Quantum Information Processing

, 18:272 | Cite as

An encryption protocol for NEQR images based on one-particle quantum walks on a circle

  • Bassem Abd-El-Atty
  • Ahmed A. Abd El-Latif
  • Salvador E. Venegas-AndracaEmail author


Quantum walks are generalizations of random walks that have extensive applications in various fields including cryptography, quantum algorithms, and quantum networking. Discrete quantum walks can be seen as nonlinear mappings between quantum states and position probability distributions, and this mathematical property may be thought of as an imprint of chaotic behavior and consequently used to generate encryption keys. In this paper, we introduce encryption and decryption algorithms for NEQR images based on discrete quantum walks on a circle. We present full quantum circuits of proposed encryption and decryption algorithms together with digital computer simulations of most common attacks on encrypted images. Our numerical results show that our quantum image encryption and decryption scheme has high efficiency and high security with high large key space.


Discrete-time quantum walks Quantum walks on a circle chaotic systems Quantum image processing Quantum image encryption 



All authors would like to thank Professor Dan Li at Nanjing University of Aeronautics and Astronautics and Professor Mohamed Amin at Menoufia University for their insightful comments and valuable feedback. SEVA gratefully acknowledges the unconditional support of his family as well as the financial support of Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias and CONACyT (SNI number 41594 as well as Fronteras de la Ciencia project number 1007).


