Multi-variate side-channel attacks allow to break higher-order masking protections by combining several leakage samples.
But how to optimally extract all the information contained in all possible $d$-tuples of points?
In this article, we introduce preprocessing tools that answer this question.
We first show that maximizing the higher-order CPA coefficient is equivalent to finding the maximum of the covariance.
We apply this equivalence to the problem of trace dimensionality reduction by linear combination of its samples.
Then we establish the link between this problem and the Principal Component Analysis. In a second step we present the optimal solution for the problem of maximizing the covariance.
We also theoretically and empirically compare these methods.
We finally apply them on real measurements, publicly available under the DPA Contest v4, to evaluate how the proposed techniques improve the second-order CPA (2O-CPA).
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