A novel efficient image encryption algorithm based on affine transformation combine with linear fractional transformation

  • Dawood ShahEmail author
  • Tariq Shah
  • Sajjad Shaukat Jamal


Algebraic structures and their hardware–software implementation gain considerable attention in the field of information security and coding theory. Research progress in the applications of arithmetic properties of algebraic structures is being frequently made. These structures are mostly useful in improvement of the cryptographic algorithms. A novel technique is given to design a cryptosystem responsible for lossless for image encryption. The proposed scheme is for the RGB image whose pixels are considered as 24 binary bits, accordingly a unique arrangement for the construction of S-boxes over a Galois field \( GF\left( {2^{9} } \right) \) is employed. Consequently, it generates multiple different S-boxes with excellent cryptographic characteristic and hence confusing process of the cryptosystem has been working. Whereas the diffusion process in this cryptosystem is based on Affine transformation over a unit elements of an integers modulo ring \( {\mathbb{Z}}_{n} \). The scrambling of the image data through the Affine transformation escalate the security asset, avoid computational effort and abbreviated the time complexity. In addition, the simulation test and comparative scrutinize illustrate that the proposed scheme is highly sensitive, large keyspace, excellent statistical properties and secure against differential attacks. Therefore, the proposed algorithm is valuable for confidential communication. Furthermore, due to the arithmetic properties of algebraic structures, the proposed scheme would be easily implemented, secure and fast enough to be utilized in real-world applications.


Galois field Substitution boxes Affine transformation LFT 



The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant Number R.G.P. 2/58/40.


