== Analyzing the Data ==
How Suppose that we've picked <math>I</math> points of interest, which are at samples <math>s_i</math> (<math>0 \le i < I</math>). Then, our goal is to find a mean and covariance matrix for every operation (every choice of subkey or intermediate Hamming weight). Let's say that there are <math>K</math> of these operations (maybe 256 subkeys or 17 possible Hamming weights). For now, we'll look at a single operation <math>k</math> (<math>0 \le k < K</math>). The steps are:* Find every power trace <math>t</math> that falls under the category of "operation <math>k</math>". (ex: find every power trace where we used a subkey of 0x01.) We'll say that there are <math>T_k</math> of these, so <math>t_{j, s_i}</math> means the value at trace <math>j</math> and POI <math>i</math>.* Find the average power <math>\mu_i</math> at every point of interest. This calculation will look like: <math>\mu_i = \frac{1}{T_k} \sum_{j=1}^{T_k} t_{j, s_i}</math> * Find the variance <math>v_i</math> of the power at each point of interest. One way of calculating this is: <math>v_i = \frac{1}{T_k} \sum_{j=1}^{T_k} (t_{j, s_i} - \mu_i)^2</math> * Find the covariance <math>c_{i, i^*}</math> between the power at every pair of POIs (<math>i</math> and <math>i^*</math>). One way of calculating this is: <math>c_{i, i^*} = \frac{1}{T_k} \sum_{j=1}^{T_k} (t_{j, s_i} - \mu_i) (t_{j, s_{i^*}} - \mu_{i^*})</math> * Put together the mean and covariance matricesas: <math>\mathbf{\mu} = \begin{bmatrix}\mu_1 \\\mu_2 \\\mu_3 \\\vdots\end{bmatrix}</math> <math>\mathbf{\Sigma} = \begin{bmatrix}v_1 & c_{1,2} & c_{1,3} & \dots \\c_{2,1} & v_2 & c_{2,3} & \dots \\c_{3,1} & c_{3,2} & v_3 & \dots \\\vdots & \vdots & \vdots & \ddots \end{bmatrix}</math> These steps must be done for every operation <math>k</math>. At the end of this preprocessing, we'll have <math>K</math> mean and covariance matrices, modelling each of the <math>K</math> different operations that the target can do.
= Using the Template =
