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Asymmetric multi-image wavelength multiplexing cryptosystem using qz algorithm and unequal modulus decomposition

OPTICS AND LASER TECHNOLOGY 2024年 172卷

作者： Saini, Indu Sharma, Neelam Banasthali Vidyapith Dept Comp Sci Aliyabad Rajasthan India

An asymmetric multi-image cryptosystem based on wavelength multiplexing is proposed in this paper. Multiimage characteristic of the encryption process has been realized by the qz synthesis process. Unequal modulus decomposition is used to make the scheme robust to any basic cryptographic attacks. The encryption process has a complex setup that involves phase retrieval algorithms but the decryption is quite simple and fast and can be realized on the hybrid system. Although the scheme is validated on grayscale multi-images, it can also process color images. Cryptosystem is exposed to statistical attacks, noise attacks, and data loss attacks, apart from basic cryptographic attacks. Simulation established the fortitude of the cryptosystem to these attacks. We also analyzed the key space and key sensitivity, and the results show that the cryptosystem has a large key space to resist any brute-force attack. Comparative analysis with similar schemes on multi-image reveals that the cryptosystem has several merits and it is secure against basic and special attacks.

关键词： Wavelength multiplexing qz algorithm Unequal modulus decomposition Multi-image

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ON A MODIFICATION OF THE qz algorithm WITH FAST GIVENS ROTATIONS

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COMPUTING 1987年第3期38卷 247-259页

作者： MARCHY, HP 1.Institut für angewandte Mathematik Universit?t Zürich R?mistrasse 74 CH-8001 Zürich Switzerland

It is explained, how Householder reflections can be replaced by fast Givens rotations in the first and second step of the qz algorithm. A count of the required operations for the two steps shows, that the modified first step does need less operations, but that the modified second step needs only insignificantly less operations in comparison with the original qz algorithm. On the base of this fact a modification of the second step is proposed. The resulting new algorithm is called Fqz algorithm, which has the same numerical properties as the qz algorithm, if a correct scaling is applied.

关键词： 65 F 15 Eigenvalues Givens rotations fast Givens rotations qz algorithm rounding error

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Multishift variants of the qz algorithm with aggressive early deflation

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2007年第1期29卷 199-227页

作者： Kagstroem, Bo Kressner, Daniel Umea Univ Dept Comp Sci S-90187 Umea Sweden Dept Math Zagreb 10000 Croatia

New variants of the qz algorithm for solving the generalized eigenvalue problem are proposed. An extension of the small-bulge multishift QR algorithm is developed, which chases chains of many small bulges instead of only one bulge in each qz iteration. This allows the effective use of level 3 BLAS operations, which in turn can provide efficient utilization of high performance computing systems with deep memory hierarchies. Moreover, an extension of the aggressive early deflation strategy is proposed, which can identify and de. ate converged eigenvalues long before classic deflation strategies would. Consequently, the number of overall qz iterations needed until convergence is considerably reduced. As a third ingredient, we reconsider the deflation of infinite eigenvalues and present a new deflation algorithm, which is particularly effective in the presence of a large number of infinite eigenvalues. Combining all these developments, our implementation significantly improves existing implementations of the qz algorithm. This is demonstrated by numerical experiments with random matrix pairs as well as with matrix pairs arising from various applications.

关键词： generalized eigenvalue problem generalized Schur form qz algorithm multishifts aggressive early deflation blocked algorithms

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Chebyshev tau-qz algorithm methods for calculating spectra of hydrodynamic stability problems

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APPLIED NUMERICAL MATHEMATICS 1996年第4期22卷 399-434页

作者： Dongarra, JJ Straughan, B Walker, DW OAK RIDGE NATL LAB DIV COMP SCI & MATHMATH SCI SECTOAK RIDGETN 37831 UNIV GLASGOW DEPT MATHGLASGOW G12 8QWLANARKSCOTLAND

The Chebyshev tau method is examined in detail for a variety of eigenvalue problems arising in hydrodynamic stability studies, particularly those of Orr-Sommerfeld type. We concentrate on determining the whole of the top end of the spectrum in parameter ranges beyond those often explored. The method employing a Chebyshev representation of the fourth derivative operator, D-4, is compared with those involving the second and first derivative operators, D-2 and D, respectively. The latter two representations require use of the qz algorithm in the resolution of the singular generalised matrix eigenvalue problem which arises. Physical problems explored are those of Poiseuille flow, Couette flow, pressure gradient driven circular pipe flow, and Couette and Poiseuille problems for two viscous, immiscible fluids, one overlying the other.

关键词： eigenvalue problems Orr-Sommerfeld equations multilayer flows Chebyshev polynomials qz algorithm

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AN EXTENSION OF THE qz algorithm BEYOND THE HESSENBERG-UPPER TRIANGULAR PENCIL

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2013年 40卷 17-35页

作者： Vandebril, Raf Watkins, David S. Katholieke Univ Leuven Dept Comp Sci B-3001 Heverlee Belgium Washington State Univ Dept Math Pullman WA 99164 USA

Recently, an extension of the class of matrices admitting a Francis type of multishift QR algorithm was proposed by the authors. These so-called condensed matrices admit a storage cost identical to that of the Hessenberg matrix and share all of the properties essential for the development of an effective implicit QR type method. This article continues along this trajectory by discussing the generalized eigenvalue problem. The novelty does not lie in the almost trivial extension of replacing the Hessenberg matrix in the pencil by a condensed matrix, but in the fact that both pencil matrices can be partially of condensed form. Again, the storage cost and crucial features of the Hessenberg-upper triangular pencil are retained, giving rise to an equally viable qz-like method. The associated implicit algorithm also relies on bulge chasing and exhibits a sort of bulge hopping from one to the other matrix. This article presents the reduction to a condensed pencil form and an extension of the qz algorithm. Relationships between these new ideas and some known algorithms are also discussed.

