## 4 Algorithm Characterization
### Falsification Accuracy Analysis  
Due to being affected by the image processing algorithm, lens stains, image distortion, positioning algorithm, and other factors, the coordinates of sub-light source imaging in the image will deviate within a certain range. This deviation causes the distribution of the light source array to change, which could lead to an error in the chirality of the light source. As shown in Figure 3, the algorithm should calculate the chirality in the order of \"sub-light source 1, sub-light source 2, and sub-light source 3\". If the imaging coordinates of sub-light source 1 deviate, and its imaging coordinates are considered to be at light source 1a, the judgment algorithm calculates the chirality in the order of \"sub-light source 1a, sub-light source 2, and sub-light source 3\". In the case shown in Figure 3a, the algorithm will produce a result that is opposite to the real situation, resulting in a false judgment regarding the authenticity of the light source array. However, the chirality does not change in Figure 3b. Deviations in the imaging coordinates of the sub-light source can cause errors in the authentication algorithm. Therefore, it is necessary to estimate the accuracy of the algorithm.  
The focus and other information used in the experiment is derived from common mobile phone models on the market, including the OPPO Reno (ultra-wide angle focus 16 mm), OnePlus 9 Pro (focus 23 mm), Xiaomi Mi 10 Pro (focus 25 mm), Xiaomi Mi 9 (focus 26 mm), Huawei P50 Pro (focus 27 mm), and so on.  
When the light source emission sequence is obtained at the receiving end, there will be some errors due to noise. The bit error rate used in this paper is \(3.8\times 10^{-3}\), which is  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Coordinates of light source array (m) & (10, 10, 3), (10.1, 10.17, 3), (10.2, 10, 3) \\ Position range (m) & \(20\times 20\) \\ Receiver height (m) & 1 \\ Light emission sequence & (1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0) \\ \hline \end{tabular}
\end{table}
Table 2: The parameters applied to determine the correct rate of the delayed addition algorithm.  
Figure 3: Diagram of sub-light source imaging deviation.(**a**) The situation where the imaging deviation causes the wrong effect on the algorithm. (**b**) The situation where the imaging deviation does not affect the algorithm incorrectly.  
obtained via the hardware verification of CMOS image sensors. In an environment with mirror interference, the receiver position is randomly selected within a 20 m \(\times\) 20 m range. When the receiver points to the light source array, the relationship between the image coordinate deviation of the light source, the focal distance of the camera, and the accuracy of the authentication algorithm is as shown in Figure 4. The abscissa \(\sigma\) is the standard deviation of the coordinate distribution of the light source imaging, and the ordinate \(P_{e}\) is the probability of an error when the algorithm determines the chirality.  
As can be seen from Figure 4, the greater the imaging deviation, the greater the probability of the sub-light source imaging coordinate deviating from the real imaging coordinate, and the greater the probability of errors in identifying the real and pseudo-light source. It can also be seen that the larger the focal length, the lower the error probability of the pseudo-detection algorithm. This is mainly because the larger the focal length, the larger the image of the light source array on the imaging plane and the easier it is to identify the stripes photographed. Therefore, a lower error probability for the pseudo-detection algorithm is achieved.