Chaotic image encryption using hopfield and hindmarsh–rose neurons implemented on FPGA

sensors Article Chaotic Image Encryption Using Hopfield and Hindmarsh–Rose Neurons Implemented on FPGA Esteban Tlelo-Cuautle 1,*,† , Jonathan Daniel Díaz-Muñoz 1,†, Astrid Maritza González-Zapata 1,†, Rui Li 2,†, Walter Daniel León-Salas 3,†, Francisco V. Fernández 4,† , Omar Guillén-Fernández 1,† and Israel Cruz-Vega 1,† 1Department of Electronics, INAOE, Puebla 72840, Mexico; [email protected] (J.D.D.-M.); [email protected] (A.M.G.-Z.); [email protected] (O.G.-F.); [email protected] (I.C.-V.) 2School of Automation Engineering, UESTC, Chengdu 611731, China; [email protected] 3School of Engineering Technology, Purdue University, 401 N. Grant St., West Lafayette, IN 47907, USA; [email protected] 4Instituto de Microelectrónica de Sevilla, CSIC and Universidad de Sevilla, 41092 Sevilla, Spain; [email protected] *Correspondence: [email protected]; Tel.: +52-222-2470-517 † These authors contributed equally to this work. Received: 11 February 2020; Accepted: 24 February 2020; Published: 28 February 2020   Abstract: Chaotic systems implemented by artificial neural networks are good candidates for data encryption. In this manner, this paper introduces the cryptographic application of the Hopfield and the Hindmarsh–Rose neurons. The contribution is focused on finding suitable coefficient values of the neurons to generate robust random binary sequences that can be used in image encryption. This task is performed by evaluating the bifurcation diagrams from which one chooses appropriate coefficient values of the mathematical models that produce high positive Lyapunov exponent and Kaplan–Yorke dimension values, which are computed using TISEAN. The randomness of both the Hopfield and the Hindmarsh–Rose neurons is evaluated from chaotic time series data by performing National Institute of Standard and Technology (NIST) tests. The implementation of both neurons is done using field-programmable gate arrays whose architectures are used to develop an encryption system for RGB images. The success of the encryption system is confirmed by performing correlation, histogram, variance, entropy, and Number of Pixel Change Rate (NPCR) tests. Keywords: chaos; Hopfield neuron; Hindmarsh–Rose neuron; Lyapunov exponent; image encryption; correlation; FPGA 1. Introduction Image encryption is one of the well-known mechanisms to preserve confidentiality over a reliable unrestricted public channel. However, public channels are vulnerable to attacks and hence efficient encryption algorithms must be developed for secure data transfer. In [ 1 ], the authors surveyed ten conventional and five chaos-based encryption techniques to encrypt three test images of different sizes based on various performance metrics, and the important conclusion was that none of the conventional schemes were designed especially for images and hence none of them have any dependence on the initial image. In this manner, the topic of image encryption remains open and several researchers are proposing the use of chaotic systems to mask information that can be transmitted in a secure channel. In this direction, this paper highlights the usefulness of Hopfield and Hindmarsh–Rose neural networks to generate chaotic behavior, and its suitability to design random number generators (RNGs) that are implemented using field-programmable gate arrays (FPGAs). Sensors 2020,20, 1326; doi:10.3390/s20051326 www.mdpi.com/journal/sensors Sensors 2020,20, 1326 2 of 22 In [ 2 ] J.J. Hopfield introduced the neuron model that nowadays is known as the Hopfield neural network. Ten years later, a modified model of Hopfield neural network was proposed in [ 3 ], and applied in information processing. Immediately, the Hopfield neural network was adapted to generate chaotic behavior in [ 4 ] where the authors explored bifurcation diagrams. In [ 5 ] the simplified Hopfield neuron model was designed to use a sigmoid as activation