Involutory Matrix Research Articles

Fourier bases on general self-similar Sierpinski measures

Let |ρ|>1 be a real number and let Q be a 2 × 2 orthogonal and involutory matrix. Let R={0,u,v}⊂Z2 such that det(u,v)≠0 and 3∤det(u,v). In this paper, we prove that the Hilbert space L2(μρQ,R) generated by the general self-similar Sierpinski measure μρQ,R has an orthonormal basis of the form EΛ=e2πi〈λ,x〉:λ∈Λwith Λ⊂R2, i.e., μρQ,R is a spectral measure, if and only if there exist three integers d,m,n such that d2∣(m2+n2) and ρQ=3md3nd3nd−3md in the standard expression. This provides new and different characterization of spectrality of self-similar measures in the plane and gives a counterexample to Łaba–Wang’s conjecture in two dimension.

On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation

For well-formed generalized Pythagorean scales, it is explained how to fill in a bidimensional table, referred to as a scale keyboard, to represent the scale tones, arranged bidimensionally as iterates and cardinals, together with the elementary intervals between them. In the keyboard, generalized diatonic and chromatic intervals are easily identified. Two factor decompositions of the scale tones, which are particular cases of duality, make evident several properties of the sequence of intervals composing the octave, such as the number of repeated adjacent intervals and the composition of the generic step-intervals. The keyboard is associated with two matrix forms. When they are mutually transposed, the keyboard is reversible, as in the 12-tone Pythagorean scale. In this case, the relationship between the two main factor decompositions is given by an involutory matrix.

The Hill Cipher Algorithm for Data Encryption and Decryption

The science of encoding and decoding signals is known as cryptography. Cryptography is frequently used in people's daily life to keep sensitive data, such as Mastercard data, safe. Many regular exercises can be easily viewed by unintentional outsiders via the Internet. Hill cipher is a simple linear transformation represented by a matrix that is a traditional cryptography based on linear algebra. It is one of the symmetric key algorithms with various data encryption advantages. However, the inverse of the plaintext encryption key matrix is not always available. The encrypted text cannot be decoded if the key matrix is not invertible. The key matrix employed for encryption in the Involutory matrix generation method is invertible. As a result, we do not need to find the inverse of the key matrix during decryption.

On the forcing matching numbers of prisms of graphs

Let G be a graph on n vertices. Denote by f(G) the minimum size of a matching M in G such that M is uniquely extendable to a perfect matching in G. The prism of G is defined as G□K2, that is, the Cartesian product of G with the complete graph on 2 vertices. Assume that there exists an involutory matrix A with zero diagonal such that the graph described by the zero/non-zero pattern of off-diagonal entries of A is isomorphic to G. It is shown that f(G□K2)=n2 if G is bipartite. This generalizes a known result. The quantity f for prisms of some families of graphs are determined.

DIVERSITY AES IN MIXCOLUMNS STEP WITH 8X8 CIRCULANT MATRIX

In AES MixColumns operation, the branch number of circulant matrix is raised from 5 to 9 with 8´8 circulant matrices that can be enhancing the diffusion power. An efficient method to compute the circulant matrices in AES MixColumns transformation for speeding encryption is presented. Utilizing 8´8 involutory matrix multiplication is required 64 multiplications and 56 additions in in AES Mix-Columns transformation. We proposed the method with diversity 8´8 circulant matrices is only needed 19 multiplications and 57 additions. It is not only to encryption operations but also to decryption operations. Therefore, 8´8 circlant matrix operation with AES key sizes of 128bits, 192bits, and 256 bits are above 29.1%, 29.3%, and 29.8% faster than using 4´4 involutory matrix operation (16 multiplications, 12 additions), respectively. 8´8 circulant matrix encryption/decryption speed is above 78% faster than 8´8 involutory matrix operation. Ultimately, the proposed method for evaluating matrix multiplication can be made regular, simple and suitable for software implementations on embedded systems.

On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices

AbstractThe singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, {1 \over \sigma }). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices.

Accelerated stability transformation method for chaos control of discrete dynamical systems

This paper proposes the accelerated stability transformation method (ASTM) of chaos control to enhance the efficiency of stabilizing the unstable fixed points or periodic orbits of discrete dynamical systems, with fewer iterative number than the original STM. For each step of iteration, the ASTM scheme utilizes the STM formulation twice to control the step size in the oscillation direction and relaxes greatly the step size of direction normal to the oscillation direction and thus reduces computational efforts of chaos control remarkably. Four examples of nonlinear maps including the hyperchaotic system indicate that the proposed ASTM scheme is more efficient and accurate than STM scheme for stabilizing the unstable fixed points embedded in chaotic attractor. The convergence rate of ASTM scheme is closely related to the involutory matrix C and control parameter q. Moreover, two effective selecting rules of involutory matrix C are suggested by analyzing the eigenvalues of Jacobian matrix of the discrete systems.