关键词： condensed matrices generalized eigenvalues qz algorithm QR algorithm extended Krylov

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Performance of the qz algorithm in the presence of infinite eigenvalues

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2000年第2期22卷 364-375页

作者： Watkins, DS Washington State Univ Dept Pure & Appl Math Pullman WA 99164 USA

The implicitly shifted (bulge-chasing) qz algorithm is the most popular method for solving the generalized eigenvalue problem Av = lambda Bv. This paper explains why the qz algorithm functions well even in the presence of infinite eigenvalues. The key to rapid convergence of qz (and QR) algorithms is the effective transmission of shifts during the bulge chase. In this paper the mechanism of transmission of shifts is identified, and it is shown that this mechanism is not disrupted by the presence of infinite eigenvalues. Both the qz algorithm and the preliminary reduction to Hessenberg-triangular form tend to push the infinite eigenvalues toward the top of the pencil. Thus they should be deflated at the top.

关键词： matrix pencil eigenvalue computation qz algorithm

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qz algorithm with two-sided generalized Rayleigh quotient shifts

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2023年第4期30卷 e2475-e2475页

作者： Chen, Xiao Shan Xu, Hongguo South China Normal Univ Sch Mat Guangzhou Peoples R China Univ Kansas Dept Math Lawrence KS USA

We generalize the recently proposed two-sided Rayleigh quotient single-shift and the two-sided Grassmann-Rayleigh quotient double-shift used in the QR algorithm and apply the generalized versions to the qz algorithm. With such shift strategies the qz algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the qz algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single-shift and Francis double-shift.

关键词： generalized Schur form matrix pair qz algorithm qz iteration two-sided generalized Rayleigh quotient single-shift two-sided generalized 2D Grassmann-Rayleigh quotient double-shift

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Blind watermarking scheme for medical and non-medical images copyright protection using the qz algorithm

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EXPERT SYSTEMS WITH APPLICATIONS 2024年 241卷

作者： Shen, Yuxin Tang, Chen Fan, Zirui Wu, Tianbo Lei, Zhenkun Tianjin Univ Sch Elect & Informat Engn Tianjin 300072 Peoples R China Dalian Univ Technol State Key Lab Struct Anal Ind Equipment Dalian 116024 Peoples R China

The widely used watermarking technique is to hide the watermark information through adding process, which means the watermark is added with the host image to realize embedding. The constraints of these addition-based watermarking systems mainly are non-blind, they are vulnerable to forgery attacks and have the false-positive problem. To solve these problems, a novel blind watermarking scheme based on the qz algorithm for medical and normal images is first presented in this paper. In the presented scheme, we take the watermark and the host image as two inputs of the qz algorithm to realize embedding, rather than the adding process. The outputs of the qz algorithm include four parts: two upper quasitriangular matrices and two unitary matrices. Then the three parts of the outputs are synthesized by the vector synthesis and inverse discrete wavelet transform-singular value decomposition (DWT-SVD) to obtain the watermarked host image. Due to the characteristics of the qz algorithm, our scheme is blind. The performance is tested and compared by applying our scheme to both medical and non -medical images. The experimental results show that our scheme is robust against various image-processing at-tacks, forgery attacks, and false-positive problem. And from the comparison results, the proposed scheme out-performs the existing addition-based schemes in terms of robustness against the Median filter, Low-pass filter, Gaussian blur attacks, and forgery attacks, while maintaining good invisibility.

关键词： Forgery Attacks False -Positive Problem qz algorithm Discrete Wavelet Transform-Singular Value Decomposition (DWT-SVD)

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algorithm 1019: A Task-based Multi-shift QR/qz algorithm with Aggressive Early Deflation

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2022年第1期48卷 11-11页

作者： Myllykoski, Mirko Umea Univ Dept Comp Sci SE-90187 Umea Sweden Umea Univ HPC2N SE-90187 Umea Sweden

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (qz algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED), and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

关键词： Eigenvalue problem real Schur form QR algorithm qz algorithm multi-shift aggressive early deflation task-based StarPU shared memory distributed memory MPI GPU

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A new deflation criterion for the qz algorithm

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2024年第1期31卷

作者： Steel, Thijs Vandebril, Raf Langou, Julien Katholieke Univ Leuven Dept Comp Sci Celestijnenlaan 200A B-3001 Leuven Belgium CU Denver Dept Math & Stat Sci Denver CO USA

The qz algorithm computes the generalized Schur form of a matrix pencil. It is an iterative algorithm and, at some point, it must decide when to deflate, that is when a generalized eigenvalue has converged and to move on to another one. Choosing a deflation criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might not have full accuracy. Additionally, the criterion should not be computationally expensive to evaluate. There are two commonly used criteria: the elementwise criterion and the normwise criterion. This paper introduces a new deflation criterion based on the size of and the gap between the eigenvalues. We call this new deflation criterion the strict criterion. This new criterion for qz is analogous to the criterion derived by Ahues and Tisseur for the QR algorithm. Theoretical arguments and numerical experiments suggest that the strict criterion outperforms the normwise and elementwise criteria in terms of accuracy. We also provide an example where the accuracy of the generalized eigenvalues using the elementwise or the normwise criteria is less than two digits whereas the strict criterion leads to generalized eigenvalues which are almost accurate to the working precision. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.

关键词： eigenvalues generalized eigenvalues numerical analysis qz algorithm stopping criterion

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