scholarly journals A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map

2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Abdul Razaq ◽  
Awais Yousaf ◽  
Umer Shuaib ◽  
Nasir Siddiqui ◽  
Atta Ullah ◽  
...  
Keyword(s):  
Finite Field ◽  
Irreducible Polynomial ◽  
Projective Line ◽  
Statistical Analyses ◽  
Substitution Box ◽  
Basic Tool ◽  
The Matrix ◽  
Specific Order ◽  
The Cost

The substitution box is a basic tool to convert the plaintext into an enciphered format. In this paper, we use coset diagram for the action of PSL(2,Z) on projective line over the finite field GF29 to construct proposed S-box. The vertices of the cost diagram are elements of GF29 which can be represented by powers of α, where α is the root of irreducible polynomial px=x9+x4+1 over Z2. Let GF⁎29 denote the elements of GF29 which are of the form of even powers of α. In the first step, we construct a 16×16 matrix with the elements of GF⁎29 in a specific order, determined by the coset diagram. Next, we consider h:GF⁎29⟶GF28 defined by hα2n=ωn to destroy the structure of GF28. In the last step, we apply a bijective map g on each element of the matrix to evolve proposed S-box. The ability of the proposed S-box is examined by different available algebraic and statistical analyses. The results are then compared with the familiar S-boxes. We get encouraging statistics of the proposed box after comparison.

scholarly journals Construction of New S-Box Using Action of Quotient of the Modular Group for Multimedia Security

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Imran Shahzad ◽  
Qaiser Mushtaq ◽  
Abdul Razaq
Keyword(s):  
Finite Field ◽  
Modular Group ◽  
Fibonacci Sequence ◽  
Projective Line ◽  
Multimedia Security ◽  
New Technique ◽  
Substitution Box ◽  
Nonlinear Component ◽  
Coset Diagrams ◽  
A New Technique

Substitution box (S-box) is a vital nonlinear component for the security of cryptographic schemes. In this paper, a new technique which involves coset diagrams for the action of a quotient of the modular group on the projective line over the finite field is proposed for construction of an S-box. It is constructed by selecting vertices of the coset diagram in a special manner. A useful transformation involving Fibonacci sequence is also used in selecting the vertices of the coset diagram. Finally, all the analyses to examine the security strength are performed. The outcomes of the analyses are encouraging and show that the generated S-box is highly secure.

scholarly journals A highly nonlinear substitution-box (S-box) design using action of modular group on a projective line over a finite field

PLoS ONE ◽  
2020 ◽  
Vol 15 (11) ◽  
pp. e0241890 ◽  
Author(s):  
Nasir Siddiqui ◽  
Fahim Yousaf ◽  
Fiza Murtaza ◽  
Muhammad Ehatisham-ul-Haq ◽  
M. Usman Ashraf ◽  
...  
Keyword(s):  
Finite Field ◽  
Image Encryption ◽  
Adjacency Matrix ◽  
Secure Communication ◽  
Modular Group ◽  
Galois Field ◽  
Projective Line ◽  
Substitution Box ◽  
Differential Probability ◽  
Highly Nonlinear

Cryptography is commonly used to secure communication and data transmission over insecure networks through the use of cryptosystems. A cryptosystem is a set of cryptographic algorithms offering security facilities for maintaining more cover-ups. A substitution-box (S-box) is the lone component in a cryptosystem that gives rise to a nonlinear mapping between inputs and outputs, thus providing confusion in data. An S-box that possesses high nonlinearity and low linear and differential probability is considered cryptographically secure. In this study, a new technique is presented to construct cryptographically strong 8×8 S-boxes by applying an adjacency matrix on the Galois field GF(28). The adjacency matrix is obtained corresponding to the coset diagram for the action of modular group PSL(2,Z) on a projective line PL(F7) over a finite field F7. The strength of the proposed S-boxes is examined by common S-box tests, which validate their cryptographic strength. Moreover, we use the majority logic criterion to establish an image encryption application for the proposed S-boxes. The encryption results reveal the robustness and effectiveness of the proposed S-box design in image encryption applications.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

scholarly journals On types of matrices and centralizers of matrices and permutations

