What you are describing is data (A/B, 1⁄2) such that parts of the data are independent from the secret X/Y, but the whole data is not independent from the secret. That’s an issue that is sort of unusual for any statistical approach, because it should be clear that only the whole leaked data should be considered.
The problem with Pearson correlation criterion is that it does not measure independence at all (even for parts of the data), but measures correlation which is just a single statistic of the two variables. It’s as if you compared two distributions by comparing their means.
Let’s say leaked data is X = −2, −1, 1, 2 equiprobably, and secret data is Y = X^2. Zero correlation just implies E(XY) - E(X)E(Y) = 0, which is the case, but it is clear that one can fully restore the secret from the leaked, they are not independent at all.
Can you give an example to explain? It’s the best example I could give based on the description in the OP.
What you are describing is data (A/B, 1⁄2) such that parts of the data are independent from the secret X/Y, but the whole data is not independent from the secret. That’s an issue that is sort of unusual for any statistical approach, because it should be clear that only the whole leaked data should be considered.
The problem with Pearson correlation criterion is that it does not measure independence at all (even for parts of the data), but measures correlation which is just a single statistic of the two variables. It’s as if you compared two distributions by comparing their means.
Let’s say leaked data is X = −2, −1, 1, 2 equiprobably, and secret data is Y = X^2. Zero correlation just implies E(XY) - E(X)E(Y) = 0, which is the case, but it is clear that one can fully restore the secret from the leaked, they are not independent at all.
See more at https://en.wikipedia.org/wiki/Correlation_and_dependence#Correlation_and_independence