(x) = ∑i aixi. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.

Application of formal languages in polynomial transformations of instances between NP-complete problems

2013 ◽  
Vol 14 (8) ◽  
pp. 623-633
Author(s):  
Jorge A. Ruiz-Vanoye ◽  
Joaquín Pérez-Ortega ◽  
Rodolfo A. Pazos Rangel ◽  
Ocotlán Díaz-Parra ◽  
Héctor J. Fraire-Huacuja ◽  
...  
Keyword(s):  
Formal Languages ◽  
Complete Problems ◽  
Polynomial Transformations ◽  
Np Complete

Polynomials with Integer Coefficients

1992 ◽  
pp. 289-322
Author(s):  
Maurice Mignotte
Keyword(s):  
Integer Coefficients

On the Computation of Minimal Free Resolutions with Integer Coefficients

2019 ◽  
pp. 51-72
Author(s):  
Soda Diop ◽  
Guy Mobouale Wamba ◽  
Andre Saint Eudes Mialebama Bouesso ◽  
Djiby Sow
Keyword(s):  
Free Resolutions ◽  
Integer Coefficients ◽  
Minimal Free Resolutions

scholarly journals On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field

2021 ◽  
Author(s):  
A. Haddley ◽  
R. Nair
Keyword(s):  
Continued Fraction ◽  
Residue Class ◽  
Natural Extension ◽  
Abelian Topological Group ◽  
Ring Of Integers ◽  
Residue Class Field ◽  
Finite Set ◽  
Compact Abelian Topological Group ◽  
Invertible Elements ◽  
Average Behaviour

AbstractLet $${\mathcal {M}}$$ M denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote $$k^{\times }$$ k × , and a uniformizer we denote $$\pi $$ π . In this paper, we consider the map $$T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}$$ T v : M → M defined by $$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$ T v ( x ) = π v ( x ) x - b ( x ) , where b(x) denotes the equivalence class to which $$\frac{\pi ^{v(x)}}{x}$$ π v ( x ) x belongs in $$k^{\times }$$ k × . We show that $$T_v$$ T v preserves Haar measure $$\mu $$ μ on the compact abelian topological group $${\mathcal {M}}$$ M . Let $${\mathcal {B}}$$ B denote the Haar $$\sigma $$ σ -algebra on $${\mathcal {M}}$$ M . We show the natural extension of the dynamical system $$({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)$$ ( M , B , μ , T v ) is Bernoulli and has entropy $$\frac{\#( k)}{\#( k^{\times })}\log (\#( k))$$ # ( k ) # ( k × ) log ( # ( k ) ) . The first of these two properties is used to study the average behaviour of the convergents arising from $$T_v$$ T v . Here for a finite set A its cardinality has been denoted by $$\# (A)$$ # ( A ) . In the case $$K = {\mathbb {Q}}_p$$ K = Q p , i.e. the field of p-adic numbers, the map $$T_v$$ T v reduces to the well-studied continued fraction map due to Schneider.

scholarly journals File Concealment using the Paillier Method and RGB Intensity Based Steganography

2018 ◽  
Author(s):  
Andysah Putera Utama Siahaan ◽  
Solly Aryza
Keyword(s):  
Large Scale ◽  
Residue Class ◽  
Public Key Cryptography ◽  
Extraction Process ◽  
Color Intensity ◽  
Stego Image ◽  
Large Scale Data ◽  
Paillier Cryptosystem ◽  
Scale Data ◽  
Rgb Image

Steganography is related to the addition of information to a given medium (referred to as cover media) without making visible changes to it. Most of the proposed steganography techniques cannot be applied to store large-scale data. In the new technique for RGB image steganography, color intensity (R-G-B) is used to determine the number of bits you want to store in each pixel. Meanwhile, to improve the security of stored confidential files, cryptographic methods will be applied. The Paillier cryptosystem invented by Pascal Paillier in 1999 is a probabilistic asymmetric algorithm for public key cryptography. The security of the Paillier algorithm depends on the problem of calculating the n-residue class that is believed to be very difficult to compute. This problem is known as the Composite Residuosity (CR) and is the basis of this Paillier cryptosystem. The software created can save secret files into a digital image into a stego image. The secret file can be extracted out through the extraction process.

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