Column Permutations Research Articles

Equivalence of Butson-type Hadamard matrices

Two matrices H_1 and H_2 with entries from a multiplicative group G are said to be monomially equivalent, denoted by H_1cong H_2, if one of the matrices can be obtained from the other via a sequence of row and column permutations and, respectively, left- and right-multiplication of rows and columns with elements from G. One may further define matrices to be Hadamard equivalent if H_1 cong phi (H_2) for some phi in mathrm {Aut}(G). For many classes of Hadamard and related matrices, it is straightforward to show that these are closed under Hadamard equivalence. It is here shown that also the set of Butson-type Hadamard matrices is closed under Hadamard equivalence.

A Heuristic Approach to the Consecutive Ones Submatrix Problem

Given a matrix, the Consecutive Ones Submatrix (C1S) problem which aims to find the permutation of columns that maximizes the number of columns having together only one block of consecutive ones in each row is considered here. A heuristic approach will be suggested to solve the problem. Also, the Consecutive Blocks Minimization (CBM) problem which is related to the consecutive ones submatrix will be considered. The new procedure is proposed to improve the column insertion approach. Then real world and random matrices from the set covering problem will be evaluated and computational results will be highlighted.

Learning doubly stochastic and nearly idempotent affinity matrix for graph-based clustering

In graph-based clustering, a relevant affinity matrix is crucial for good results. Double stochasticity of the affinity matrix has been shown to be an important condition, both in theory and in practice. In this paper, we emphasize idempotency as another key condition. In fact, a theorem from Sinkhorn, R. (1968) allows us to exhibit the bijective relationship between the set of doubly stochastic and idempotent matrices of order n (modulo permutation of rows and columns) on the one hand, and the set of possible partitions of a set of n objects on the other hand. Thereby, both properties are necessary and sufficient conditions for properly modeling the clustering or graph partitioning tasks using matrices. Yet, this leads to a NP-hard discrete optimization problem. In this context, our main contribution is the introduction of a new relaxed model that efficiently learns a double stochastic and nearly idempotent affinity matrix for graph-based clustering. Our approach leverages existing properties between doubly stochastic and idempotent matrices on the one hand, and their associated Laplacian matrices on the other hand. The resulting optimization problem is bi-convex and can be addressed by an Alternating Direction Method of Multipliers scheme. Furthermore, our model requires less parameters to set in contrast to most of recent works. The experimental results we obtained using several real-world benchmarks, exhibit the interest of our method and the importance of taking into account idempotency in graph-based clustering.

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 Easttom (2018) 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×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 Sn and a subgroup S2n respectively on columns and rows of Boolean functions (GF(2n)→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 Hussain et al. (2012); Hussain et al. (2012); Hussain et al. (2018); Anees and Chen (2020) which involved group action of S8 only on columns of Boolean functions (GF(28)→GF(2) ) of s-box. For n=4, we have used an initial 4 × 4 s-box constructed by Carlisle Adams and Stafford Tavares (Adams and Tavares, 1990) to generated (4!)2 clone copies. For n=8, it can be seen (Hussain et al. (2012); Hussain et al. (2012); Hussain et al. (2018); Anees and Chen (2020)) 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.

New McEliece cryptosystem based on non-permutation equivalent polar codes

In this paper, we propose a new McEliece public-key cryptosystem based on polar codes with non-permutation equivalent. Due to the fact that polar code is a decreasing monomial code, any permutation of columns of its generator matrix can be found. Thus, any McEliece cryptosystem based on only the permutation equivalent of the standard polar code seems to be insecure against some structural attacks. In contrast, our proposed scheme is able to resist against these attacks and achieves good security level against the generic decoding attacks.

A novel multiple color image encryption scheme based on algebra M(2, F2[u]/〈u8〉) and chaotic map

The substance security of computerized images ends up being a huge issue for researchers and architects. Thus, it is important to obscure and conceal image data before an impart over the web for security. Due to the operational strength of the matrix algebra M(2,F2[u]〈u8〉) and a unique attempt on a 4D chaotic map, this study recommends novel multiple colored image encryption techniques. This technique is applied in two stages. Initially, a double permutation model is built to permute a group of images, and hence row and column permutations are incorporated. Finally, chaotic sequences are produced by the 4D chaotic system to perform the exclusive OR operation on permuted images and chaotic successions. The proposed cryptosystem is assessed by using different analyses. The outcomes of our investigation support the fact that the proposed technique is satisfying in terms of high security and better adequacy when contrasted with the present algorithms.