function, and three neurons were used to generate chaotic behavior. In addition, the authors performed an optimization process updating the weights of the neurons interconnections. The Hopfield neuron was combined with a chaotic map in [ 6 ] to be applied in chaotic masking. More recently, the authors in [ 7 ] proposed an image encryption algorithm using the Hopfield neural network. In the same direction, the authors in [ 8 ] detailed the behavior of Hindmarsh–Rose neuron to generate chaotic behavior. Its bifurcation diagrams were described in [ 9 ], and the results were used to select the values of the model to improve chaotic behavior. Hindmarsh–Rose neurons were synchronized in [ 10 ], optimizing the scheme of Lyapunov function with two gain coefficients. In this way, the synchronization region is estimated by evaluating the Lyapunov stability. Two Hindmarsh–Rose neurons were synchronized in [ 11 ], and the system was used to mask information in continuous time. To show that the neurons generate chaotic behavior, one must compute Lyapunov exponents, and for the Hindmarsh–Rose neuron they were evaluated by the TISEAN package in [12]. The Hopfield neural network has been widely applied in chaotic systems [ 13 – 15 ]. This network consists of three neurons, and the authors in [ 13 ] proposed a simplified model by removing the synaptic weight connection of the third and second neuron in the original Hopfield network. Numerical simulations were carried out considering values from the bifurcation diagrams, and Lyapunov exponents were evaluated to conclude that the simplified model exhibits rich nonlinear dynamical behaviors including symmetry breaking, chaos, periodic window, antimonotonicity and coexisting self-excited attractors. An FPGA-based modified Hopfield neural network was introduced in [ 14 ], to generate multiple attractors, but there are no details of their hardware design from computer arithmetic. The authors in [ 15 ] showed the existence of hidden chaotic sets in a simplified Hopfield neural network with three neurons. Similar to the Hopfield neural network, the Hindmarsh–Rose neuron is quite useful, for example: using the Hindmarsh–Rose neuron model, the authors in [ 16 ] showed that in the parameter region close to the bifurcation value, where the only attractor of the system is the limit cycle of tonic spiking type, the noise can transform the spiking oscillatory regime to the bursting one. The fractional-order version of the Hindmarsh–Rose neuron was used in [ 17 ], for the synchronization of fractional-order chaotic systems. In [ 18 ], based on two-dimensional Hindmarsh–Rose neuron and non-ideal threshold memristor, a five-dimensional neuron model of two adjacent neurons coupled by memristive electromagnetic induction, was introduced. In a similar way, the authors in [ 19 ] showed the effects of time delay on burst synchronization transitions of a neural network which was locally modeled by Hindmarsh–Rose neurons. On the one hand, the main drawback of those works was the lack of statistical tests according to the National Institute of Standard and Technology (NIST), as done in other chaotic systems given in [ 20 – 22 ], to guarantee the randomness of the chaotic sequences. On the other hand, and in addition to NIST tests, the authors in [ 23 ] recommend to improve the key space when using chaotic maps, thus enhancing the image encryption schemes. In this work we show the application of neural networks in the design of random number generators (RNGs), whose binary sequences are applied to implement an image encryption scheme [ 24 ]. This idea has been previously exploited, for example: the Hopfield neural network was used in [ 25 ] to design a RNG, but showed low randomness. In this manner, this paper introduces the selection of the best coefficients of both Hopfield and Hindmarsh–Rose neurons, from the bifurcation diagram, to generate robust chaotic sequences that improve NIST tests and enhance chaotic encryption of