Reduced Encoding Complexity for LDPC Codes Using Partially Random and Involutory Matrix Concept in the Generation of Parity Check Matrix

This study assesses the usefulness of the PhouChomVoy protected area by the local community to analyze the factors that influence individuals’ participation in the payment for environmental service program. This study uses a statistical analysis and econometric approach with data on 244 individuals. The results from the econometric equation show that the relevant factors are age, gender, status as the head of household, and income.

Chaos control of a multi-dimensional chaotic mapping system by modified stability transformation method

For the multi-dimensional chaotic mapping system with various forms of bifurcations, it is found that the spectral radius around the static bifurcation point is approximately equal to one and the convergence speed for traditional stability transformation method (STM) which is used to stabilize the unstable fixed points is fairly slow. In this paper, a modified STM is proposed to efficiently stabilize a 3D chaotic mapping system to stable fixed points. Firstly, according to the information of the fixed point, the stability matrix C is derived, demonstrating that it is unnecessarily an involutory matrix for STM that can also stabilize chaotic system to the fixed point. Then, the critical parameter qcrit that satisfies convergence condition and the optimal parameter qopt corresponding to best performance for STM are determined respectively. Moreover, STM is combined with Newton method (NT) to overcome the disadvantage of slow convergence around the static bifurcation point without requiring a priori information of fixed point. It is indicated that the number of iterations, the absolute and relative errors between the convergent value and analytical fixed point for combined STM-NT decrease enormously comparing with that of traditional STM. Finally, numerical analysis verifies the high efficiency of modified STM proposed in this paper.

On a matrix group constructed from an [formula omitted]-potent matrix

Let R∈Cn×n be a {k}-involutory matrix (that is, Rk=In) for some integer k≥2, and let s be a nonnegative integer. A matrix A∈Cn×n is called an {R,s+1,k}-potent matrix if A satisfies RA=As+1R. In this paper, a matrix group corresponding to a fixed {R,s+1,k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group GA that is associated with a generalized group invertible matrix A.

Generalized centro-invertible matrices with applications

Centro-invertible matrices are introduced by R.S. Wikramaratna in 2008. For an involutory matrix R , we define the generalized centro-invertible matrices with respect to R to be those matrices A such that R A R = A − 1 . We apply these matrices to a problem in modular arithmetic. Specifically, algorithms for image blurring/deblurring are designed by means of generalized centro-invertible matrices. In addition, if R 1 and R 2 are n × n involutory matrices, then there is a simple bijection between the set of all centro-invertible matrices with respect to R 1 and the set with respect to R 2 .

Linear maps on matrix algebra Jordan derivable at involutory matrices

Let be a positive integer and be a unital commutative ring with 2 invertible. By , we mean the algebra consisting of all matrices over , and by , we denote the identity matrix in . We call an involutory matrix if . The Jordan product ‘’ is defined by . A linear transformation on is called derivable (resp., Jordan derivable) at involutory matrices if (resp., ) for any involutory matrix . In this paper, we show that a linear transformation on is derivable, or Jordan derivable, at involutory matrices if it is an inner derivation.

A Secure Encryption Technique based on Advanced Hill Cipher For a Public Key Cryptosystem

In the present era of Information Technology, Transmission of information in a secured manner is the primary concern of all agencies. Security is highly essential, as intruders are very keen to rob the information with all their might and intelligence. Cryptography, then, not only protects data from theft or alteration, but can also be used for user authentication. The objective of this paper is to encrypt a text using a technique different from the conventional Hill Cipher. The Advanced Hill Cipher uses an involutory matrix and permuted key. In this paper, we implement a second level of encryption using permutation approach, which makes the cipher highly secure. This encryption scheme is highly reliable as it uses tamper detection of the cipher text ensuring successful decryption of the cipher. All of these functions and transformations that are introduced ensure that this cipher is a very strong one and it cannot be broken by any cryptanalytic attack.