Exponential multivalued forbidden configurations

The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.

Iterated local search for consecutive block minimization

A 1-block in a binary matrix is any maximal sequence of 1s occurring consecutively in the same row. Consecutive block minimization (CBM) requires a permutation of the matrix columns that minimizes the number of 1-blocks. CBM is NP-hard for matrices with a maximum of two 1s per row and an arbitrary number of 1s per column. First, we prove that it is NP-hard even when it is restricted to binary matrices having a maximum of two 1s per row and a maximum of three 1s per column. In these circumstances, a reasonable and traditional approach to solve the problem is to design a heuristic capable of finding good configurations within acceptable computation times. This paper proposes an Iterated Local Search (ILS) metaheuristic. A binary matrix satisfies the strong consecutive ones property (is strongly C1P) if its non-zero entries occur consecutively in each row. It is C1P if there is a permutation of its columns such that the resulting matrix is strongly C1P. In this study, two necessary conditions for a binary matrix to be C1P are provided, each allowing to define a neighborhood for local search (LS). In addition, a third large-scale neighborhood is proposed where the local optimal solution can be directly obtained by solving a Traveling Salesman Problem (TSP). It is noteworthy that the three neighborhoods are being proposed for the first time. The ILS algorithm is tested on a dataset from the OR-library, composed of thirty large-scale set covering instances of varying sizes and densities. Comparison with two TSP solvers as well as with a recently published metaheuristic, from both accuracy and computing time points of view, allows to conclude that the proposed method competes favorably with the methods available in the literature.

Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows.

Strong Chordality of Graphs with Possible Loops

We unify two popular graph classes, strongly chordal graphs and chordal bigraphs, by introducing an umbrella class that contains both classes and maintains their essential properties. This is done by allowing loops at vertices. Considering loops often has little impact on a class of graphs; it however makes a big difference in this case. We call the new class \itstrongly chordal graphs with possible loops. When all vertices have loops, we recover the usual strongly chordal graphs; when all vertices are loopless, we obtain the usual chordal bigraphs. Moreover, there is a surprizing wealth of graphs in the new class that have loops at some vertices and not at others. These graphs also admit the elegant algorithms previously only applied in the extreme two cases. Formulated in the language of adjacency matrices, we study the class of symmetric 0, 1 matrices that admit a simultaneous row and column permutation avoiding the $\Gamma$ matrix $[ \begin{smallmatrix} 1 \ 1 \\ 1 \ 0 \end{smallmatrix}]$. We give ordering characterizations, matrix characterizations, and forbidden subgraph characterizations of the new class, and illustrate its usefulness by solving the minimum domination problem in this general context. This implies solutions of both the minimum dominating set in strongly chordal graphs and the minimum total dominating set in chordal bigraphs.

Generalized Differential Index Modulation for Pilot-Free Communications

Short packet transmission is a key ingredient for the Internet-of-Things (IoT) communications that require extremely low latency. The pilot-based short packet transmission makes lower spectral efficiency and larger transmission delays. To resolve this problem, in this article, we present a generalized differential subcarrier index modulation (GDIM) for pilot-free short packet transmission. The central idea of GDIM is to map information bits to possible indices that can be constructed by joint row and column permutations of a unitary matrix. We show that the proposed GDIM achieves higher spectral efficiency than that attained by the existing differential modulation techniques, including differential subcarrier index modulation and differential Alamouti code. We present a noncoherent maximum-likelihood (ML) detection method and a low-complexity variant of it for GDIM. Furthermore, by analyzing the average bit-error-rate when applying GDIM, we propose an algorithm to optimize GDIM constellation sets. Simulation results demonstrate that the proposed GDIM outperforms the existing differential modulation techniques in terms of bit-error-rates for numerous target spectral efficiencies.

