# Single Trace Analysis against HyMES by Exploitation of Joint Distributions of Leakages
## 2 Related Works
### Code-Based Cryptography
#### 2.2.1 McEliece Cryptosystem
The power consumption model is set to \(\alpha=1,\ \beta=0\), and \(\sigma=0.1\), where \(\sigma\) is the noise standard deviation. Figure 1 shows the similarity between the theoretical distribution and the estimated distribution using HMD. Note that the smaller distance indicates high similarity between the two distributions. When \(g^{\prime}_{31}=1499\), the highest similarity was found at 0.0001016 and the second highest was found at 0.007898 when \(g^{\prime}_{31}=508\). This indicates that the difference between the distance of the correct key and the distance of the wrong key is not very large, so that when the noise value becomes large enough, the analysis failure rate rises. To solve this problem, two random variables \(C=mod(\log[\alpha_{i}]+\log[h_{i}[t-2]])\) and \(D=\exp[C]\) can be respectively added to \(A\) and \(B\) to form a joint distribution using four random variables in total. The use of four random variables magnifies the difference between the correct key and wrong key, which allows the correct key to be guessed with a higher success rate. Figure 2 shows the result of HMD between the theoretical joint distribution and the estimated distribution constructed by using four random variables. The experimental result shows that the distance at \(g^{\prime}_{31}=1499\) is 0.01776 and that the distance at \(g^{\prime}_{31}=927\) is 0.3911. We found that the difference between the distance to the correct guessing key and the wrong key is larger when using four random variables than when using two random variables. As mentioned earlier, many random variables elaborates the joint distributions of the leakages. Figure 3 shows the success rate of the attack according to the noise standard deviation. Recall that the attack is based on the joint distributions of leakages constructed by usi

\[h_{0}[t-2]=g_{t-1}+(\alpha_{0}\times h_{0}[t-1])\]0 Harmonic Mean Distance (HMD) using two random variables \(A,B\). \[h_{0}[t-2]=g_{t-1}+(\alpha_{0}\times h_{0}[t-1])\]1 Harmonic Mean Distance using four random variables \(A,B,C,D\). The success rate of the attack was measured by increasing the noise standard deviation from 0 to 2.5 in increments of 0.1. We analyzed 10,000 case according to the noise standard deviation. The experimental result using IPD shows that even if the noise standard deviation is zero, the success rate is still zero. The result when \(\chi^{2}\)PD is used shows that the success rate of the attack is 1 until the noise standard deviation reaches 0.3, and the success rate of the attack starts to decrease rapidly after 0.4. The result when HMD is used is the best among the results of the three methods. When the noise standard deviation is 0.4, the success rate of the attack begins to decrease to 0.9814. Figure 4 shows the experimental results using four random variables. The other experimental conditions are the same as those shown in \[h_{0}[t-2]=g_{t-1}+(\alpha_{0}\times h_{0}[t-1])\]2 The success rate of the attack according to the noise standard deviation is highest when HMD is used. The success rate of the attack remains at 1 up to the noise standard deviation of 0.6, and then starts to decrease from 0.7 to 0.9823. Figure 4 shows that using four random variables improves the performance of the attack. However, in practice, environments with a noise standard deviation of 0.6 are extremely rare and can be difficult to analyze. The success rate of the attack is not high even in an environment where the noise standard deviation is small because of the errors between the estimated Hamming weight and the actual value used in constructing the estimated joint distribution. Table 1 shows the success rate of the slice method for the data with a maximum Hamming weight of 11. The success rate was calculated based on the noise standard deviation for the case of \(\alpha=1\) and

\[h_{0}[t-2]=g_{t-1}+(\alpha_{0}\times h_{0}[t-1])\]3 Success rate according to noise standard deviation (using 4 random variables).