images. Section 2describes both the Hopfield and the Hindmarsh–Rose neuron models, showing their chaotic behavior. Section 3shows simulation results of the cases that generate better chaotic time series, applying the 4th-order Runge-Kutta method. Bifurcation diagrams are generated to select appropriate values that improve the generation of chaotic times series that are evaluated using TISEAN, in order to Sensors 2020,20, 1326 3 of 22 verify the Lyapunov exponents. Section 4details the FPGA-based implementation of both Hopfield and Hindmarsh–Rose neuron models. Section 5shows the selection of the series with the highest values of the positive Lyapunov exponent, which are used to generate binary sequences, and whose randomness is evaluated by performing NIST tests. Section 6shows the application of the generated binary sequences to encrypt an image in a chaotic secure communication system, and the success of the RGB image encryption system is confirmed by performing correlation, histogram, variance, entropy, and Number of Pixel Change Rate (NPCR) tests. Finally, Section 7summarizes the main results of this work. 2. Mathematical Models of Hopfield and Hindmarsh–Rose Neurons This section describes the mathematical models of both the Hopfield and the Hindmarsh–Rose neural networks. For instance, the complex dynamics of the Hopfield-type neural network with three neurons are analyzed in [ 26 ], as well as the observation of the stable points, limit circles, single-scroll chaotic attractors and double-scrolls chaotic attractors. By varying the parameters, the numerical simulations performed in [ 27 ] show that the simple Hopfield neural networks can display chaotic attractors and periodic orbits for different parameters, and they associate different values of the Lyapunov exponents and bifurcation plots. The Hindmarsh–Rose neural network is analyzed in [ 28 ], and by using the polynomial model previously introduced in [ 29 ], the authors perform a detailed bifurcation analysis of the full fast-slow system for bursting patterns. 2.1. Hopfield Neuron The Hopfield neural network can be modeled by Equation (1), where v represents the state variables, c is a proportional constant, W is the weights matrix, and f(v) is associated to the activation function [5]. ˙ v=−cv +W f (v). (1) Commonly, c is made equal to one, and when a chaotic behavior is desired, the activation function is a hyperbolic tangent function and the weights matrix W is modified, whose size depends on the number of neurons. In this work, W has size 3 × 3, meaning that the Hopfield neural network has three state variables, associated with each neuron. That way, Equations (2)–(4), describe the model of the three neurons, as shown in [ 5 ]. In this case, v in (1) is replaced by a three elements vector, including three state variables, namely: x , y , and z ; and the control parameter p is set to 0.0997. This value is found by exploring values that maximize the positive Lyapunov exponent (LE+). Figures 1 and 2show the chaotic time series and the attractors obtained by applying the 4th-order Runge–Kutta method.    ˙ x ˙ y ˙ z  =−   x y z  +W   f(x) f(y) f(z)  (2)    f(x) f(y) f(z)  =   tanh(x) tanh(y) tanh(z)  (3) W=   2−1.2 0 1.9 +p1.71 1.15 −4.75 0 1.1   . (4) The equilibrium points are obtained by applying the Newton Raphson method Gi+1= Gi−[J(Gi)]−1f(Gi) , where J(Gi) is the Jacobian, because the neural network has nonlinear terms as tanh() . The Equilibrium points are: EP1=(0, 0, 0) , EP2=(0.4932, 0.3658, −3.2666) , EP3= (−0.4932, −0.3658, 3.2666) . The eigenvalues are obtained for each equilibrium point evaluating Sensors 2020,20, 1326 4 of 22 |λI−J|= 0, so that the ones associated to EP1 are: λ1 =1.9416 and λ2,3 =-0.0658 ± j 1.8793. The eigenvalues associated to EP2and EP3are: λ1=-0.9870 and λ2,3=0.5381 ±j 1.2861. (a) (b) (c) Figure 1. Simulation results of the chaotic time series of the state variables: ( a ) x , ( b ) y , and ( c ) z , of the Hopfield neural network given in [ 5 ]. The initial conditions are: x0 = 1.951738939809982, y0=−1.207112821944644, and z0=−0.284321234701517. Figure 2. Attractors generated by plotting the state variables shown in Figure 1. 