To summarize, orthogonal transformation regression solves the problem that the original linear regression singular matrix cannot be inversed. As shown in Figure 8d, the minimum correlation coefficient between the vectors of the matrix X T X in our proposed orthogonal transformation regression is close to 0, indicating that the matrix X T X does not have collinearity. As shown in Figure 8c, when matrix X T X is a singular matrix, although ridge regression solves the problem of accurate correlation to a certain extent, there is still a strong correlation between vector 2 and vector 1, and the correlation coefficient reaches 0.99999984. As shown in Figure 8b, when the matrix X T X is a singular matrix, vector 2 of linear regression is completely related to vector 1. As shown in Figure 8a, when the matrix X T X is a nonsingular matrix, each vector is only completely related to its own vector. The correlation coefficient matrix is shown in Figure 8, and yellow indicates complete correlation. In Figure 8, the correlation coefficients between vectors of a matrix X T X of linear regression, and orthogonal transformation regression as well as matrix X T X + λ I of ridge regression are obtained, respectively. Since the orthogonal transformation nonlinear regression projects the original data of the characteristic matrix X from the original space to the new space, the results in Figure 7d show that there are no eigenvectors with strong autocorrelation in the new space, and all eigenvectors have autocorrelation estimation R( k) closer to 0, showing good randomness. The results in Figure 7b,c show that there is a group of eigenvectors with strong autocorrelation in both linear regression and ridge regression, and the autocorrelation estimation R( k) of other eigenvectors is concentrated at about 0.3. Figure 7b–d shows the autocorrelation test results of characteristic matrix X when the matrix X T X is a singular matrix. It can be seen that the autocorrelation of the eight groups of eigenvectors is very weak, in micro correlation. Figure 7a shows the autocorrelation test results of characteristic matrix X when the matrix X T X is a nonsingular matrix. Under the condition of high noise and high order template analysis, the established template has good universality.īased on the above theoretical analysis, the eigenvectors of the three methods can be tested by autocorrelation. The experimental results show that the second-order template analysis based on orthogonal transformation nonlinear regression can complete key recovery without sacrificing the performance of regression estimation. In order to verify the data fitting effect of the constructed template, a comparative experiment of template analysis based on regression, Gaussian, and clustering was carried out on SAKURA-G. The irreversibility of a singular matrix and the inaccuracy of the model are solved by orthogonal transformation and adding a negative direction to the calculation of the regression coefficient matrix. Therefore, this paper proposes a second-order template analysis method based on orthogonal transformation nonlinear regression. However, linear regression may have the problem of irreversibility of a singular matrix in the modeling stage of template analysis and the problem of poor data fit in the template analysis after the cryptographic algorithm is masked. As one of many leakage models, the XOR operation leakage proposed by linear regression has typical representative significance in side-channel analysis. SCA decrypts the key information in the encryption device by establishing an appropriate leakage model. In recent years, side-channel analysis technology has been one of the greatest threats to information security.
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