  1. 1.
    Patel, K.D., Belani, S.: Image encryption using different techniques: a review. Int. J. Emerg. Technol. Adv. Eng. 1(1), 30 (2011)Google Scholar
  2. 2.
    El-Latif, A.A.A., Li, L., Zhang, T., Wang, N., Song, X., Niu, X.: Digital image encryption scheme based on multiple chaotic systems. Sens. Imaging Int. J. 13(2), 67 (2012)ADSCrossRefGoogle Scholar
  3. 3.
    El-Latif, A.A.A., Li, L., Niu, X.: A new image encryption scheme based on cyclic elliptic curve and chaotic system. Multimed. Tools Appl. 70(3), 1559 (2014)CrossRefGoogle Scholar
  4. 4.
    Kocarev, L., Shiguo, L. (eds.): Chaos-Based Cryptography. Theory, Algorithms and Applications. Studies on Computational Intelligence. Springer, New York (2011)Google Scholar
  5. 5.
    Sobottka, M., de Oliveira, L.: Periodicity and predictability in chaotic system. Am. Math. Mon. 113(5), 415 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley, Boston (1990)Google Scholar
  7. 7.
    Banks, J., Dragan, V., Jones, A.: Chaos: A Mathematical Introduction. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  8. 8.
    Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos-based cryptosystems. Int. J. Bifurc. Chaos 16(8), 2129 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kocarev, L.: Chaos-based cryptography: a brief overview. IEEE Circuits Syst. Mag. 1(3), 6 (2001)CrossRefGoogle Scholar
  10. 10.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University, Cambridge (2000)zbMATHGoogle Scholar
  11. 11.
    Wittek, P.: Quantum Machine Learning. Academic Press, Cambridge (2014)zbMATHGoogle Scholar
  12. 12.
    Schulda, M., Sinayskiy, I., Petruccione, F.: An introduction to quantum machine learning. Contemp. Phys. 56(2), 172 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., Lloyd, S.: Quantum machine learning. Nature 549(7671), 195–202 (2017)ADSCrossRefGoogle Scholar
  14. 14.
    Lanzagorta, M.: Quantum Radar. Morgan and Claypool, San Rafael (2011)CrossRefGoogle Scholar
  15. 15.
    Yan, F., Iliyasu, A.M., Venegas-Andraca, S.E.: A survey of quantum image representations. Quantum Inf. Process. 15, 1 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Abura’ed, N., Khan, F., Bhaskar, H.: Advances in the quantum theoretical approach to image processing applications. ACM Comput. Surv. 49(4), 1–49 (2017)CrossRefGoogle Scholar
  17. 17.
    Abd-El-Atty, B., Venegas-Andraca, S.E., El-Latif, A.A.A.: Quantum information protocols for cryptography. In: Hassanien, A.E., Elhoseny, M., Kacprzyk, J. (eds.) Quantum Computing: An Environment for Intelligent Large Scale Real Application, pp. 3–23. Springer, Cham (2018)CrossRefGoogle Scholar
  18. 18.
    Yan, F., Chen, K., Venegas-Andraca, S., Zhao, J.: Quantum image rotation by an arbitrary angle. Quantum Inf. Process. 16, 282 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Vlasov, A.: Quantum Computations and Image Recognition. arXiv:quant-ph/9703010 (1997)
  20. 20.
    Beach, G., Lomont, C., Cohen, C.: Quantum image processing. In: Proceedings of The 2003 IEEE Workshop on Applied Imagery Pattern Recognition, pp. 39–44 (2003)Google Scholar
  21. 21.
    Venegas-Andraca, S., Bose, S.: Quantum computation and image processing: new trends in artificial intelligence. In: Proceedings of the International Conference on Artificial Intelligence IJCAI-03, pp. 1563–1564 (2003)Google Scholar
  22. 22.
    Venegas-Andraca, S., Bose, S.: Storing, processing and retrieving an image using quantum mechanics. In: Proceedings of the SPIE Conference Quantum Information and Computation, pp. 137–147 (2003)Google Scholar
  23. 23.
    Zhang, W.W., Gao, F., Liu, B., Wen, Q.Y., Chen, H.: A watermark strategy for quantum images based on quantum Fourier transform. Quantum Inf. Process. 12(2), 793 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Song, X.H., Wang, S., Liu, S., El-Latif, A.A.A., Niu, X.M.: A dynamic watermarking scheme for quantum images using quantum wavelet transform. Quantum Inf. Process. 12(12), 3689 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Song, X., Wang, S., El-Latif, A.A.A., Niu, X.: Dynamic watermarking scheme for quantum images based on Hadamard transform. Multimed. Syst. 20(4), 379 (2014)CrossRefGoogle Scholar
  26. 26.
    Miyake, S., Nakamae, K.: A quantum watermarking scheme using simple and small-scale quantum circuits. Quantum Inf. Process. 15(5), 1849–1864 (1849)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Yang, Y.G., Xia, J., Jia, X., Zhang, H.: Novel image encryption/decryption based on quantum Fourier transform and double phase encoding. Quantum Inf. Process. 12(11), 3477 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Song, X., Wang, S., El-Latif, A.A.A., Niu, X.: Quantum image encryption based on restricted geometric and color transformations. Quantum Inf. Process. 13(8), 1765 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Zhou, N.R., Hua, T.X., Gong, L.H., Pei, D.J., Liao, Q.H.: Quantum image encryption based on generalized Arnold transform and double random-phase encoding. Quantum Inf. Process. 14(4), 1193 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Gong, L.H., He, X.T., Cheng, S., Hua, T.X., Zhou, N.R.: Quantum image encryption algorithm based on quantum image XOR operations. Int. J. Theor. Phys. 55(7), 3234 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Liang, H.R., Tao, X.Y., Zhou, N.R.: Quantum image encryption based on generalized affine transform and logistic map. Quantum Inf. Process. 15(7), 2701 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Tan, R.C., Lei, T., Zhao, Q.M., Gong, L.H., Zhou, Z.H.: Quantum color image encryption algorithm based on a hyper-chaotic system and quantum fourier transform. Int. J. Theor. Phys. 10, 5368–5384 (2016)zbMATHCrossRefGoogle Scholar
  33. 33.