2.2. Hindmarsh–Rose Neuron The Hindmarsh–Rose neural network can be modeled by three state variables, as given by Equation (5) . This model is used to analyze the charge and discharge of a neuron, and in addition, Sensors 2020,20, 1326 5 of 22 when it provides chaotic behavior, its applications can be extended to cryptography, as shown in this work. ˙ x=−s(−ax3+x2)−y−bz ˙ y=ϕ(x2−y) ˙ z=ε(sa1x+b1−kz). (5) In Equation (5) , x is associated to the membrane voltage, y is the recovery variable associated to the current, and z is the slow and adaptable current. The coefficients a , b , a1 , b1 , k , s are parameters of the neuron, and ϕ and ε are associated to the time scale. Their values are set to: ϕ= 1, a= 0.5, b= 1, a1=− 0.1, k= 0.2, b1=− 0.045, ε= 0.02, and s=− 1.605 [ 11 ]. Figure 3shows the time series of the state variable xof Hindmarsh–Rose neuron, and Figure 4shows the phase-space portraits. Figure 3. Chaotic time series of state variable x of the Hindmarsh–Rose neuron using x0= 0.1169282607, y0=0.03563851071, and z0=0.01034665217, as initial conditions. Figure 4. Phase-space portraits of the Hindmarsh–Rose neuron given in Equation (5) , using x0= 0.1169282607, y0=0.03563851071, and z0=0.01034665217, as initial conditions. Sensors 2020,20, 1326 6 of 22 The equilibrium points from Equation (5) are: EP1= ( 0.325817, 0.106157, 0.0364682 ) , EP2= ( 0.214038 −j 0.902614, − 0.7689 −j 0.38638, − 0.053234 −j 0.7243 ) , and EP3= ( 0.214038 + j 0.902614, − 0.7689 +j 0.38638, − 0.053234 +j 0.7243 ) . The eigenvalues associated to each equilibrium point are: λ1 = 0.261784, λ2 = 0.0204526, λ3 =- 0.495936 for EP1 ; and λ1 = 2.2075 ± j 1.5659, λ2 =- 0.002488 ±j 0.0001127, and λ3=−0.67089 ±j 0.4012, for EP2and EP3. 3. Bifurcation Diagrams and Selection of the Best Values to Generate Enhanced Chaotic Time Series Bifurcation diagrams are quite useful to find appropriate values of the mathematical models of the neurons, and in this paper, they are generated to find the best Lyapunov exponents and Kaplan–Yorke dimension, which are considered as appropriate metrics to enhance the generation of chaotic time series. In the case of the Hopfield neuron, the state variable x is selected to plot the bifurcation with respect to the control parameter p . This process must be performed following the variation of all the coefficients of the mathematical model and for all the state variables in various iterations until some dynamical characteristics of the chaotic system, like the positive Lyapunov exponent (LE+) and Kaplan–Yorke dimension [ 30 ], are improved. Varying the weights matrix W , one can find better characteristics. For example, in this paper the nine elements in W were varied in the ranges and steps listed in Table 1. All these cases generated different bifurcation diagrams from which the feasible values were selected. For example, Figure 5shows the bifurcation diagram varying W( 3, 1 ) , where it can be appreciated that the feasible values to generate chaotic behavior must be chosen with values lower than − 4.5. In this manner, after exploring the bifurcation diagrams by varying the values in the ranges given in Table 1, three feasible sets of values are given in Table 2, where it can be appreciated their variations with respect to the original values given in [ 5 ]. The chaotic time series associated with those sets of values, obtained by applying the 4th-order Runge–Kutta method are shown in Figure 6for the state variable x . Those chaotic time series are used to evaluate Lyapunov exponents and Kaplan–Yorke dimension using TISEAN. Figure 5. Bifurcation diagram varying W(3, 1)in the range given in Table 1. Sensors 2020,20, 1326 7 of 22 Table 1. Variation conditions of the elements in W. Matrix Element Variation Range Step W(1, 1)[−3, 3] 0.01 W(1, 