# Chaos-Based Physical Unclonable Functions
## 3 Behavioral Modeling  
The chaotic circuit described with a 1D map must have a region of operation, in which the inclination coefficient of subsequent values of the state variable \((\delta)\) is \(1<k<2\), where \(\delta[m+1]=k\delta[m]+q\) and \(k\), \(q\) are the constants which depend on the circuit parameters. The circuit shown in Figure 1 reveals a chaotic behavior due to the presence of \(C\) and \(C^{\prime}\) regions with \(k>1\), as shown in Figure 3. These regions of operation result from the delay in the feedback loop (FB), which causes the logical state inconsistency in \(C\) and \(C^{\prime}\) ranges of operation. In order to extensively verify the chaotic behavior of the system proposed in Figure 1, we have implemented its behavioral model in a Matlab Simulink environment (Matlab R2017a, The MathWorks, Inc., Natick, MA, USA). In the model, we have assumed linear and lumped output inverter resistance \(R_{o}\) and input capacitance \(C_{o}\) (of adjacent inverter attached in the ring), as shown in Figure 5.  
These two lumped components are responsible for the finite rising/falling edges of signals. Subsequently, we have assumed the sigmoid transfer function (hyperbolic tangent) of direct current inverter characteristics, constant signal delay \(t_{pd}\), and a Gaussian noise process \(N(t)\) affecting the threshold level of the inverter (its transfer function). The \(N(t)\) process of our model represents the phase walk of SCROs in their physical implementation, and allowed us to obtain non-deterministic circuit behavior, available in the physical implementations either in FPGA or CPLD. All the parameters \(R_{o}\), \(C_{o}\), \(t_{pd}\), \(N(t)\) (standard deviation) and inclination of the transfer function have been adjusted to obtain identical operation of SCROs in the Simulink model and physical implementation. The detailed method of parameter extraction is explained in [48]. This way, we have obtained identical phase-walk (and jitter), \(f_{1}\) and \(f_{2}\) frequencies of the model and the circuit physically implemented in a programmable device. With the use of the Simulink, we simulated the chaotic behavior of the circuit proposed in Figure 1. For this reason, we performed multiple transient analysis, in which the DLa1, DLb1, DLa2, DLb2, FB parameters (i.e., \(t_{pd}\) and \(R_{o}\), \(C_{o}\)) were subjected to 1% dispersion (Monte Carlo analysis). During the analysis, the series of \(\delta[m]\) values were registered for subsequent \(K\) sets of randomly generated DLa1, DLb1, DLa2, DLb2, FB parameters. Each transient analysis for a particular parameter set was repeated \(N\) times (realizations) in order to observe the non-deterministic circuit behavior of a particular parameter set. The experiment scheme can be described according to formula (3):  
Figure 5: Model of an inverter. \[\delta_{0}[N,M]=\left\{ \begin{array}{llll}\delta_{k=0,n=0}[0],&\delta_{k=0,n=0}[1 ],&\delta_{k=0,n=0}[2],&\cdots&\delta_{k=0,n=0}[M]\\ \delta_{k=0,n=1}[0],&\delta_{k=0,n=1}[1],&\delta_{k=0,n=1}[2],&\cdots&\delta_{ k=0,n=1}[M]\\ \delta_{k=0,n=2}[0],&\delta_{k=0,n=2}[1],&\delta_{k=0,n=2}[2],&\cdots&\delta_{ k=0,n=2}[M]\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \delta_{k=0,n=N}[0],&\delta_{k=0,n=N}[1],&\delta_{k=0,n=N}[2],&\cdots&\delta_ {k=0,n=N}[M],\\ \end{array} \right. \tag{3}\]  
\[\delta_{1}[N,M]=\left\{\begin{array}{llll}\delta_{k=1,n=0}[0],&\delta_{k=1,n =0}[1],&\delta_{k=1,n=0}[2],&\cdots&\delta_{k=1,n=0}[M]\\ \delta_{k=1,n=1}[0],&\delta_{k=1,n=1}[1],&\delta_{k=1,n=1}[2],&\cdots&\delta_ {k=1,n=1}[M]\\ \delta_{k=1,n=2}[0],&\delta_{k=1,n=2}[1],&\delta_{k=1,n=2}[2],&\cdots&\delta_ {k=1,n=2}[M]\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \delta_{k=1,n=N}[0],&\delta_{k=1,n=N}[1],&\delta_{k=1,n=N}[2],&\cdots&\delta_ {k=1,n=N}[M],\\ \vdots&\vdots&\ddots&\vdots\\ \delta_{K}[N,M]=\left\{