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Added info on correlation + attacking a subkey
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= Calculating the Pearson's Correlation Coefficient =Once we have a way to model our power consumption, we need a way to compare our power estimate to our measured traces. A helpful tool for finding this relationship is through Pearson's correlation coefficient, which is
<math> \rho_{X,Y}= \frac{\text{cov}(X, Y)}{\sigma_X \sigma_Y}= \frac{E[(X - \mu_X)(Y - \mu_Y)]}{\sqrt{E[(X - \mu_X)^2] E[(Y - \mu_Y)^2]}}</math> This correlation coefficient will always be in the range [-1, 1]. It describes how closely the random variables <math>X</math> and <math>Y</math> are related:* If <math>Y</math> always increases when <math>X</math> increases, it will be 1;* If <math>Y</math> always decreases when <math>X</math> increases, it will be -1;* If <math>Y</math> is totally independent of <math>X</math>, it will be 0. = Attacking with Correlation =After taking our measurements, we'll have <math>D</math> power traces <math>t</math>, and each of these traces will have <math>T</math> data points. Using subscript notation, <math>t_{d, j}</math> will refer to point <math>j</math> in trace <math>d</math> (<math>1 \le d \le D, 1 \le j \le T</math>). We'll also estimate the Subkeys power consumption in each trace using our model. We'll say that there are <math>I</math> different subkeys that we want to try. Then, <math>h_{d, i}</math> will refer to our power estimate in trace <math>d</math>, assuming that the subkey is <math>i</math> (<math>1 \le d \le D, 1 \le i \le I</math>). With this data, we can see how well our model and measurements match for each guess <math>i</math> and time <math>j</math>. We'll do this by finding how <math>t</math> and <math>h</math> correlate over the <math>D</math> traces. One way of calculating this is: <math>r_{i,j}=\frac{{\sum\nolimits_{d = 1}^D {\left[ {\left( {{h_{d,i}} - \overline {{h_i}} } \right)\left( {{t_{d,j}} - \overline {{t_j}} } \right)} \right]} }}{{\sqrt {\sum\nolimits_{d = 1}^D {{{\left( {{h_{d,i}} - \overline {{h_i}} } \right)}^2}} \sum\nolimits_{d = 1}^D {{{\left( {{t_{d,j}} - \overline {{t_j}} } \right)}^2}} } }}</math> There is an alternative form of the correlation equation that we can use for ''online'' calculations - it allows us to add one trace at a time without re-summing all of the past data. This form is: <math>r_{i,j} = \frac{D \sum_{d=1}^D h_{d,i}t_{d,j} - \sum_{d=1}^D h_{d,i} \sum_{d=1}^D t_{d,j}}{\sqrt{\Big(\big(\sum_{d=1}^D h_{d,i}\big)^2 - D\sum_{d=1}^D h_{d,i}^2\Big)\Big(\big(\sum_{d=1}^D t_{d,j}\big)^2 - D\sum_{d=1}^D t_{d,j}^2\Big)}}</math> Note that these two sums are equivalent. = Picking a Subkey =The last step is to use the values of <math>r_{i,j}</math> to decide which subkey matches our traces most closely. There are two steps to this:* For each subkey <math>i</math>, find the highest value of <math>|r_{i,j}|</math>. This will discard the time information - we want to know how good our guess was, but we don't care where our guess matched the trace. * Looking at the maximum values for each subkey, find the highest value of <math>|r_i|</math>. The location <math>i</math> of this maximum is our best guess: it correlated more closely with the traces than any other guess.Note that we're only working with absolute values here because we don't care about the sign of the relationship. All we need to know is that a linear correlation exists.
= Example: AES-128 =
