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1.1. Cryptographic criteria for Boolean functions 9⊕ ⊕ ⊕s i+L−1 · · · s i+1 s ix (t)1x (t)n−1x (t)nfk (t)Figure 1.1: The filter modelLFSR 1LFSR 2.LFSR nx (t)1x (t)2x (t)nfk (t)c (t)⊕m (t)Figure 1.2: The combiner modelDefinition 1.1.1 (Support). Let f : F n 2 → F 2 be a Boolean function in n variables. The supportof f, denoted by supp(f), is the set of x ∈ F n 2 such that f(x) = 1:supp(f) = {x ∈ F n 2 | f(x) = 1} .The support of a Boolean function provides a complete characterization of it. Most familiespresented in Section 1.2 will be given in this way.Definition 1.1.2 (Hamming weight). Let f : F n 2 → F 2 be a Boolean function in n variables.The Hamming weight of f, denoted by w H (f), is the cardinality of its support (or the Hammingweight of its value vector).We can now define a first criterion of interest for cryptographic Boolean functions.Definition 1.1.3 (Balancedness [38, 4.1.3]). A Boolean function f : F n 2 → F 2 in n variables issaid to be balanced if it has Hamming weight 2 n−1 .Hence, a Boolean function is said to be balanced if it takes as often the values 0 and 1.Balancedness is needed to avoid statistical dependence between the input and the output of thestream cipher and to prevent distinguishing attacks [38, 4.1.3].The notion of balancedness can be generalized as follows.Definition 1.1.4 (Resiliency [243], [38, 4.1.3]). A Boolean function f : F n 2 → F 2 in n variablesis said to be m-resilient if any of its restrictions with at most m-input fixed is balanced.