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Numerical algorithms for corner-modified symmetric Toeplitz linear system with applications to image encryption and decryption

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Abstract

The corner-modified symmetric Toeplitz matrix can be considered as a symmetric Toplitz matrix plus two rank-one matrices. Based on the decomposition of Toeplitz inversion and order-reduction algorithm, two new algorithms for the product of symmetric Toplitz inversion and the vector are proposed. Also, we give two fast algorithms for the solution of corner-modified symmetric Toeplitz linear system. The structured perturbation analysis of the symmetric Toplitz matrix inversion is presented next. We exploit the proposed methods for image encryption and decryption, which can be performed via the fast (corner-modified) symmetric Toeplitz matrix–vector multiplications, and (corner-modified) symmetric Toeplitz linear solvers. Numerical experiments demonstrate that our proposed algorithms are accurate and effective, especially in high-order systems (often over \(2^{12}\)).

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Authors and Affiliations

  1. School of Information Science and Technology, Linyi University, Linyi, 276000, China

    Xing Zhang

  2. College of Information Technology, The University of Suwon, Hwaseong-si, 445-743, Korea

    Xing Zhang & Heejung Byun

  3. School of Automation and Electrical Engineering, Linyi University, Linyi, 276000, China

    Yanpeng Zheng

  4. School of Mathematics and Statistics, Linyi University, Linyi, 276000, China

    Zhaolin Jiang

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  1. Xing Zhang
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  2. Yanpeng Zheng
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Correspondence to Yanpeng Zheng or Zhaolin Jiang.

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The research was supported by National Natural Science Foundation of China (Grant No. 12001257), the Natural Science Foundation of Shandong Province (Grant No. ZR2020QA035) and the PhD Research Foundation of Linyi University (Grant No. LYDX2018BS067).

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Zhang, X., Zheng, Y., Jiang, Z. et al. Numerical algorithms for corner-modified symmetric Toeplitz linear system with applications to image encryption and decryption. J. Appl. Math. Comput. 69, 1967–1987 (2023). https://doi.org/10.1007/s12190-022-01819-7

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  • DOI: https://doi.org/10.1007/s12190-022-01819-7

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