Statistical and Linear Independence of Binary Random Variables

Linear cryptanalysis makes use of statistical models that consider linear approximations over block cipher and random permutation as binary random variables. In this note we develop conditions under which linear independence of binary random variables and statistical independence of their correlations are equivalent properties. As an application we obtain that the correlations of linear combinations of the components of a random n-bit to m-bit transformation are statistically independent if and only if these linear combinations are linearly independent.