2)[−2, 2] 0.01 W(1, 3)[−0.5, 0.5] 0.01 W(2, 1)[−2.9, 2.9] 0.01 W(2, 2)[−2.6, 2.6] 0.01 W(2, 3)[−1.7, 1.7] 0.01 W(3, 1)[−5, 5] 0.01 W(3, 2)[−0.5, 0.5] 0.01 W(3, 3)[−1.9, 1.9] 0.01 Table 2. Proposed sets of values to generate enhanced chaotic behavior using Hopfield neuron. Matrix Element Sets of Values Original [5] HNNset1 HNNset2 HNNset3 W(1, 1)2 2 2 1.98 W(1, 2)−1.2 −1.26 −1.26 −1.18 W(1, 3)0 0 0 0.01 W(2, 1)1.9997 1.93 1.93 1.94 W(2, 2)1.71 1.71 1.71 1.72 W(2, 3)1.15 1.2 1.2 1.2 W(3, 1)−4.75 −4.76 −4.76 −4.74 W(3, 2)0 0.06 0.06 0.03 W(3, 3)1.1 1.1 −0.11 1.1 Table 3lists all Lyapunov exponents values and their associated Kaplan–Yorke dimension for each case from Table 2. Table 3. Lyapunov exponents and Kaplan–Yorke dimension DKY associated to each set of values from Table 2. Sets of Values Lyapunov Exponents DKY Original [5] 9.443281 ×10−3−2.407246 ×10−3−6.072266 ×10−22.115872 HNNset1 1.769144 ×10−2−2.681429 ×10−3−8.082816 ×10−22.185703 HNNset2 1.294477 ×10−2−1.194553 ×10−5−7.149536 ×10−22.164349 HNNset3 1.488237 ×10−2−9.456641e×10−4−7.905735 ×10−22.176286 In the case of the Hindmarsh–Rose neural network model given in (5) , one can count eight coefficients that can be varied. In this case, a heuristic process was performed with the goal of improving the chaotic behavior. Each coefficient ϕ , a , b , a1 , k , b1 , ε , and s , was varied in steps of 0.001 and observing the degradation of chaotic behavior. After performing the variations and observing the bifurcation diagrams, two sets of values were found, which are listed in Table 4. In this manner, Figure 7shows the chaotic time series associated with these sets of values. The simulation of the chaotic time series was performed using the initial conditions x0= 0.1169282607, y0= 0.03563851071, and z0= 0.01034665217, and those series were introduced to TISEAN to evaluate Lyapunov exponents and Kaplan–Yorke dimension that are given in Table 5. Since the maximum exponent is positive, then chaotic behavior is guaranteed. In this case, the sets Sensors 2020,20, 1326 8 of 22 of values HRNset1 is the best because the Kaplan–Yorke dimension is 3, i.e., the ideal value for a three-dimensional dynamical system. (a) (b) (c) (d) Figure 6. Chaotic time series generated by using the sets of values listed in Table 2: ( a ) original, (b) HNNset1, (c) HNNset2, and (d) HNNset3, plotting the state variable x. Table 4. Proposed sets of values to generate enhanced chaotic behavior using the Hindmarsh–Rose neuron. Coefficient Sets of Values Original [11] HRNset1 HRNset2 a0.5 0.5 0.5 b1 0.99 1.005 ϕ1 1 1 a1−0.1 −0.095 −0.1 k0.2 0.15 0.198 b1−0.045 −0.045 −0.045 ε0.02 0.02 0.02 s−1.605 −1.610 −1.607 Sensors 2020,20, 1326 9 of 22 (a) (b) (c) Figure 7. Chaotic time series generated by using the sets of values listed in Table 4: ( a ) original, (b) HRNset1, and (c) HRNset2, plotting the state variable x. Table 5. Lyapunov exponents and Kaplan–Yorke dimension DKY associated to each set of values from Table 4. Sets of Values Lyapunov Exponents DKY Original [11] 6.212 ×10−3−6.2947 ×10−5−6.3243 ×10−32.9723 HRNset1 7.3873 ×10−36.5024 ×10−4−7.7942 ×10−33.0000 HRNset2 3.5572 ×10−3−1.7318 ×10−4−5.2275 ×10−32.6473 Sensors 2020,20, 1326 16 of 22 Baboon images, the NPCR values are: 99.2672 when using state variable x , 99.2886 when using the state variable y, and 99.3420 when using z. All NPCR results pass the criterium given in [38]. NPCR =∑ ij D(i,j) T×100% (12) D(i,j) = (0 1 C1(i,j) = C2(i,j) C1(i,j)6=C2(i,j). (13) The binary sequences from the FPGA implementation of the Hindmarsh–Rose neural network shown in Figure 7, were used as PRNG to encrypt the Lena, Fruits and Baboon images, which results are shown in Figure 14. Table 11. Variance of the histograms for the Lena, Fruits and Baboon images using the sequences generated by HNNset2. Encryption Color Variance Lena Fruits Baboon Original Image HNNset2 Original Image HNNset2 Original Image HNNset2 x,y,z R 1,017,334.70 824.77 2,559,535.12 1005.51 331,358.91 