Cryptographic Boolean functions must be balanced (i.e., their output must be uniformly distributed) for avoiding statistical dependence between their input and their output (such statistical dependence can be used in attacks).
Moreover, any combining function \(f (x)\) (see combination generator), used for generating the pseudorandom sequence in a stream cipher, must stay balanced if we keep constant some coordinates \(x_i\) of x (at most m of them, where m is as large as possible). We say that f is then m-resilient. More generally, a (non necessarily balanced) Boolean function, whose output distribution probability is unaltered when any m of its input bits are kept constant, is called mth order correlation-immune. The notion of correlation-immune function is related to the notion of orthogonal array (see [1]). Only resilient functions are of practical interest as cryptographic functions.
The notion of correlation immunity was introduced by Siegenthaler in [5]; it is related to an...
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Carlet, C. (2005). Correlation Immune and Resilient Boolean Functions. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_81
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