  1. Alvarez, G., & Li, S. J. (2006). Some basic cryptographic requirements for a chaos-based cryptosystem. International Journal of Bifurcation and Chaos, 16(8), 2129–2151.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Amin, M., Faragallah, O. S., & Abd El-Latif, A. A. (2010). A chaotic block cipher algorithm for image cryptosystems. Communications in Nonlinear Science and Numerical Simulation, 15(11), 3484–3497.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Awad, A., & Awad, D. (2010). Efficient image chaotic encryption algorithm with no propagation error. ETRI Journal, 32(5), 774–783.CrossRefGoogle Scholar
  4. Behnia, S., Akhshani, A., Mahmodi, H., & Akhavan, A. (2008). A novel algorithm for image encryption based on a mixture of chaotic maps. Chaos, Solitons & Fractals, 35(2), 408–419.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Belazi, A., El-Latif, A. A. A., & Belghith, S. (2016). A novel image encryption scheme based on substitution-permutation network and chaos. Signal Processing, 128, 155–170.CrossRefGoogle Scholar
  6. Chai, X., Fu, X., Gan, Z., Lu, Y., & Chen, Y. (2018). A color image cryptosystem based on dynamic DNA encryption and chaos. Journal of Signal Processing, 155(2019), 44–62.Google Scholar
  7. Chai, X. L., Gan, Z. H., Lu, Y., Zhang, M. H., & Chen, Y. R. (2016). A novel color image encryption algorithm based on genetic recombination and the four-dimensional memristive hyperchaotic system. Chinese Physics B, 25(10), 100503.CrossRefGoogle Scholar
  8. Chiaraluce, F., Ciccarelli, L., Gambi, E., Pierleoni, P., & Reginelli, M. (2002). A new chaotic algorithm for video encryption. IEEE Transactions on Consumer Electronics, 48(4), 838–844.CrossRefGoogle Scholar
  9. Daemen, J., & Rijmen, V. (2002). The design of Rijndael: AES—the advanced encryption standard. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  10. Dong, C. (2014). Color image encryption using one-time keys and coupled chaotic systems. Signal Processing: Image Communication, 29, 628–640.Google Scholar
  11. Fridrich, J. (1998). Symmetric ciphers based on two-dimensional chaotic maps. International Journal of Bifurcation and Chaos, 8(6), 1259–1284.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Gan, Z., Chai, X., Yuan, K., & Lu, Y. (2018a). A novel image encryption algorithm based on LFT based S-boxes and chaos. Multimedia Tools and Applications, 77(7), 8759–8783.CrossRefGoogle Scholar
  13. Gan, Z., Chai, X., Zhang, M., & Lu, Y. (2018b). A double color image encryption scheme based on three-dimensional Brownian motion. Multimedia Tools and Applications, 77(21), 27919–27953.CrossRefGoogle Scholar
  14. Gao, T. G., & Chen, Z. Q. (2008). A new image encryption algorithm based on hyper-chaos. Physics Letters A, 372(4), 394–400.zbMATHCrossRefGoogle Scholar
  15. Hussain, I., & Gondal, M. A. (2014). An extended image encryption using chaotic coupled map and S-box transformation. Nonlinear Dynamics, 76(2), 1355–1363.CrossRefGoogle Scholar
  16. Hussain, I., Shah, T., & Gondal, M. A. (2012). Image encryption algorithm based on PGL (2, GF (28)) S-boxes and TD-ERCS chaotic sequence. Nonlinear Dynamics, 70(1), 181–187.MathSciNetCrossRefGoogle Scholar
  17. Li, S., Chen, G., & Zheng, X. (2005). Chaos-based encryption for digital images and videos. Multimedia security handbook, chapter 4 (pp. 133–167). Boca Raton: CRC Press.Google Scholar
  18. Li, S. J., Li, C. Q., Chen, G. R., Bourbakis, N. G., & Lo, K. T. (2008). General quantitative cryptanalysis of permutation-only multimedia ciphers against plain-image attacks. Signal Processing: Image Communication, 23(3), 212–223.Google Scholar
  19. Li, C., Zhang, L. Y., Ou, R., Wong, K.-W., & Shu, S. (2012). Breaking a novel color image encryption algorithm based on chaos. Nonlinear Dynamics, 70(4), 2383–2388.MathSciNetCrossRefGoogle Scholar
  20. Lian, S. G., Sun, J. S., & Wang, Z. Q. (2005). A block cipher based on a suitable use of the chaotic standard map. Chaos, Solitons & Fractals, 26(1), 117–129.zbMATHCrossRefGoogle Scholar
  21. Liu, H. J., & Kadir, A. (2015). Asymmetric color image encryption scheme using 2D discrete-timemap. Signal Processing, 113, 104–112.CrossRefGoogle Scholar
  22. Liu, H., Kadir, A., & Gong, P. (2015). A fast color image encryption scheme using one-time S-boxes based on complex chaotic system and random noise. Optics Communications, 338, 340–347.CrossRefGoogle Scholar
  23. Liu, Y., Zhang, L. Y., Wang, J., Zhang, Y., & Wong, K.-W. (2016). Chosen-plaintext attack of an image encryption scheme based on modified permutation-diffusion structure. Nonlinear Dynamics, 84(4), 2241–2250.zbMATHCrossRefGoogle Scholar
  24. Naseer, Y., Shah, D., & Shah, T. (2019a). A novel approach to improve multimedia security utilizing 3D mixed chaotic map. Microprocessors and Microsystems. Scholar
  25. Naseer, Y., Shah, T., Shah, D., & Hussain, S. (2019b). A novel algorithm of constructing highly nonlinear sp-boxes. Cryptography, 3(1), 6.CrossRefGoogle Scholar