Encryption Approach for Images using Bits Rotation Reversal and Extended Hill Cipher Techniques

method for image encryption with two stages has been described in the present study. In first stage, each pixel of image is converted to its equivalent eight bit binary number and in that eight bit number, number of bits equal to the length of password are rotated and then reversed. In second stage, extended hill cipher technique is applied by using involutory matrix generated by same password given in second stage to make encryption more secured. Performance of the proposed technique is evaluated by Statistical Analysis, Differential Analysis and also quantifying Mean Square Error (MSE) & Peak Signal to Noise Ratio (PSNR). The proposed approach is implemented for different images using MATLAB. Decryption involves the reverse process of encryption.

On the construction of involutory rhotrices

An involutory matrix is a matrix that is its own inverse. Such matrices are of great importance in matrix theory and algebraic cryptography. In this note, we extend this involution to rhotrices and present their properties. We have also provided a method of constructing involutory rhotrices.

Relations between [formula omitted]-potent matrices and different classes of complex matrices

In this paper, {K,s+1}-potent matrices are considered. A matrix A∈Cn×n is called {K,s+1}-potent when KAs+1K=A where K is an involutory matrix and s∈{1,2,3,…}. Specifically, {K,s+1}-potent matrices are analyzed considering their relations to different classes of complex matrices. These classes of matrices are: {s+1}-generalized projectors, {K}-Hermitian matrices, normal matrices, and matrices B∈Cn×n (anti-)commuting with K or such that KB is involutory, Hermitian or normal. In addition, some new relations for K-generalized centrosymmetric matrices have been derived.

Characterizations of [formula omitted]-potent matrices and applications

Recently, situations where a matrix coincides with some of its powers have been studied. This kind of matrices is related to the generalized inverse matrices. On the other hand, it is possible to introduce another class of matrices that involve an involutory matrix, generalizing the well-known idempotent matrix, widely useful in many applications. In this paper, we introduce a new kind of matrices called { K , s + 1 } -potent, as an extension of the aforementioned ones. First, different properties of { K , s + 1 } -potent matrices have been developed. Later, the main result developed in this paper is the characterization of this kind of matrices from a spectral point of view, in terms of powers of the matrix, by means of the group inverse and, via a block representation of a matrix of index 1. Finally, an application of the above results to study linear combinations of { K , s + 1 } -potent matrices is derived.

Eigenvector-free solutions to AX = B with PX = XP and XH = sX constraints

The matrix equation AX=B with PX=XP and XH=sX constraints is considered, where P is a given Hermitian involutory matrix and s=±1. By an eigenvalue decomposition of P, we equivalently transform the constrained problem to two well-known constrained problems and represent the solutions in terms of the eigenvectors of P. Using Moore–Penrose generalized inverses of the products generated by matrices A, B and P, the involved eigenvectors can be released and eigenvector-free formulas of the general solutions are presented. Similar strategy is applied to the equations AX=B, XC=D with the same constraints.

The centro-invertible matrix: A new type of matrix arising in pseudo-random number generation

This paper defines a new type of matrix (which will be called a centro-invertible matrix) with the property that the inverse can be found by simply rotating all the elements of the matrix through 180 degrees about the mid-point of the matrix. Centro-invertible matrices have been demonstrated in a previous paper to arise in the analysis of a particular algorithm used for the generation of uniformly-distributed pseudo-random numbers.An involutory matrix is one for which the square of the matrix is equal to the identity. It is shown that there is a one-to-one correspondence between the centro-invertible matrices and the involutory matrices. When working in modular arithmetic this result allows all possible k by k centro-invertible matrices with integer entries modulo M to be enumerated by drawing on existing theoretical results for involutory matrices.Consider the k by k matrices over the integers modulo M. If M takes any specified finite integer value greater than or equal to two then there are only a finite number of such matrices and it is valid to consider the likelihood of such a matrix arising by chance. It is possible to derive both exact expressions and order-of-magnitude estimates for the number of k by k centro-invertible matrices that exist over the integers modulo M. It is shown that order (√N) of the N=M(k2) different k by k matrices modulo M are centro-invertible, so that the proportion of these matrices that are centro-invertible is order (1/√N).

Advanced Hill Cipher Handling the Entire Plaintext as a Single Block

In this paper we have devoted our attention to the study of large block cipher by employing the basic concepts of the advanced Hill cipher. In this the computation of the inverse of a matrix is very simple as we have confined our attention to an involutory matrix, i.e., a matrix whose inverse is the same as the original matrix. The avalanche effect and the cryptanalysis carried out in this investigation, to check the strength of the cipher, prominently indicate that the cipher is a potential one, and it can be applied to a plaintext of any length Keywords: key, involutory matrix, cryptanalysis, avalanche effect, Ciphertext, block cipher