1014.40 G 455, 718.80 965.59 1,131,488.32 930.56 571,232.16 934.05 B 1,377,355.87 1071.79 338280.76 891.06 319,770.47 1178.41 x R 1,017,334.70 824.77 2,559,535.12 1005.51 331,358.91 1014.40 G 455,718.80 1033.72 1,131,488.32 1096.43 571,232.16 1017.02 B 1, 377, 355.87 798.66 338,280.76 1233.43 319,770.47 1064.48 y R 1,017,334.70 1188.50 2,559,535.12 963.14 331,358.91 1064.51 G 455,718.80 965.59 1,131,488.32 930.56 571,232.16 934.05 B 1,377,355.87 1107.14 338,280.76 1026.64 319,770.47 1058.48 z R 1,017,334.70 1079.58 2,559,535.12 1136.72 331,358.91 1261.09 G 455,718.80 1055.34 1,131,488.32 1033.53 571,232.16 1133.23 B 1,377,355.87 1071.79 338,280.76 891.06 319,770.47 1178.41 Table 12. Entropies of the original Lena, Fruits and Baboon images and the encrypted ones using the sets of values HNNset2 from Table 2. Encryption Color Entropy Lena Fruits Baboon Original Image HNNset2 Original Image HNNset2 Original Image HNNset2 x,y,z R 7.2531 7.9994 7.0556 7.9994 7.7067 7.9993 G 7.5940 7.9993 7.3527 7.9994 7.4744 7.9994 B 6.9684 7.9993 7.7134 7.9993 7.7522 7.9992 x R 7.2531 7.9994 7.0556 7.9993 7.7067 7.9993 G 7.5940 7.9993 7.3527 7.9992 7.4744 7.9993 B 6.9684 7.9995 7.7134 7.9992 7.7522 7.9993 y R 7.2531 7.9992 7.0556 7.9993 7.7067 7.9993 G 7.5940 7.9993 7.3527 7.9994 7.4744 7.9994 B 6.9684 7.9992 7.7134 7.9993 7.7522 7.9993 z R 7.2531 7.9993 7.0556 7.9992 7.7067 7.9991 G 7.5940 7.9993 7.3527 7.9993 7.4744 7.9992 B 6.9684 7.9993 7.7134 7.9994 7.7522 7.9992 Sensors 2020,20, 1326 17 of 22 Figure 13. Histograms of the Lena image encrypted using HNNset2 from Table 2. Original images on the left and the R, G and B encryption on the right. (a) (b) (c) Figure 14. Image encryption of ( a ) Lena, ( b ) Fruits and ( c ) Baboon, using the binary sequences from the FPGA implementation of Hindmarsh–Rose neuron with HRNset1 from Table 7. The original image is on the left, the encrypted in the center and the recovered on the right column. Sensors 2020,20, 1326 18 of 22 The correlation analysis between the images and the sets of values from Table 7, provides the values given in Table 13. The first row x , y , z , means that the chaotic time series of state x is used to encrypt R (red), y to G (green) and z to B (blue). The second row means that all R, G and B are encrypted using the data from x, and so on. Table 13. Correlations between the chaotic channel and the RGB Lena, Fruits and Baboon images using the sequences generated by the Hindmarsh–Rose neuron for the state variables x,y, and z. Image Encryption Sets of Values Original [11] HRNset1 HRNset2 Lena x,y,z0.00016 0.00002 0.00147 x0.00083 0.00075 0.00110 y0.00093 0.00010 0.00205 z0.00008 0.00314 0.00024 Fruits x,y,z0.00007 0.00066 0.00011 x0.00291 0.00245 0.00001 y0.00036 0.00207 0.00328 z0.00162 0.00307 0.00089 Baboon x,y,z0.00078 0.00102 0.00160 x0.00084 0.00069 0.00206 y0.00120 0.00002 0.00119 z0.00016 0.00414 0.00111 Figure 15 shows the histogram of the Lena image before and after encryption using HRNset1 from Table 4. The variance of the histograms for the three images (Lena, Fruits and Baboon) using HRNset1, is calculated according to [36], and the results are shown in Table 14. Table 14. Variance of the histograms for the Lena, Fruits and Baboon images using the sequences generated by HRNset1. Encryption Color Variance Lena Fruits Baboon Original Image HRNset1 Original Image HRNset1 Original Image HRNset1 x,y,z R 1017334.60 969.95 2,559,535.12 954.50 331,358.90 996.01 G 455,718.80 1002.58 1,131,488.32 1020.69 571,232.15 1051.16 B 1,377,355.80 995.09 338,280.76 904.91 319,770.47 1165.13 x R 1,017,334.60 969.95 2,559,535.12 954.50 331,358.90 996.01 G 455,718.80 1189.90 1,131,488.32 997.30 571,232.15 1143.03 B 1,377,355.80 1126.64 338,280.76 914.46 319,770.47 1071.51 y R 1,017,334.60 1005.61 2,559,535.12 943.03 331,358.90 1109.45 G 455,718.80 1002.58 1,131,488.32 1020.69 571,232.15 1051.16 B 1,377,355.80 1025.56 338,280.76 1025.20 319,770.47 940.07 z R 1,017,334.60 1053.77 2,559,535.12 1044.60 