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In this work, we study the spectral property of generalized Sierpinski-type self-affine measures μ M , D on R 2 generated by an expanding integer matrix M ∈ M 2 ( Z ) with det ⁡ ( M ) ∈ 3 Z and a non-collinear integer digit set D = { ( 0 , 0 ) t , ( α 1 , α 2 ) t , ( β 1 , β 2 ) t } with α 1 β 2 − α 2 β 1 ∈ 3 Z . We give the sufficient and necessary conditions for μ M , D to be a spectral measure, i.e., there exists a countable subset Λ ⊂ R 2 such that E ( Λ ) = { e 2 π i 〈 λ , x 〉 : λ ∈ Λ } forms an orthonormal basis for L 2 ( μ M , D ) . This completely settles the spectrality of the self-affine measure μ M , D .

A novel color image encryption algorithm based on a cross 2D hyperchaotic map is proposed in this paper. The cross 2D hyperchaotic map is constructed using one nonlinear function and two chaotic maps with a cross structure. Chaotic behaviors are illustrated using bifurcation diagrams, Lyapunov exponent spectra, phase portraits, etc. In the color image encryption algorithm, the keys are generated using hash function SHA-512 and the information of the plain color image. First, the color plain image is converted to a combined bit-level matrix and permuted by the chaos-based row and column combined cycle shift scrambling method. Then, the scrambled integer matrix is diffused according to the selecting sequence which depends on the chaotic sequence. Last, decompose the diffusion matrix to get the encrypted color image. Simulation experiments and security evaluations show that the algorithm can encrypt the color image effectively and has good security to resist various kinds of attacks.

We describe a representation for periodic tilings of the plane by regular polygons. Our approach is to represent explicitly a small subset of seed vertices from which we systematically generate all elements of the tiling by translations. We represent a tiling concretely by a ( 2 + n ) × 4 integer matrix containing lattice coordinates for two translation vectors and n seed vertices. We discuss several properties of this representation and describe how to exploit the representation elegantly and efficiently for reconstruction, rendering, and automatic crystallographic classification by symmetry detection.

We construct a nonnegative integer matrix to evaluate the matrix of nonlinearity characteristics for the coordinate polynomials of a product of transformations of a binary vector space. The matrix of the characteristics of the transformation is defined by the degrees of nonlinearity of the derivatives of all coordinate functions in each input variable. The entries of the evaluation matrix are expressed in terms of the characteristics of the coordinate polynomials of the multiplied transformations. Calculation of the evaluation matrix is easier than calculating the exact values of the characteristics. The estimation method is extended to an arbitrary number of multiplied transformations. Computational examples are given that in particular show the accuracy of the obtained estimates and the domain of their nontriviality.

This short note extends a recent result (Bonifas et al, On sub-determinants and the diameter of polyhedra, Discrete Computational Geometry, 52, 2014) of an upper bound of the diameter of a convex polytope defined by an integer matrix to a similar upper bound of the diameter of a convex polytope defined by a real matrix. It also shows, by an example, that the new bound may be better than the ones of Bonifas et al., Comment: 7 pages

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

Let $$\mu _{M, D}$$ be the self-affine measure generated by an expanding integer matrix $$M\in M_{2}(\mathbb {Z})$$ and an integer three-element digit set $$D=\{(0,0)^T, (\alpha ,\beta )^T,(\gamma ,\eta )^T\}$$ . In this paper, we show that if $$3\mid \det (M)$$ and $$3\not \mid \alpha \eta -\beta \gamma $$ , then $$L^2(\mu _{M,D})$$ has an orthogonal basis of exponential functions if and only if $$M^*\varvec{u}\in 3\mathbb {Z}^2$$ , where $$\varvec{u}=(\eta -2\beta ,\; 2\alpha -\gamma )^T$$ .

International audience; We rely on aggregate separation bounds for univariate polynomials to introduce novel worst-case separation bounds for the isolated roots of zero-dimensional, positive-dimensional, and overde- termined polynomial systems. We exploit the structure of the given system, as well as bounds on the height of the sparse (or toric) resultant, by means of mixed volume, thus establishing adaptive bounds. Our bounds improve upon Canny’s Gap theorem [9]. Moreover, they exploit sparseness and they apply without any assumptions on the input polynomial system. To evaluate the quality of the bounds, we present polynomial systems whose root separation is asymptotically not far from our bounds.We apply our bounds to three problems. First, we use them to estimate the bitsize of the eigenvalues and eigenvectors of an integer matrix; thus we provide a new proof that the problem has polynomial bit complexity. Second, we bound the value of a positive polynomial over the simplex: we improve by at least one order of magnitude upon all existing bounds. Finally, we asymptotically bound the number of steps of any purely subdivision-based algorithm that isolates all real roots of a polynomial system.

A matrix is Bohemian if its elements are taken from a finite set of integers. An upper Hessenberg matrix is normalized if all its subdiagonal elements are ones, and hollow if it has only zeros along the main diagonal. All possible determinants of families of normalized and hollow normalized Bohemian upper Hessenberg matrices are enumerated. It is shown that in the case of hollow matrices the maximal determinants are related to a generalization of Fibonacci numbers. Several conjectures recently stated by Corless and Thornton follow from these results.

As a non-linear precoding alternative to Tomlinson-Harashima precoding (THP), in this paper, so-called lattice reduction aided precoding (LRP) is considered as a crosstalk precompensation technique for downstream transmission in G.fast DSL networks. First, a practically achievable bit-rate expression for LRP is proposed in function of the precoder and integer matrix. The problem then consists of a joint precoder and integer matrix design in order to maximize the weighted sum-rate (WSR) under per-line power constraints. For a fixed integer matrix, zero-forcing (ZF) precoder matrix design simplifies to gain scaling optimization with complex gain scalars, for which a successive lower bound maximization method is presented. Additionally, it is established that the achievable ZF-LRP sum-rate is upper bounded by the achievable ZF-THP sum-rate at high SNR. For computing the optimal precoder matrix, on the other hand, an efficient method is developed by leveraging on the equivalence between the WSR maximization and the weighted sum of mean squared error (MSE) minimization, leading to a locally-optimal MMSE-LRP solution. Simulations with a measured G.fast cable binder are provided to compare the proposed LRP schemes with THP schemes.

As atomic operations, secure matrix-based computations using homomorphic encryption (HE) have attracted much attention in cloud-based machine learning. However, most existing secure matrix computation solutions that focus on HE schemes suffer efficiency loss as the size of the matrix, which greatly limits their applications in the big data environment. To address these issues, this paper proposes seIMC, an integer matrix computation scheme based on the Gentry-Sahai-Waters (GSW) scheme, to cope with privacy protection and secure computation of large-scale data. In detail, we translate the GSW scheme to encrypt an integer matrix modulo $q$ (i.e., a large positive integer), and homomorphically compute matrix addition and multiplication, which is a natural extension of HAO scheme. Besides, the correctness and security analysis of seIMC are shown, and complexity analysis is also given in this study. Furthermore, the proposed schemes are implemented, including public-key encryption and private-key encryption schemes. Compared with existing secure matrix computation schemes, the proposed scheme performs better in execution time. Finally, seIMC is applied to solve the problem of the number of ways in which any two participants make friends through $k$ steps in an encrypted social network. Experiments show that when the cloud server processes an integer matrix of 1000 people with a security level of 90, namely, 1 million data volumes, it only takes approximately 1.9 minutes for each homomorphic matrix multiplication. Hence, the practicality of the proposed seIMC in privacy protection under a big data environment is highly proven.

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.