scholarly journals Splitting of PG(1,27) by Sets, Orbits, and Arcs on the Conic

2021 ◽  
pp. 1979-1985
Author(s):  
Emad Bakr Abdulkareem
Keyword(s):  
Finite Field ◽  
Projective Line ◽  
Companion Matrix ◽  
Standard Frame

This research aims to give a splitting structure of the projective line over the finite field of order twenty-seven that can be found depending on the factors of the line order. Also, the line was partitioned by orbits using the companion matrix. Finally, we showed the number of projectively inequivalent -arcs on the conic  through the standard frame of the plane PG(1,27)

Commutators of Matrices with Prescribed Determinant

1968 ◽  
Vol 20 ◽  
pp. 203-221 ◽  
Author(s):  
R. C. Thompson
Keyword(s):  
Finite Field ◽  
Multiplicative Group ◽  
Companion Matrix ◽  
Identity Matrix ◽  
Commutative Field

Let K be a commutative field, let GL(n, K) be the multiplicative group of all non-singular n × n matrices with elements from K, and let SL(n, K) be the subgroup of GL(n, K) consisting of all matrices in GL(n, K) with determinant one. We denote the determinant of matrix A by |A|, the identity matrix by In, the companion matrix of polynomial p(λ) by C(p(λ)), and the transpose of A by AT. The multiplicative group of nonzero elements in K is denoted by K*. We let GF(pn) denote the finite field having pn elements.

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.

Erdös-Ko-Rado theorem in some linear groups and some projective special linear group

2014 ◽  
Vol 51 (1) ◽  
pp. 83-91
Author(s):  
Milad Ahanjideh ◽  
Neda Ahanjideh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Prime Number ◽  
Linear Group ◽  
Projective Line ◽  
Special Linear Group ◽  
Linear Groups ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Derangement Graph

Let V be the 2-dimensional column vector space over a finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document} (where q is necessarily a power of a prime number) and let ℙq be the projective line over \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. In this paper, it is shown that GL2(q), for q ≠ 3, and SL2(q) acting on V − {0} have the strict EKR property and GL2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GLn(q) acting on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$ \end{document} has the EKR property and the derangement graph of PSL2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.

scholarly journals Certain Types of Linear Codes over the Finite Field of Order Twenty-Five

2021 ◽  
pp. 4019-4031
Author(s):  
Emad Bakr Al-Zangana ◽  
Elaf Abdul Satar Shehab
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Weight Distribution ◽  
Incidence Matrix ◽  
Linear Codes ◽  
Projective Line ◽  
Three Dimensions ◽  
Hamming Weight ◽  
Maximum Distance Separable ◽  
Maximum Distance Separable Codes

The aim of the paper is to compute projective maximum distance separable codes, -MDS of two and three dimensions with certain lengths and Hamming weight distribution from the arcs in the projective line and plane over the finite field of order twenty-five. Also, the linear codes generated by an incidence matrix of points and lines of  were studied over different finite fields.  

scholarly journals Results on the Projective Plane over a Finite Field of Order Seventeen

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Najm A. M. AL-Seraji ◽  
Hussam. H. Jawad
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Projective Line ◽  
Projective Mapping ◽  
Cubic Curve ◽  
Inflection Points ◽  
Stabilizer Group

The main goal of this research is to find the projective mapping that transforms a geometric formation called an i -set onto an arc such that the domain of the mapping is a subset of the projective line PG (1,q), q=17 , such that a5-set is called a pentad, a6-set is a hexad, a7-set is a heptad, a8-set is an octad, and a9 -set is a nonad, mapped onto a conicY2-XZ. The research also aims to find the stabilizer group of points on a non-singular cubic curve, with or without rational inflection points, on the projective plane over a finite field of order seventeen, and to give some examples.

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.

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.

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