  26. Norouzi, B., Mirzakuchaki, S., Seyedzadeh, S.-M., & Mosavi, M.-R. (2014). A simple, sensitive and secure image encryption algorithm based on hyper-chaotic system with only one round diffusion. Multimedia Tools and Applications, 71(3), 1469–1497.CrossRefGoogle Scholar
  27. Patidar, V., Pareek, N. K., & Sud, K. K. (2009). A new substitution–diffusion-based image cipher using chaotic standard and logistic maps. Communications in Nonlinear Science and Numerical Simulation, 14(7), 3056–3075.CrossRefGoogle Scholar
  28. Pisarchik, A. N., Flores-Carmona, N. J., & Carpio-Valadez, M. (2006). Encryption and decryption of images with chaotic map lattices. Chaos, 16(3), 033118.MathSciNetzbMATHCrossRefGoogle Scholar
  29. Saberi, K. M., Mohammad, D., Rahim, M., & Yaghobi, M. (2014). Using 3-cell chaotic map for image encryption based on biological operations. Nonlinear Dynamics, 75(3), 407–416.CrossRefGoogle Scholar
  30. Shah, T., & Shah, D. (2019). Construction of highly nonlinear S-boxes for degree 8 primitive irreducible polynomials over ℤ 2. Multimedia Tools and Applications, 78(2), 1219–1234.CrossRefGoogle Scholar
  31. Shah, D., ul Haq, T., & Shah, T. (2018). Image encryption based on action of projective general linear group on a galois field GF (28). In 2018 international conference on applied and engineering mathematics (ICAEM).
  32. Tuchman, W., IV. (1979). Hellman presents no shortcut solutions to the DES’. IEEE Spectrum, 16(7), 40–41.CrossRefGoogle Scholar
  33. ur Rehman, A., Liao, X. F., Ashraf, R., Ullah, S., & Wang, H. W. (2018). A color image encryption technique using exclusive-OR with DNA complementary rules based on chaos theory and SHA-2. Optik, 159, 348–367.CrossRefGoogle Scholar
  34. Ur Rehman, A., Liao, X., Kulsoom, A., & Abbas, S. A. (2015). Selective encryption for gray images based on chaos and DNA complementary rules. Multimedia Tools and Applications, 74(13), 4655–4677.CrossRefGoogle Scholar
  35. Wang, X., Teng, L., & Qin, X. (2012). A novel colour image encryption algorithm based on chaos. Signal Processing, 92(4), 1101–1108.MathSciNetCrossRefGoogle Scholar
  36. Wang, X., & Wang, Q. (2014). A novel image encryption algorithm based on dynamic S-boxes constructed by chaos. Nonlinear Dynamics, 75(3), 567–576.CrossRefGoogle Scholar
  37. Wang, Y., Wong, K. W., Liao, X. F., & Xiang, T. (2009a). A block cipher with dynamic S-boxes based on tent map. Communications in Nonlinear Science and Numerical Simulation, 14(7), 3089–3099.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Wang, Y., Wong, K. W., Liao, X. F., Xiang, T., & Chen, G. R. (2009b). A chaos-based image encryption algorithm with variable control parameters. Chaos, Solitons & Fractals, 41(4), 1773–1783.zbMATHCrossRefGoogle Scholar
  39. Wang, X. Y., Yang, L., Liu, R., & Kadir, A. (2015). A chaotic image encryption algorithm based on perceptron model. Nonlinear Dynamics, 62(3), 615–621.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Wong, K. W., Kwok, B. S. H., & Law, W. S. (2008). A fast image encryption scheme based on the chaotic standard map. Physics Letters A, 372(15), 2645–2652.zbMATHCrossRefGoogle Scholar
  41. Wu, X. J., Kan, H. B., & Kurths, J. (2015). A new color image encryption scheme based on DNA sequences and multiple improved 1D chaotic maps. Applied Soft Computing, 37, 24–39.CrossRefGoogle Scholar
  42. Wu, J. H., Liao, X. F., & Yang, B. (2017). Color image encryption based on chaotic systems and elliptic curve ElGamal scheme. Signal Processing, 141, 109–124.CrossRefGoogle Scholar
  43. Zhang, Y., Li, C., Li, Q., Zhang, D., & Shu, S. (2012). Breaking a chaotic image encryption algorithm based on perceptron model. Nonlinear Dynamics, 69(3), 1091–1096.MathSciNetzbMATHCrossRefGoogle Scholar
  44. Zhang, Y., Li, Y., Wen, W., Wu, Y., & Che, J.-X. (2015). Deciphering an image cipher based on 3-cell chaotic map and biological operations. Nonlinear Dynamics, 82(4), 1831–1837.MathSciNetCrossRefGoogle Scholar
  45. Zhang, X., Mao, Y., & Zhao, Z. (2014a). An efficient chaotic image encryption based on alternate circular S-boxes. Nonlinear Dynamics, 78(1), 359–369.CrossRefGoogle Scholar
  46. Zhang, Y., Xiao, D., Wen, W., & Li, M. (2014b). Breaking an image encryption algorithm based on hyper-chaotic system with only one round diffusion process. Nonlinear Dynamics, 76(3), 1645–1650.CrossRefGoogle Scholar
  47. Zhu, Z. L., Zhang, W., Wong, K. W., & Yu, H. (2011). A chaos-based symmetric image encryption scheme using a bit-level permutation. Information Sciences, 181(6), 1171–1186.CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Dawood Shah
    • 1
    Email author
  • Tariq Shah
    • 1
  • Sajjad Shaukat Jamal
    • 2
  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  2. 2.Department of Mathematics, College of ScienceKing Khalid UniversityAbhaSaudi Arabia

Personalised recommendations