Novel Scheme for Image Encryption and Decryption Based on a Hermite-Gaussian Matrix

  • Mohammed AlsaediEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 943)


Media security is an issue of great concern over the internet and during wireless transmissions. In this paper, a novel scheme for image encryption and decryption is proposed based on a Hermite-Gaussian matrix and an array of subkeys. The proposed scheme includes a secret key that is processed to extract an array of subkeys, which are employed with the extracted phase part of the inverse Fourier transform of a Hermite-Gaussian matrix to encrypt and decrypt a grayscale image. After key generation and the production of two columns of subkeys, the Hermite-Gaussian matrix is multiplied by the first group of subkeys and subjected to a modulus operation (the remainder) with the second group, and the resulting matrix is verified for singularity. If the singularity test is passed, then the resulting image is multiplied by the original image and the output is subjected to a modulus operation and is used for the following subkeys. However, if the singularity test fails, then a new subkey is chosen and the process is repeated until all subkeys are tested and used to produce the encrypted image. For subsequent decryption, the reverse process is implemented to recover the original image from the encrypted one. Statistical analysis shows that the proposed scheme is robust and is strong against attacks. The correlation factor, among other tests, shows that reshuffling the image pixels reduces the correlation between neighboring pixels to very low values (i.e., <1%).


Encryption Decryption Hermite-Gaussian matrix Subkey regeneration Two dimensional array Hermite polynomial Orthogonal functions 


  1. 1.
    Ünal Çavuşoǧlu, S., Kaçar, S., Pehlivan, I., Zengin, A.: Secure image encryption algorithm design using a novel chaos based S-Box. Chaos, Solitons Fractals 95, 92–101 (2017)CrossRefGoogle Scholar
  2. 2.
    Ye, G., Zhao, H., Chai, H.: Chaotic image encryption algorithm using wave-line permutation and block diffusion. Nonlinear Dyn. 83(4), 2067–2077 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Zhang, Z., Li, S., Ge, R., Yuan, M., Ma, Y.: A novel 1D hybrid chaotic map-based image compression and encryption using compressed sensing and Fibonacci-Lucas transform. Math. Probl. Eng. Article ID 7683687. Hindawi Publishing Corp (2016)Google Scholar
  4. 4.
    Zhu, H., Zhang, X., Yu, H., Zhao, C., Zhu, Z.: A novel image encryption scheme using the composite discrete chaotic system. Entropy 18(8), 276 (2016)CrossRefGoogle Scholar
  5. 5.
    Zhou, N.R., Hua, T.X., Hua, G.L., Ju, P.D., Hong, L.Q.: Quantum image encryption based on generalized Arnold transform and double random-phase encoding. Quantum Inf. Process. 14, 1193–1213 (2015). Scholar
  6. 6.
    Khushboo, T., Tripathi, B.P., Sharma, B.K.: Implementation of Hermite normal form in NTRU matrix formulation algorithm. Int. J. Innov. Sci. Res. 12(1), 58–64 (2014)Google Scholar
  7. 7.
    Román, P.E., Asahi, T., Casassus, S.: Hermite-Gaussian functions for image synthesis. In: APSIPA (2014). ISBN 978-616-361-823-8Google Scholar
  8. 8.
    Hanna, M.T., Seif, N.P.A., Ahmad, W.A.E.M.: Hermite–Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the direct utilization of the orthogonal projection matrices on its eigenspaces. IEEE Trans. Signal Process. 54(7), 2815–2819 (2006)CrossRefGoogle Scholar
  9. 9.
    Wang, L., Wu, Y., Dai, M.: Some aspects of Gaussian-Hermite moments in image analysis. In: ICNC (2007). 7695-2875-9/07Google Scholar
  10. 10.
    Wang, L., Suo, H., Dai, M.: Fingerprint image segmentation based on Gaussian-Hermite moments. In: 1st International Conference on Advanced Data Mining and Applications (ADMA), pp. 446–454 (2005)CrossRefGoogle Scholar
  11. 11.
    Wang, L., Dai, M.: Extraction of singular points in fingerprint by the distribution of Gaussian-Hermite moments. In: Proceedings of IEEE International Conference on Distributed Framework for Multimedia Applications (DFMA 2005), Besancon (France), pp. 206–209 (2005)Google Scholar
  12. 12.
    Enderlein, J., Pampaloni, F.: Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes. J. Opt. Soc. Am. 21(8), 1553–1558 (2004)CrossRefGoogle Scholar
  13. 13.
    Martens, J.-B.: The Hermite transform theory. IEEE Trans. Acoust. Speech Signal Process. 38, 1595–1606 (1990)CrossRefGoogle Scholar
  14. 14.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1, p. 560. Wiley, New York (1989)CrossRefGoogle Scholar
  15. 15.
    Andrews, L.C.: Special Functions for Engineers and Applied Mathematicians. MacMillan Publishing Co., New York (1985)Google Scholar
  16. 16.
    Abramowitz, M., Stegun, I.S.: Handbook of Mathematical Functions. Dover Publications Inc., New York (1965)zbMATHGoogle Scholar
  17. 17.
    Arfken, G.: Mathematical Methods for Physicists, 3rd edn. Academic Press, New York (1985)zbMATHGoogle Scholar
  18. 18.
    Gertner, I., Geri, G.A.: Image representation using Hermite Gaussian. Biol. Cybern. 71, 147–151 (1994)CrossRefGoogle Scholar
  19. 19.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES like cryptosystems. In: Proceedings of the 10th Annual International Cryptology Conference on Advances in Cryptology. Springer (1991)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.College of Computer Science and EngineeringTaibah UniversityMedinaKingdom of Saudi Arabia

Personalised recommendations