Genetic algorithms with permutation-based representation for computing the distance of linear codes

Abstract Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This result enables the use of permutations as representation scheme, in contrast to the usual discrete representation, which makes the search of the optimum polynomial time dependent from the base field. In particular, we have implemented genetic and CHC algorithms using this representation as a proof of concept. Experimental results have been carried out employing codes over fields with two and eight elements, which suggests that evolutionary algorithms with our proposed permutation encoding are competitive with regard to existing methods in the literature. As a by-product, we have found and amended some inaccuracies in the Magma Computational Algebra System concerning the stored distances of some linear codes.

Color image encryption via Hénon-zigzag map and chaotic restricted Boltzmann machine over Blockchain

A color image encryption algorithm using the Hénon-zigzag map and chaotic restricted Boltzmann machine (CRBM) is proposed in this paper. The proposed pseudo-random number generator, chaotic restricted Boltzmann machine (CRBM), can simultaneously generate three pseudo-random number sequences. The algorithm includes the permutation phase and the diffusion phase. In the Hénon-zigzag map-based permutation phase, zigzag map is used to modulate two pseudo-random number sequences generated by Hénon map to obtain two new pseudo-random number sequences. The mixing of these two chaotic maps makes the security of the permutation phase significantly improve. Subsequently the two pseudo-random number sequences are used for row permutation and column permutation, respectively. In the diffusion phase, through multiple iterations of CRBM of the 3 × 3 architecture, three pseudo-random number sequences are generated by the state values of three neurons in the visible layer. Then these three pseudo-random number sequences are used for bitxor operation with the R, G and B components of the scrambled image, respectively. A series of numerical experiments and analyses on encrypted images prove that the proposed algorithm is more secure than state-of-the-art algorithms. Furthermore, based on the combined use of blockchain and the proposed algorithm, a novel image encryption/decryption system is proposed. The system has two features: asymmetric encryption/decryption of images and authoritative verification of the integrity of encrypted images. It may provide a better understanding of blockchain in digital image encryption.

Efficiency of elementary non-unitary transformations together with virtual matrices in the organization of calculations of matrix algebra

Modern applied problems of matrix algebra require algorithms for efficient and reliable operation with a large number of equations. The purpose of such algorithms is to transform these equations with a wide range of numerical values of their coefficients. The transformations are aimed at simplifying the form of matrices in order to make them one of the canonical ones. Many numerical methods, schemes, algorithms have been developed to achieve these goals. The quality of the transformation results is certainly influenced not only by the method itself, but also by the order in which individual operations are performed. In this paper, an acceptable (in our opinion) version of working with known methods and a scheme for their use is proposed when the issues of accuracy and stability, saving RAM and a rational number of operations are consistent with each other. An elementary non-unitary matrix is accepted as the basic one in these transformations.To limit the uncontrolled growth of the coefficients, a certain order of operations is assigned using Virtual matrices, which are based on the management of row / column permutations. Chains of transformations using elementary matrices form cyclical processes and can be universal. The paper gives examples of using the proposed schemes for two important typical matrix transformations.

DeepTable: a permutation invariant neural network for table orientation classification

Tables are a common way to present information in an intuitive and concise manner. They are used extensively in media such as scientific articles or web pages. Automatically analyzing the content of tables bears special challenges. One of the most basic tasks is determination of the orientation of a table: In column tables, columns represent one entity with the different attribute values present in the different rows; row tables are vice versa, and matrix tables give information on pairs of entities. In this paper, we address the problem of classifying a given table into one of the three layouts horizontal (for row tables), vertical (for column tables), and matrix. We describe DeepTable, a novel method based on deep neural networks designed for learning from sets. Contrary to previous state-of-the-art methods, this basis makes DeepTable invariant to the permutation of rows or columns, which is a highly desirable property as in most tables the order of rows and columns does not carry specific information. We evaluate our method using a silver standard corpus of 5500 tables extracted from biomedical articles where the layout was determined heuristically. DeepTable outperforms previous methods in both precision and recall on our corpus. In a second evaluation, we manually labeled a corpus of 300 tables and were able to confirm DeepTable to reach superior performance in the table layout classification task. The codes and resources introduced here are available at https://github.com/Marhabibi/DeepTable.

