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M. T. Mustafa

Publications and source records attributed to M. T. Mustafa.

At least 19 recordsLinked to original sources
arXiv2022

Algorithms for Simultaneous Block Triangularization and Block Diagonalization of Sets of Matrices

In a recent paper, a new method was proposed to find the common invariant subspaces of a set of matrices. This paper invstigates the more general problem of putting a set of matrices into block triangular or block-diagonal form simultaneously. Based on common invariant subspaces, two algorithms for simultaneous block triangularization and block diagonalization of sets of matrices are presented. As an alternate approach for simultaneous block diagonalization of sets of matrices by an invertible matrix, a new algorithm is developed based on the generalized eigen vectors of a commuting matrix. Moreover, a new characterization for the simultaneous block diagonalization by an invertible matrix is provided. The algorithms are applied to concrete examples using the symbolic manipulation system Maple.

math.GM↗
arXiv2021

A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties

The s-box plays the vital role of creating confusion between the ciphertext and secret key in any cryptosystem, and is the only nonlinear component in many block ciphers. Dynamic s-boxes, as compared to static, improve entropy of the system, hence leading to better resistance against linear and differential attacks. It was shown in [2] that while incorporating dynamic s-boxes in cryptosystems is sufficiently secure, they do not keep non-linearity invariant. This work provides an algorithmic scheme to generate key-dependent dynamic $n\times n$ clone s-boxes having the same algebraic properties namely bijection, nonlinearity, the strict avalanche criterion (SAC), the output bits independence criterion (BIC) as of the initial seed s-box. The method is based on group action of symmetric group $S_n$ and a subgroup $S_{2^n}$ respectively on columns and rows of Boolean functions ($GF(2^n)\to GF(2)$) of s-box. Invariance of the bijection, nonlinearity, SAC, and BIC for the generated clone copies is proved. As illustration, examples are provided for $n=8$ and $n=4$ along with comparison of the algebraic properties of the clone and initial seed s-box. The proposed method is an extension of [3,4,5,6] which involved group action of $S_8$ only on columns of Boolean functions ($GF(2^8)\to GF(2)$ ) of s-box. For $n=4$, we have used an initial $4\times 4$ s-box constructed by Carlisle Adams and Stafford Tavares [7] to generated $(4!)^2$ clone copies. For $n=8$, it can be seen [3,4,5,6] that the number of clone copies that can be constructed by permuting the columns is $8!$. For each column permutation, the proposed method enables to generate $8!$ clone copies by permuting the rows.

cs.CR↗
arXiv2020

Quantization of the Szekeres spacetime through generalized symmetries

We present the effect of the quantum corrections on the Szekeres spacetime, a system important for the study of the inhomogeneities of the pre-inflationary era of the universe. The study is performed in the context of canonical quantisation in the presence of symmetries. We construct an effective classical Lagrangian and impose the quantum version of its classical integrals of motion on the wave function. The interpretational scheme of the quantum solution is that of Bohmian mechanics, in which one can avoid the unitarity problem of quantum cosmology. We discuss our results in this context.

gr-qc↗
arXiv2020

An image encryption algorithm based on chaotic Lorenz system and novel primitive polynomial S-boxes

Nowadays, the chaotic cryptosystems are gaining more attention due to their efficiency, the assurance of robustness and high sensitivity corresponding to initial conditions. In literature, on one hand there are many encryption algorithms that only guarantee security while on the other hand there are schemes based on chaotic systems that only promise the uncertainty. Due to these limitations, each of these approaches cannot adequately encounter the challenges of current scenario. Here we take a unified approach and propose an image encryption algorithm based on Lorenz chaotic system and primitive irreducible polynomial S-boxes. First, we propose 16 different S-boxes based on projective general linear group and 16 primitive irreducible polynomials of Galois field of order 256, and then utilize these S-boxes with combination of chaotic map in image encryption scheme. Three chaotic sequences can be produced by the Lorenz chaotic system corresponding to variables $x$, $y$ and $z$. We construct a new pseudo random chaotic sequence $k_i$ based on $x$, $y$ and $z$. The plain image is encrypted by the use of chaotic sequence $k_i$ and XOR operation to get a ciphered image. To demonstrate the strength of presented image encryption, some renowned analyses as well as MATLAB simulations are performed.

cs.CR↗
arXiv2020

A novel encryption algorithm using multiple semifield S-boxes based on permutation of symmetric group

With the tremendous benefits of internet and advanced communications, there is a serious threat from the data security perspective. There is a need of secure and robust encryption algorithm that can be implemented on each and diverse software and hardware platforms. Also, in block symmetric encryption algorithms, substitution boxes are the most vital part. In this paper, we investigate semifield substitution boxes using permutation of symmetric group on a set of size 8 S_8 and establish an effective procedure for generating S_8 semifield substitution boxes having same algebraic properties. Further, the strength analysis of the generated substitution boxes is carried out using the well-known standards namely bijectivity, nonlinearity, strict avalanche criterion, bit independence criterion, XOR table and differential invariant. Based on the analysis results, it is shown that the cryptographic strength of generated substitution boxes is on par with the best known $8\times 8$ substitution boxes. As application, an encryption algorithm is proposed that can be employed to strengthen any kind of secure communication. The presented algorithm is mainly based on the Shannon idea of (S-P) network where the process of substitution is performed by the proposed S_8 semifield substitution boxes and permutation operation is performed by the binary cyclic shift of substitution box transformed data. In addition, the proposed encryption algorithm utilizes two different chaotic maps. In order to ensure the appropriate utilization of these chaotic maps, we carry out in-depth analyses of their behavior in the context of secure communication and apply the pseudo-random sequences of chaotic maps in the proposed image encryption algorithm accordingly. The statistical and simulation results imply that our encryption scheme is secure against different attacks and can resist linear and differential cryptanalysis.

