## 6 Start-Up Value Probability vs. SD Relation  
The aim of this section is to find a relation between the simulation values of cell imbalance (\(SD\) parameter described in Section 5) and the start-up probabilities based on experimental measurements (\(SUP_{1}\) parameter described in Section 4). This will allow determining the threshold value of the \(SD\) parameter for a cell to start up to a fixed value with a probability higher than the desired target value. This threshold value can be very useful information at the early design stages, since \(SD\) can be easily obtained by simulation, and because it allows determining the percentage of cells that will be suitable to be used in the target PUF application of the design. In Section 4, we showed experimentally that most cells (about 92%) started up to a given value with a probability greater than 91%. This means that not only cells with a high \(SD\) absolute value, but also cells with a moderate \(SD\) parameter will start up to a given logic state with relatively high probability. It is important to note that the cell distributions we are dealing with are probability distributions. This means that it is possible that a single cell with very high \(SD\) absolute value does not always start up to the same value. However, cells belonging to this subset of cells are the ones that are more probable to perform reliably regarding its start-up value. The correspondence between the probability distribution of the experimental start-up values (\(SUP_{1}\)) in Figure 3a and the probability distribution of the \(SD\) parameter in Figure 5 leads to a series of data-points, which must be fitted with an analytical function. The input of this function is \(SD\) and it provides the \(SUP_{1}\) as the output. This function needs to be chosen and meet the following requirements:  
1. \(SD\) = 0 must lead to \(SUP_{1}\) = 0.5 (start-up at logic '1' probability equal to 0.5). 2. \(SD\) \(\gg\) 0 V must lead to \(SUP_{1}\) =  
Figure 6: Relative number of bit-cells (cumulative frequency on the vertical axis) that present a value of \(\mid\)SD \(\mid\) equal or greater than SD\({}_{\text{th}}\) (horizontal axis). \(SD\ll 0\) V must lead to \(SUP_{1}=0\), and thus \(SUP_{0}=1\)  
These requirements are in principle satisfied by the logistic function, a common S-shape curve (sigmoid curve) that has been extensively used to model research problems in many fields [36]. Its expression in terms of \(SUP_{1}\) and \(SD\) is given by:  
\[SUP_{1}=\frac{a}{1+e^{-k\ SD}} \tag{3}\]  
where \(a\) is the curve maximum value when \(SD\gg 0\) V, and \(k\) is the curve steepness parameter. So, in our case, due to the second requirement, \(a=1\), and \(k\) must be determined by fitting with the experimental \(SUP_{1}\) data. A logistic curve with \(a=1\) is shown in Figure 7a. The fitting process tries to fit the simulated \(SUP_{1}\) distribution, obtained from Equation (3) when using the \(SD\) values resulting from the Monte Carlo analysis, with the experimental \(SUP_{1}\) distribution. When the fitting was performed, the following result was obtained:  
\[k=195.4\ V^{-1} \tag{4}\]  
The sigmoid curve in Figure 7a plots Equation (3) with \(a=1\) and this fitted parameter value. Figure 8a shows the relative frequency of cells as a function of \(SUP_{1}\) for both the experimental data and the simulated data once transformed into start-up probabilities using Equation (3) and fitted. As can be seen, the matching between the two sets of data is not good enough. This is because the experimental data (Figure 3a) show an abrupt transition around 0.09 and 0.91 \(SUP_{1}\) values, which a function with only one fitting parameter is not capable of correctly modeling. Figure 7: (a) Plot of the sigmoid function defined in Equation (3) after a fitting process (a = 1; k = 195.4 V\({}^{-1}\)); (b) Sigmoid function that meets the requirements of the proposed model.