Abstract: A novel color image encryption algorithm based on dynamic deoxyribonucleic acid (DNA) encoding and chaos is presented. A three-neuron fractional-order discrete Hopfield neural network (FODHNN) is employed as a pseudo-random chaotic sequence generator. Its initial value is obtained with the secret key generated by a five-parameter external key and a hash code of the plain image. The external key includes both the FODHNN discrete step size and order. The hash is computed with the SHA-2 function. This ensures a large secret key space and improves the algorithm sensitivity to the plain image. Furthermore, a new three-dimensional projection confusion method is proposed to scramble the pixels among red, green, and blue color components. DNA encoding and diffusion are used to diffuse the image information. Pseudo-random sequences generated by FODHNN are employed to determine the encoding rules for each pixel and to ensure the diversity of the encoding methods. Finally, confusion II and XOR are used to ensure the security of the encryption. Experimental results and the security analysis show that the proposed algorithm has better performance than those reported in the literature and can resist typical attacks.

一种基于离散分数阶混沌系统和DNA序列运算的新型彩色图像加密算法

[1]Abdeljawad T, Banerjee S, Wu GC, 2019. Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption. Optik, in press.

[2]Agarwal RP, El-Sayed AMA, Salman SM, 2013. Fractional-order Chua’s system: discretization, bifurcation and chaos. Adv Differ Equat, 2013:320.

[3]Ahgue AO, de Nkapkop JD, Effa JY, et al., 2018. A DNA-based chaos algorithm for an efficient image encryption application. Int Symp on Electronics and linebreak Telecommunications, p.1-4.

[4]Al-Hazaimeh OM, Al-Jamal MF, Alhindawi N, et al., 2019. Image encryption algorithm based on Lorenz chaotic map with dynamic secret keys. Neur Comput Appl, 31(7):2395-2405.

[5]Angstmann CN, Henry BI, Jacobs BA, et al., 2017. Discretization of fractional differential equations by a piecewise constant approximation. Math Model Nat Phenom, 12(6):23-36.

[6]Atici FM, Şengül S, 2010. Modeling with fractional difference equations. J Math Anal Appl, 369(1):1-9.

[7]Chai XL, Fu XL, Gan ZH, et al., 2019. A color image cryptosystem based on dynamic DNA encryption and chaos. Signal Process, 155:44-62.

[8]Chen GR, Mao YB, Charles KC, 2004. A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos Sol Fract, 21(3):749-761.

[9]Chen JL, Lei C, Lin SL, et al., 2015. Preparation and structural characterization of a partially depolymerized beta-glucan obtained from Poria cocos sclerotium by ultrasonic treatment. Food Hydrocoll, 46:1-9.

[10]Chen JX, Zhu ZL, Fu C, et al., 2015. An image encryption scheme using nonlinear inter-pixel computing and swapping based permutation approach. Commun Nonl Sci Numer Simul, 23(1-3):294-310.

[11]Chen JX, Zhu ZL, Zhang LB, et al., 2018. Exploiting self-adaptive permutation-diffusion and DNA random encoding for secure and efficient image encryption. Signal Process, 142:340-353.

[12]Chen LP, Wu RC, He YG, et al., 2015. Robust stability and stabilization of fractional-order linear systems with polytopic uncertainties. Appl Math Comput, 257:274-284.

[13]Chen LP, Cao JD, Wu RC, et al., 2017. Stability and synchronization of fractional-order memristive neural networks with multiple delays. Neur Netw, 94:76-85.

[14]Deng SJ, Zhan YP, Xiao D, et al., 2011. Analysis and improvement of a hash-based image encryption algorithm. Commun Nonl Sci Numer Simul, 16(8):3269-3278.

[15]El Raheem ZF, Salman SM, 2014. On a discretization process of fractional-order logistic differential equation. J Egypt Math Soc, 22(3):407-412.

[16]Enayatifar R, Abdullah AH, Isnin IF, 2014. Chaos-based image encryption using a hybrid genetic algorithm and a DNA sequence. Opt Lasers Eng, 56:83-93.

[17]Enayatifar R, Sadaei HJ, Abdullah AH, et al., 2015. A novel chaotic based image encryption using a hybrid model of deoxyribonucleic acid and cellular automata. Opt Lasers Eng, 71:33-41.