English

We generalize the Monge property of real matrices for interval matrices. We define two classes of interval matrices with the Monge property—in a strong and a weak sense. We study the fundamental properties of both types. We show several different characterizations of the strong Monge property. For the weak Monge property, we give a polynomial description and several sufficient and necessary conditions. For both classes, we study closure properties. We further propose a generalization of an algorithm by Deineko and Filonenko which for a given matrix returns row and column permutations such that the permuted matrix is Monge if the permutations exist.

Symmetric keys image encryption and decryption using 3D chaotic maps with DNA encoding technique

In present digital era, multimedia like images, text, documents and videos plays a vital role, therefore due to increase in usage of digital data; there comes high demand of security. Encryption is a technique used to secure and protect the images from unfair means. In cryptography, chaotic maps play an important role in forming strong and effective encryption algorithm. In this paper 3D chaotic logistic map with DNA encoding is used for confusion and diffusion of image pixels. Additionally, three symmetric keys are used to initialize 3D chaos logistic map, which makes the encryption algorithm strong. The symmetric keys used are 32 bit ASCII key, Chebyshev chaotic key and prime key. The algorithm first applies 3D non-linear logistic chaotic map with three symmetric keys in order to generate initial conditions. These conditions are then used in image row and column permutation to create randomness in pixels. The third chaotic sequence generated by 3D map is used to generate key image. Diffusion of these random pixels are done using DNA encoding; further XOR logical operation is applied between DNA encoded input image and key image. Analysis parameters like NPCR, UACI, entropy, histogram, chi-square test and correlation are calculated for proposed algorithm and also compared with different existing encryption methods.

Design of BCH Polarization Kernels With Reduced Processing Complexity

The letter introduces a construction of extended BCH polarization kernels, which admit simple computation of the probabilities or log-likelihood ratios arising in the successive cancellation decoding algorithm. The kernels are obtained by finding a column permutation of extended BCH kernels with natural column order, such that the total complexity of the associated trellises is minimized.

An effective image encryption algorithm based on compressive sensing and 2D-SLIM

This paper presents an image encryption algorithm based on the compressive sensing and a hyperchaotic map, which includes the permutation, compression and diffusion processes. Firstly, the plain image is transformed using discrete wavelet transform, and the coefficient matrix is obtained. Then, the row and column permutation are applied to achieve a good scrambling effect. Secondly, the permuted coefficient matrix is measured by a measurement matrix, and the measurement matrix is constructed using chaotic sequences generated by the 2D-SLIM map. Finally, the diffusion algorithm based on Galois field multiplication is designed to encrypt the compressed matrix, and the cipher image with small size and high security is obtained. What’s more, the SHA-512 hash value of the plain image is introduced to set the initial values and parameters of the 2D-SLIM map to establish the correlation between the plain image and the algorithm, and it makes the algorithm can resist the known/chosen plaintext attacks. Furthermore, the row and column permutation before compressive sensing successfully improves the compression effect with low compression radio, and the diffusion operation improves the security of this algorithm. Simulations and performance analyses validate the good compression performance and high security of the proposed algorithm.

On unsupervised simultaneous kernel learning and data clustering

A novel optimization framework for joint unsupervised clustering and kernel learning is derived. Sparse nonnegative matrix factorization of kernel covariance matrices is utilized to categorize data according to their information content. It is demonstrated that a pertinent kernel covariance matrix for clustering can be constructed such that it is block diagonal within arbitrary row and column permutations, while each diagonal block has rank one. To achieve this, a linear combination of a dictionary of kernels is sought such that it has rank equal to the number of clusters while a certain kernel eigenvalue is maximized by a novel difference of convex functions formulation. We establish that the proposed algorithm converges to a stationary solution. Numerical tests with different datasets demonstrate the effectiveness of the proposed scheme whose performance is very close to supervised methods, and performs better than unsupervised alternatives without the need of painstaking parameter tuning.