Geodesic
Statistical independence of the Boolean function superposition.~II
Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 14-15.

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Let $x,y,z$ be sets of different Boolean variables, $f(x,y)$, $f_1(x,y)$, $f_2(x,y)$, $f_1(x,y)\oplus f_2(x,y)$ are Boolean functions being statistically independent on the variables in $x$, and $h(x_1,x_2,z)$, $g(x)$ are any Boolean functions. Then the function $h(f_1(x,y),f_2(x,y),z)$ is statistically independent on the variables in $x$; and the same is true for the function $f(x,y)\oplus g(x)$ iff $f$ is balanced or $g=\mathrm{const}$. 
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O. L. Kolcheva; I. A. Pankratova. Statistical independence of the Boolean function superposition.~II. Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 14-15. http://geodesic.mathdoc.fr/item/PDMA_2012_5_a5/

[1] Agibalov G. P., Pankratova I. A., “Elementy teorii statisticheskikh analogov diskretnykh funktsii s primeneniem v kriptoanalize iterativnykh blochnykh shifrov”, Prikladnaya diskretnaya matematika, 2010, no. 3(9), 51–68

[2] Kolcheva O. L., Pankratova I. A., “O statisticheskoi nezavisimosti superpozitsii bulevykh funktsii”, Prikladnaya diskretnaya matematika. Prilozhenie, 2011, no. 4, 11–12