331,358.90 1175.52 G 455,718.80 1309.57 1,131,488.32 1154.54 571,232.15 1008.48 B 1,377,355.80 995.09 338,280.76 904.91 319,770.47 1165.13 Sensors 2020,20, 1326 19 of 22 Figure 15. Histograms of the Lena image encrypted using the set of values HRNset1. Original images on the left and the R, G and B encryption on the right. The entropy is evaluated by (11) using HRNset1, and using 8 bits ( N= 8), Table 15 shows the entropies for the three images (Lena, Fruits and Baboon). Table 15. Entropies of the original Lena, Fruits and Baboon images and the encrypted ones using the sets of values HRNset1 from Table 4. Encryption Color Entropy Lena Fruits Baboon Original Image HRNset1 Original Image HRNset1 Original Image HRNset1 x,y,z R 7.2531 7.9993 7.0556 7.9993 7.7067 7.9993 G 7.5940 7.9993 7.3527 7.9993 7.4744 7.9993 B 6.9684 7.9993 7.7134 7.9994 7.7522 7.9992 x R 7.2531 7.9993 7.0556 7.9993 7.7067 7.9993 G 7.5940 7.9992 7.3527 7.9993 7.4744 7.9992 B 6.9684 7.9992 7.7134 7.9994 7.7522 7.9993 y R 7.2531 7.9993 70556 7.9994 7.7067 7.9992 G 7.594 7.9993 7.3527 7.9993 7.4744 7.9993 B 6.9684 7.9993 7.7134 7.9993 7.7522 7.9994 z R 7.2531 7.9993 7.0556 7.9993 7.7067 7.9992 G 7.5940 7.9991 7.3527 7.9992 7.4744 7.9993 B 6.9684 7.9993 7.7134 7.9994 7.7522 7.9992 Sensors 2020,20, 1326 20 of 22 The key space is equal to 160 bits because each datum is encoded by 32 bits, the initial condition for each state variable counts, the step-size of the numerical method can change and is also encoded using 32 bits. The NPCR tests results using HRNset1 for the color images, were: 99.60365 when using the state variable x , 99.49608 when using the state variable y , and 98.49586 when using the state variable z. In both FPGA-based implementations for the Hopfield and Hindmarsh–Rose neural networks, the image is transmitted from a personal computer running MatLab to the FPGA using the serial port RS-232, as described in [31]. 7. Conclusions The use of two well-known neural networks, the Hopfield and Hindmarsh–Rose ones, for image encryption applications has been described. With the help of bifurcation diagrams, new feasible sets of values were proposed in order to generate binary strings with more randomness than the ones previously published in the literature. We proposed three sets of values for the Hopfield neuron and two sets of values for the Hindmarsh–Rose neuron. The chaotic time series were analyzed by TISEAN to compute Lyapunov exponents and Kaplan–Yorke dimension. The proposed sets of values were much better than the already published ones. By applying numerical methods, we showed the descriptions of the hardware design of both neurons and the FPGA resources were listed for the Hopfield and Hindmarsh–Rose neurons, respectively. The binary strings that were generated by the FPGA-based implementations of both neurons were taken as PRNGs to perform the encryption of the RGB Lena, Fruits and Baboon images. The success of the encryption system has been confirmed by the results obtained from correlation, histogram, variance, entropy, and NPCR tests. This demonstrates that both neurons are very useful for chaotic image encryption. Author Contributions: Conceptualization E.T.-C., R.L., W.D.L.-S., and F.V.F.; methodology E.T.-C., J.D.D.-M., A.M.G.-Z., O.G.-F.; software E.T.-C., J.D.D.-M., A.M.G.-Z., and O.G.-F.; validation E.T.-C., R.L., and I.C.-V.; formal analysis E.T.-C., J.D.D.-M., and A.M.G.-Z.; investigation E.T.-C., J.D.D.-M., and A.M.G.-Z.; resources J.D.D.-M., A.M.G.-Z., and O.G.-F.; writing—original draft preparation E.T.-C., J.D.D.-M., and A.M.G.-Z.; writing—review and editing E.T.-C., J.D.D.-M., A.M.G.-Z., R.L., O.G.-F., I.C.-V., W.D.L.-S., and F.V.F.; supervision E.T.-C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. Kumari, M.; Gupta, S.; Sardana, P. A Survey of Image Encryption Algorithms. 3D Res. 2017 ,8, 37. [CrossRef] 2. Hopfield, J.J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. 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