# Multibyte Electromagnetic Analysis Based on Particle Swarm Optimization Algorithm
## 5 Experimental Results
This also can be deduced by  
Figure 4:0 Success rate comparison of single byte and multiple bytes, the red line is multiple bytes optimized by PSO, and the blue line is a single byte optimized by PSO. Figure 4:1 Success rate comparison of PSO-CEMA with or without mutation operation. Figure 4:2  
where SNR is signal-to-noise ratio, \(\rho\) is the estimated correlation coefficient. The detailed proof of Equation (15) can be seen in Appendix A. There is a positive correlation between \(SNR\) and \(|\rho|\). The larger the \(|\rho|\), the larger the \(SNR\). In other words, the more bytes we have, the larger the \(SNR\) will be. Thus, it is meaningful to use multiple bytes of the key. Figure 10 demonstrates the performance of different methods. The performance of PSO-CEMA is best and CEMA is much better than DEMA. For PSO-CEMA, the number of electromagnetic traces is approximately 4500 when the secret key is recovered correctly. With the same number of traces, CEMA's success rate is approximately 70%, which still needs more traces. When PSO-CEMA and CEMA recover the key correctly, the success rate of DEMA is very low. It still needs a large number of traces if we want to get the right key. The reason is that, in DEMA, only a single bit is used, the other 127 bits are regarded as noise. As a result, little useful information can be used in DEMA. Thus, in the latter, we didn't take DEMA into account because of its low efficiency. From an attacker aspect, it is better to recover the secret key with less electromagnetic traces. Thus, we used thousands of electromagnetic radiation traces to verify our PSO-CEMA method's efficiency. In order to highlight the advantage of PSO-CEMA over  
Figure 4:3 Success rate of different methods. Figure 4:4 The relationship between correlation and multiple key bytes. the existing CEMA method, we compared the total trace number required when the key was recovered correctly with a success rate of 100%. All electromagnetic traces are divided into different sets randomly, and we repeated our method on different sets of electromagnetic traces. The experimental results in Figure 11 show that our method has a better performance. CEMA needs 7286 \(\pm\) 1075 traces on average while our method needs 6286 \(\pm\) 859 traces on average, which achieves approximately 13.72% improvement for the corresponding case in CEMA. In order to ensure the reproducibility of our method, we applied our method to a public power traces dataset. The experimental results demonstrated a similar conclusion as the previous. Figure 12 shows the success rate of different methods. It can be seen that our method has a better performance than correlation power analysis (CPA). Our method achieves a 100% success rate with about 3500 electromagnetic traces while classical CPA requires about 4500 electromagnetic traces. We also repeated our method on different sets of power traces. All of these traces are also divided into different sets randomly, and we repeated the experiments on these different sets. The total trace number required when the key was recovered correctly is shown in Figure 4:5 The result showed that our method uses less electromagnetic traces than classical CPA. On average, CPA needs 3571 \(\pm\) 838 traces while our method only needs 2286 \(\pm\) 393 traces, which is an improvement of 35.98%. The extra experiments verified our method's efficiency and reproducibility. Figure 4:6 The minimum number of electromagnetic traces when the right key is recovered. Figure 4:7 Success rate of different methods.