cs.CR↗
arXiv2019

Invariant characterization of scalar third-order ODEs that admit the maximal contact symmetry Lie algebra

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits the maximal ten-dimensional contact symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently determine the contact transformation that does the reduction to the simplest linear equation $\bar{u}'''=0$. Furthermore, ample examples are given to illustrate our method.

math.CA↗
arXiv2018

Quantization of the Szekeres System

We study the quantum corrections on the Szekeres system in the context of canonical quantization in the presence of symmetries. We start from an effective point-like Lagrangian with two integrals of motion, one corresponding to the Hamiltonian and the other to a second rank Killing tensor. Imposing their quantum version on the wave function results to a solution which is then interpreted in the context of Bohmian mechanics. In this semiclassical approach, it is shown that there is no quantum corrections, thus the classical trajectories of the Szekeres system are not affected at this level. Finally, we define a probability function which shows that a stationary surface of the probability corresponds to a classical exact solution

gr-qc↗
arXiv2018

Linearization of third-order ordinary differential equations u'''=f(x,u,u',u'') via point transformations

The linearization problem by use of the Cartan equivalence method for scalar third-order ODEs via point transformations was solved partially in [1,2]. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation u'''=f(x,u,u',u'') which admits a four-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The method provides auxiliary functions which can be utilized to efficiently determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.

math.CA↗
arXiv2017

Invariant characterization of scalar third-order ODEs that admit the maximal point symmetry Lie algebra

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'"=f(x,u,u',u")$ which admits the maximal seven-dimensional point symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation $\bar{u}'"=0$. Moreover, examples are given to illustrate the method.

math.CA↗
arXiv2017

Invariant characterization of third-order ODEs $u'''=f(x,u,u',u'')$ that admit a five-dimensional point symmetry Lie algebra

The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function $f$ in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.

math.CA↗
arXiv2017

Classification of the Lie and Noether point symmetries for the Wave and the Klein-Gordon equations in pp-wave spacetimes

We perform a classification of the Lie and Noether point symmetries for the Klein-Gordon and for the wave equation in pp-wave spacetimes. To perform this analysis we reduce the problem of the determination of the point symmetries to the problem of existence of conformal killing vectors on the pp-wave spacetimes. We use the existing results of the literature for the isometry classes of the pp-wave spacetimes and we determine in each class the functional form of the potential in which the Klein-Gordon equation admits point symmetries and Noetherian conservation law. Finally we derive the point symmetries of the wave equation and we find that the maximum Noether algebra has dimension seven, that is the case of plane wave spacetimes.

math-ph↗
arXiv2014

Symmetry analysis of the Klein-Gordon equation in Bianchi I spacetimes

In this work we perform the symmetry classification of the Klein Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also and Noether symmetries for the Klein Gordon equation. We use these resutls in order to determine all the potentials in which the Klein Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.

math-ph↗
arXiv2014

On computing joint invariants of vector fields

A constructive version of the Frobenius integrability theorem -- that can be programmed effectively -- is given. This is used in computing invariants of groups of low ranks and recover examples from a recent paper of Boyko, Patera and Popoyvich \cite{BPP}.

math.DG↗
arXiv2014

Invariants of third-order ordinary differential equations $y'''=f(x,y,y',y'')$ via fiber preserving transformations

Bagderina \cite{Bagderina2008} solved the equivalence problem for scalar third-order ordinary differential equations (ODEs), quadratic in the second-order derivative, via point transformations. However, the question is open for the general class $y'''=f(x,y,y',y'')$ which is not quadratic in the second-order derivative. We utilize Lie's infinitesimal method to study the differential invariants of this general class under pseudo-group of fiber preserving equivalence transformations $\bar{x}=ϕ(x), \bar{y}=ψ(x,y)$. As a result, all third-order differential invariants of this group and the invariant differentiation operators are determined. This leads to simple necessary explicit conditions for a third-order ODE to be equivalent to the respective canonical form under the considered group of transformations. Applications motivated by the literature are presented.

math.CA↗
arXiv2014

Higher order self-adjoint operators with polynomial coefficients

Algebraic and analytic aspects of self-adjoint operators of order four or more with polynomial coefficients are investigated. As a consequence, a systematic way of constructing such operators is given. The procedure is applied to obtain many examples up to order 8; similar examples can be constructed for all even order operators. In particular, a complete classification of all order 4 operators is given.

math.CA↗
arXiv2013

$λ$-symmetry criteria for linearization of second order ODEs via point transformations

An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $λ$-symmetries and first integrals. This relation further leads to a new $λ$-symmetry linearization criteria for second order ODEs which provides a new approach for constructing the linearization transformations with lower complexity. The effectiveness of the approach is illustrated by obtaining the local linearization transformations for the linearizable nonlinear ODEs of the form $y''+F_1(x,y)y'+F(x,y)=0$. Examples of linearizing nonlinear ODEs which are quadratic or cubic in the first derivative are also presented.

math.CA↗