    Zhou, N., Hu, Y., Gong, L., Li, G.: Quantum image encryption scheme with iterative generalized Arnold transforms and quantum image cycle shift operations. Quantum Inf. Process. 10, 164 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    El-Latif, A.A.A., Abd-El-Atty, B., Talha, M.: Robust encryption of quantum medical images. IEEE Access 6, 1073–1081 (2017)CrossRefGoogle Scholar
  35. 35.
    Li, L., Abd-El-Atty, B., El-Latif, A.A.A., Ghoneim, A.: Quantum color image encryption based on multiple discrete chaotic systems. In: Federated Conference on Computer Science and Information Systems (FedCSIS), pp. 555–559 (2017)Google Scholar
  36. 36.
    Jiang, N., Zhao, N., Wang, L.: LSB based quantum image steganography algorithm. Int. J. Theor. Phys. 55(1), 107 (2016)zbMATHCrossRefGoogle Scholar
  37. 37.
    Abd-El-Atty, B., El-Latif, A.A.A., Amin, M.: New quantum image steganography scheme with Hadamard transformation. In: International Conference on Advanced Intelligent Systems and Informatics, pp. 342–352. Springer (2016)Google Scholar
  38. 38.
    Zhang, T.J., Abd-El-Atty, B., Amin, M., El-Latif, A.A.A.: QISLSQb: a quantum image steganography scheme based on least significant qubit. In: DEStech Transactions on Computer Science and Engineering (MCSSE) (2016)Google Scholar
  39. 39.
    Wang, S., Sang, J., Song, X., Niu, X.: Least significant qubit (LSQb) information hiding algorithm for quantum image. Measurement 73, 352 (2015)CrossRefGoogle Scholar
  40. 40.
    El-Latif, A.A.A., Abd-El-Atty, B., Hossain, M.S., Rahman, M.A., Alamri, A., Gupta, B.: Efficient quantum information hiding for remote medical image sharing. IEEE Access 6, 21075 (2018)CrossRefGoogle Scholar
  41. 41.
    Zhang, Y., Lu, K., Gao, Y., Wang, M.: NEQR: a novel enhanced quantum representation of digital images. Quantum Inf. Process. 12(12), 2833 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Le, P.Q., Dong, F., Hirota, K.: A flexible representation of quantum images for polynomial preparation, image compression and processing operations. Quantum Inf. Process. 10(1), 63 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Childs, A.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Shenvi, N., Kempe, J., Whaley, R.: A quantum random walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)ADSCrossRefGoogle Scholar
  46. 46.
    Childs, A., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speedup by quantum walk. In: Proceedings of the 35th ACM Symposium on The Theory of Computation (STOC’03), pp. 59–68. ACM (2003)Google Scholar
  47. 47.
    Santha, M.: Quantum walk based search algorithms. In: Proceedings of the 5th Theory and Applications of Models of Computation (TAMC08), Xian, LNCS 4978, pp. 31–46 (2008)Google Scholar
  48. 48.
    Childs, A., van Dam, W.: Quantum algorithms for algebraic problems. Rev. Mod. Phys. 82, 1 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Yang, Y.G., Pan, Q.X., Sun, S.J., Xu, P.: Novel image encryption based on quantum walks. Sci. Rep. 5(7784), 7784 (2015)CrossRefGoogle Scholar
  50. 50.
    Yang, Y.G., Zhao, Q.Q.: Novel pseudo-random number generator based on quantum random walks. Sci. Rep. 6, 20362 (2016)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    Li, D., Zhang, J., Guo, F.Z., Huang, W., Wen, Q.Y., Chen, H.: Discrete-time interacting quantum walks and quantum Hash schemes. Quantum Inf. Process. 12, 1–13 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Li, D., Yang, Y.G., Bi, J.L., Yuan, J.B., Xu, J.: Controlled alternate quantum walks based quantum hash function. Sci. Rep. 8(1), 225 (2018)ADSCrossRefGoogle Scholar
  53. 53.
    Yang, Y.G., Xu, P., Yang, R., Zhou, Y.H., Shi, W.M.: Quantum Hash function and its application to privacy amplification in quantum key distribution, pseudo-random number generation and image encryption. Sci. Rep. 6, 19788 (2016)ADSCrossRefGoogle Scholar
  54. 54.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687 (1993)ADSCrossRefGoogle Scholar
  55. 55.
    Nayak, A., Vishwanath, A.: Quantum walk on the line. arXiv:quant-ph/0010117
  56. 56.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of the 33th ACM Symposium on the Theory of Computation (STOC’01), pp. 50–59. ACM (2001)Google Scholar
  57. 57.
    Lovett, N., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81(4), 042330 (2010)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    Feldman, E., Hillery, M.: Modifying quantum walks: a scattering theory approach. J. Phys. A Math. Theor. 40, 11343 (2007)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Carson, G.R., Loke, T., Wang, J.B.: Entanglement dynamics of two-particle quantum walks. Quantum Inf. Process. 14(9), 3193 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Luo, H., Xue, P.: Properties of long quantum walks in one and two dimensions. Quantum Inf. Process. 14(12), 4361 (2015)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    Wong, T.G.: Quantum walk on the line through potential barriers. Quantum Inf. Process. 15(2), 675 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Konno, N., Mitsuhashi, H., Sato, I.: The discrete-time quarternionic quantum walk on a graph. Quantum Inf. Process. 15(2), 651 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. N. J. Phys. 5(1), 83 (2003)CrossRefGoogle Scholar
  64. 64.
    Gonzalez, R., Woods, R.: Digital Image Processing, 2nd edn. Prentice Hall, Upper Saddle River (1992)Google Scholar
  65. 65.
    Bitcoin Private Keys. Accessed 1 June 2019

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of ScienceMenoufia UniversityShebin El-KoomEgypt
  2. 2.School of Information Technology and Computer ScienceNile UniversitySheikh Zayed CityEgypt
  3. 3.Tecnologico de Monterrey, Escuela de Ingenieria y CienciasMonterreyMexico

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