[18]Essaid M, Akharraz I, Saaidi A, et al., 2019. A novel image encryption scheme based on permutation/diffusion process using an improved 2D chaotic system. Int Conf on Wireless Technologies, Embedded and Intelligent Systems, p.1-6.

[19]Goodrich C, Peterson AC, 2015. Discrete Fractional Calculus. Springer, New York, USA.

[20]Gottwald GA, Melbourne I, 2004. A new test for chaos in deterministic systems. Proc R Soc Lond Ser A, 460(2042):603-611.

[21]Gray RM, 2011. Entropy and Information Theory (2^nd Ed.). Springer, New York, USA.

[22]Guan ZH, Huang FJ, Guan WJ, 2005. Chaos-based image encryption algorithm. Phys Lett A, 346(1-3):153-157.

[23]Guesmi R, Farah MAB, Kachouri A, et al., 2016. A novel chaos-based image encryption using DNA sequence operation and secure Hash algorithm SHA-2. Nonl Dynam, 83(3):1123-1136.

[24]Hanis S, Amutha R, 2019. A fast double-keyed authenticated image encryption scheme using an improved chaotic map and a butterfly-like structure. Nonl Dynam, 95(1):421-432.

[25]Hopfield JJ, 1982. Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA, 79(8):2554-2558.

[26]Hu GY, Kou WL, Dong JE, et al., 2018. A novel image encryption algorithm based on cellular neural networks hyper chaotic system. IEEE 4^th Int Conf on Computer and Communications, p.1878-1882.

[27]Hua ZY, Zhou YC, Huang HJ, 2019. Cosine-transform-based chaotic system for image encryption. Inform Sci, 480:403-419.

[28]Huang LL, Park JH, Wu GC, et al., 2020. Variable-order fractional discrete-time recurrent neural networks. J Comput Appl Math, 370:112633.

[29]Kaslik E, Sivasundaram S, 2012. Nonlinear dynamics and chaos in fractional-order neural networks. Neur Netw, 32:245-256.

[30]Li CQ, Lin DD, Li JH, 2017. Cryptanalyzing an image-scrambling encryption algorithm of pixel bits. IEEE Multim, 24(3):64-71.

[31]Li RZ, Liu Q, Liu LF, 2019. Novel image encryption algorithm based on improved logistic map. IET Image Process, 13(1):125-134.

[32]Li Z, Peng CG, Li LR, et al., 2018. A novel plaintext-related image encryption scheme using hyper-chaotic system. Nonl Dynam, 94(2):1319-1333.

[33]Liu HJ, Kadir A, 2015. Asymmetric color image encryption scheme using 2D discrete-time map. Signal Process, 113:104-112.

[34]Machado JAT, 2015. Fractional order description of DNA. Appl Math Model, 39(14):4095-4102.

[35]Machado JAT, 2017. Bond graph and memristor approach to DNA analysis. Nonl Dynam, 88(2):1051-1057.

[36]Machado JAT, Costa AC, Quelhas MD, 2011. Entropy analysis of the DNA code dynamics in human chromosomes. Comput Math Appl, 62(3):1612-1617.

[37]Miller KS, Ross B, 1988. Fractional difference calculus. Proc Int Symp on Univalent Functions, Fractional Calculus and Their Applications, p.139-152.

[38]Norouzi B, Mirzakuchaki S, 2017. An image encryption algorithm based on DNA sequence operations and cellular neural network. Multim Tools Appl, 76(11):13681-13701.

[39]Ravichandran D, Praveenkumar P, Rayappan JBB, et al., 2017. DNA chaos blend to secure medical privacy. IEEE Trans NanoBiosci, 16(8):850-858.

[40]Sun SL, 2018. A novel hyperchaotic image encryption scheme based on DNA encoding, pixel-level scrambling and bit-level scrambling. IEEE Photon J, 10(2):7201714.

[41]Toughi S, Fathi MH, Sekhavat YA, 2017. An image encryption scheme based on elliptic curve pseudo random and Advanced Encryption System. Signal Process, 141:217-227.

