We find different equations to compute the correlation or correlation coefficient:

$\begin{align*} corr &= \frac{a+d}{a+b+c+d}\\ &= \frac{(a+d) - (b+d)}{a+b+c+d}\\ &= \frac{(a+d) - (b+d)}{\sqrt{(a+b)*(c+d)*(a+c)*(b+d)}} \end{align*}$

where a(0,0), b(1,0), c(0,1), d(1,1), which formula give the correct result?

  • $\begingroup$ Would you mind to use "math tags" so that the formulae are showed correctly? $\endgroup$ – Hilder Vítor Lima Pereira Sep 7 '18 at 9:17
  • $\begingroup$ I can actually answer this one, but it will not be for a day or 2 $\endgroup$ – Richie Frame Sep 7 '18 at 19:33
  • $\begingroup$ @Richie Frame , give simple example or a useful reference if you are busy $\endgroup$ – bassam Sep 7 '18 at 20:30
  • $\begingroup$ I've formatted your correlation equations to be easier to read. Please make changes as needed. $\endgroup$ – puzzlepalace Sep 7 '18 at 21:48
  • $\begingroup$ @bassam I can tell you the answer is substantially more complex than any examples you had above, but I have written an algorithm to calculate it, so I know how it works $\endgroup$ – Richie Frame Sep 7 '18 at 23:46

The math

The correlation coefficient between an array of boolean vectors is a value between 0 and 1 defined by the following equations, covariance and standard deviation:

$\operatorname{cov} (X,Y)=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{X})(y_i-\overline{Y})$

$\sigma_X=\sqrt{\frac{1}{n}\sum_{i=1}^n (x_i-\overline{X})^2}$

$\sigma_Y=\sqrt{\frac{1}{n}\sum_{i=1}^n (y_i-\overline{Y})^2}$

Where $\overline{X}$ is the average value of a set, and $X$ is the set of $x$ values, which would be the first array in your variables [0,1,0,1] , and $Y$ is the set of $y$ values, the second array [0,0,1,1]. Note the array indexing is 1-4 for the equations, but will not be so in computer code.

The correlation coefficient is the final equation:

$\rho_{X,Y}= \left|\frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}\right|$

$\rho_{X,Y}= \left|\frac{\sum_{i=1}^n (x_i-\overline{X})(y_i-\overline{Y})}{\sqrt{\sum_{i=1}^n (x_i-\overline{X})^2} \ \cdot \ \sqrt{\sum_{i=1}^n (y_i-\overline{Y})^2}}\right|$

In the scenario where $\sigma_X$ and $\sigma_Y$ are both 0, the vectors are most likely completely correlated, and a different comparison must be made. If one of them is 0, the results are probably not correlated. You will probably not run into these situations if you are analyzing an s-box for bit independence.

Your example

Because your dataset is small, it is easy to use as an example, but is still a bad example, because the correlation coefficient is 0:

The standard dev of your X and Y sets is both 0.5, but the covariance is 0:

$0 = \left|\frac{0}{0.5 * 0.5}\right|$

Now, if you change your X,Y to [0,1,0,1],[1,0,1,0], you can see that they are completely correlated [y = not(x)], and the math agrees, as the covariance calculation is -0.25:

$1 = \left|\frac{-0.25}{0.5 * 0.5}\right|$

Optimization and code

You may notice that standard deviation looks like the square root of covariance between the same set, and for this calculation it is. You may also notice you cancel out the 1/N in the final equation. The intermediate calculations of average values and subtractions are also the same, so you can combine them all into a single calculation. Careful optimization will also help compensate for any floating point rounding errors.

Optimized pseudocode for the final equation on arrays of X and Y would look something like this:

mean_X = ArithMean(X)
mean_Y = ArithMean(Y)

For i = 1 to N
    val_x = X(i) - mean_X
    val_y = Y(i) - mean_Y
    deltasum_c = deltasum_c + (val_x * val_y)
    deltasum_x = deltasum_x + (val_x * val_x)
    deltasum_y = deltasum_y + (val_y * val_y)
Next i

CoVariance = deltasum_c
StdDeviation_xy = Sqrt(deltasum_x * deltasum_y)

If StdDeviation_xy <> 0 Then COR = Abs(CoVariance / StdDeviation_xy)
Elseif deltasum_x = deltasum_y Then COR = 1
Else COR = 0
  • $\begingroup$ Your covariance and standard deviation formulae assume equiprobability of the bitstrings, which may not be true in general (and is not specified in the question). $\endgroup$ – Ginswich Sep 9 '18 at 11:59
  • $\begingroup$ @Ginswich X and Y will be the same length as per the questions intent, which is performing a bit independence analysis on a block cipher s-box $\endgroup$ – Richie Frame Sep 10 '18 at 1:21
  • $\begingroup$ If that is the case, then the question should be updated consequently to clearly state that: IMO, a tag mentioning "s-boxes" is not clear enough. I agree that in that case, it's more a problem of the question rather than of your answer (although making some mention about it in the answer wouldn't hurt, though). $\endgroup$ – Ginswich Sep 10 '18 at 6:47
  • $\begingroup$ @Ginswich It was fairly clear to me due to the tags on the question, and because the examples were given in bit pairs. Also the other question from same author is about how to compute bit independence $\endgroup$ – Richie Frame Sep 10 '18 at 15:48
  • 1
    $\begingroup$ @PaulUszak yes, the absolute value is there so you can perform standard is (a>b) comparisons, as -0.5 is a greater correlation than +0.4 $\endgroup$ – Richie Frame Oct 11 '18 at 23:48

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