2014 ◽  
Vol 17 (5) ◽  
Author(s):  
John R. Britnell ◽  
Mark Wildon
Keyword(s):  
Finite Field ◽  
Alternating Groups ◽  
Type Invariant ◽  
The Matrix ◽  
Generalized Type

AbstractIt is known that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.

scholarly journals A recognition algorithm for non-generic classical groups over finite fields

1999 ◽  
Vol 67 (2) ◽  
pp. 223-253 ◽  
Author(s):  
Azice C. Niemeyer ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Special Linear Group ◽  
Theoretical Background ◽  
Recognition Algorithm ◽  
Large Field ◽  
Matrix Group ◽  
Classical Groups ◽  
Original Algorithm ◽  
The Matrix

AbstractIn a previous paper the authors described an algorithm to determine whether a group of matrices over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm was designed to work for all sufficiently large field sizes and dimensions of the matrix group. However, it did not apply to certain small cases. Here we present an algorithm to handle the remaining cases. The theoretical background of the algorithm presented in this paper is a substantial extension of that needed for the original algorithm.

scholarly journals On the matrix Monge–Kantorovich problem

2019 ◽  
Vol 31 (4) ◽  
pp. 574-600 ◽  
Author(s):  
YONGXIN CHEN ◽  
WILFRID GANGBO ◽  
TRYPHON T. GEORGIOU ◽  
ALLEN TANNENBAUM
Keyword(s):  
Euclidean Distance ◽  
Strong Duality ◽  
Wasserstein Distance ◽  
Positive Definite ◽  
Type Result ◽  
Density Matrices ◽  
Wirtinger Inequality ◽  
The Matrix ◽  
The Cost ◽  
Kantorovich Problem

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

Representation of replacement rules in the form of a matrix

2016 ◽  
Vol 22 (2) ◽  
pp. 164-179
Author(s):  
Sergey Golovin
Keyword(s):  
Mathematical Models ◽  
Cost Structure ◽  
Verbal Description ◽  
System Component ◽  
Content Type ◽  
Maintenance Program ◽  
Repair Costs ◽  
System Components ◽  
The Matrix ◽  
The Cost

Purpose – The purpose of this paper is to represent replacement policies (rules) in the form of a matrix. Visualization of replacement rules is useful for maintenance records. Matrix representation is more effective than the verbal description usually provided, as it allows better understanding of the specifics of the different replacement rules without careful research of their mathematical models. Design/methodology/approach – This approach employs mathematical models to investigate the simple conditions (requirements) for replacement of system component with illustrative examples. When comparing the different replacement rules a cost structure is applied to takes into account the nature and technology of disassembly assembly actions for the repair unit. Findings – Representation of replacement rules in the matrix form is useful when describing planned replacement models, opportunity replacement models, group replacement models and others, as well as computer modeling of the renewal process. Forming simple conditions for the replacement of system components ensures the total average repair cost is minimized. These conditions can be applied in the early stages of creating a maintenance program for the machine. Practical implications – Replacement matrices can be specified in a technical manual for maintenance of machines to achieve reliable operation and to reduce repair costs. Replacement matrices can be put into practical use for maintenance records and may be included in the maintenance procedures library of CMMSs. Developed in the paper, the replacement matrix, the conditions for replacement of system components and the cost structure will help engineers to make decisions at the time of repair for assembly units. Originality/value – Proposed in the paper is a new approach to the visualization of the replacement rules and cost structure which simplifies the analysis of options for repair actions. The proposed technique contributes to the record of maintenance actions and the decision making process for replacement.

scholarly journals Constructing irreducible polynomials with prescribed level curves over finite fields

2001 ◽  
Vol 27 (4) ◽  
pp. 197-200
Author(s):  
Mihai Caragiu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Level Curves ◽  
Curves Over Finite Fields ◽  
Irreducibility Criterion ◽  
Absolutely Irreducible Polynomial

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomialP(X,Y)∈GF(q)[X,Y]with coefficients in the finite fieldGF(q)withqelements, with prescribed level curvesXc:={(x,y)∈GF(q)2|P(x,y)=c}.

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