[42]ur Rehman A, Liao XF, Ashraf R, et al., 2018. A color image encryption technique using exclusive-OR with DNA complementary rules based on chaos theory and SHA-2. Optik, 159:348-367.

[43]Wang JS, Long F, Ou WH, 2017. CNN-based color image encryption algorithm using DNA sequence operations. Int Conf on Security, Pattern Analysis, and Cybernetics, p.730-736.

[44]Wang XY, Zhang HL, Bao XM, 2016a. Color image encryption scheme using CML and DNA sequence operations. Biosystems, 144:18-26.

[45]Wang XY, Liu CM, Zhang HL, 2016b. An effective and fast image encryption algorithm based on chaos and interweaving of ranks. Nonl Dynam, 84(3):1595-1607.

[46]Wu GC, Baleanu D, Lin ZX, 2016. Image encryption technique based on fractional chaotic time series. J Vibr Contr, 22(8):2092-2099.

[47]Wu GC, Abdeljawad T, Liu JL, et al., 2019a. Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique. Nonl Anal Model Contr, 24(6):919-936.

[48]Wu GC, Deng ZG, Baleanu D, et al., 2019b. New variable-order fractional chaotic systems for fast image encryption. Chaos, 29(8):083103.

[49]Wu XJ, Kan HB, Kurths J, 2015. A new color image encryption scheme based on DNA sequences and multiple improved 1D chaotic maps. Appl Soft Comput, 37:24-39.

[50]Wu XJ, Wang KS, Wang XY, et al., 2017. Lossless chaotic color image cryptosystem based on DNA encryption and entropy. Nonl Dynam, 90(2):855-875.

[51]Ye GD, Pan C, Huang XL, et al., 2018. An efficient pixel-level chaotic image encryption algorithm. Nonl Dynam, 94(1):745-756.

[52]Zhang GJ, Liu Q, 2011. A novel image encryption method based on total shuffling scheme. Opt Commun, 284(12): 2775-2780.

[53]Zhang LM, Sun KH, Liu WH, et al., 2017. A novel color image encryption scheme using fractional-order hyperchaotic system and DNA sequence operations. Chin Phys B, 26(10):100504.

[54]Zhang LY, Li CQ, Wong KW, et al., 2012. Cryptanalyzing a chaos-based image encryption algorithm using alternate structure. J Syst Softw, 85(9):2077-2085.

[55]Zhang Q, Wei XP, 2013. A novel couple images encryption algorithm based on DNA subsequence operation and chaotic system. Optik, 124(23):6276-6281.

[56]Zhang Q, Liu LL, Wei XP, 2014. Improved algorithm for image encryption based on DNA encoding and multi-chaotic maps. AEU Int J Electron Commun, 68(3): 186-192.

[57]Zhang R, Qi DW, Wang YZ, 2010. Dynamics analysis of fractional order three-dimensional Hopfield neural network. Proc 6^th Int Conf on Natural Computation, p.3037-3039.

[58]Zhang Y, 2018. The unified image encryption algorithm based on chaos and cubic S-Box. Inform Sci, 450:361-377.

[59]Zhang YQ, Wang XY, 2015. A new image encryption algorithm based on non-adjacent coupled map lattices. Appl Soft Comput, 26:10-20.

[60]Zhang YQ, Wang XY, Liu J, et al., 2016. An image encryption scheme based on the MLNCML system using DNA sequences. Opt Lasers Eng, 82:95-103.

[61]Zhang YS, Xiao D, 2014. Self-adaptive permutation and combined global diffusion for chaotic color image encryption. AEU Int J Electron Commun, 68(4):361-368.

[62]Zheng XD, Xu J, Li W, 2009. Parallel DNA arithmetic operation based on n-moduli set. Appl Math Comput, 212(1):177-184.

[63]Zhou YC, Bao L, Chen CLP, 2014. A new 1D chaotic system for image encryption. Signal Process, 97:172-182.

[64]Zhou YC, Hua ZY, Pun CM, et al., 2015. Cascade chaotic system with applications. IEEE Trans Cybern, 45(9):2001-2012.

[65]Zhu ZL, Zhang W, Wong KW, et al., 2011. A chaos-based symmetric image encryption scheme using a bit-level permutation. Inform Sci, 181(6):1171-1186.