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1.2. Families of Boolean functions with good cryptographic properties 13Definition 1.2.3 (Construction of Dillon [70]). Let n = 2k ≥ 4 be an even integer and g : F 2 k →F 2 a balanced Boolean function in k variables. Let f : F 2 k × F 2 k → F 2 be the Boolean functiondefined by( ) xf(x, y) = g ,ywhere x/y is understood as xy 2k −2 , so equal to 0 when y = 0.These functions form the so-called Partial Spread class PS ap [70]. In particular, all functionsin this class are bent [70] and have algebraic degree n/2 = k [222].Definition 1.2.4 (First construction { of Tu and Deng }[264]). Let n = 2k ≥ 4 be an even integer,α a primitive element of F 2 n, A = 1, α, . . . , α 2k−1 −1and g : F 2 k → F 2 a Boolean function ink variables defined bysupp(g) ={α }s , α s+1 , . . . , α s+2k−1 −1= α s A ,for any 0 ≤ s ≤ 2 k − 2. Let f : F 2 k × F 2 k → F 2 be the Boolean function in n variables defined by{ ( )xgf(x, y) = yif x ≠ 0 ,0 otherwise .They proved that these functions are1. bent (because they belong to PS ap ),2. of algebraic degree n/2 = k [222],3. and of optimal algebraic immunity n/2 = k if Conjecture 1.2.2 is verified.The approach of Tu and Deng to prove the optimal algebraic immunity was to identify annihilatorsof the Boolean function with codewords of BCH codes [185, 186, 272]. The role of the conjectureis then to deduce from the BCH bound [185, 186, 272] that those codewords are equal to zero ifthe algebraic degrees of the corresponding annihilators are less than n/2 = k.These functions can then be modified to give rise to functions with different good cryptographicproperties as follows.Definition 1.2.5 (Second construction { of Tu and Deng } [264]). Let n = 2k ≥ 4 be an even integer,α a primitive element of F 2 n, A = 1, α, . . . , α 2k−1 −1and g : F 2 k → F 2 a Boolean function ink variables defined bysupp(g) = α s A ,for any 0 ≤ s ≤ 2 k − 2. Let f : F 2 k × F 2 k → F 2 be the Boolean function in n variables defined by⎧ ( )x ⎪⎨ gyif xy ≠ 0 ,f(x, y) = 1 if x = 0 and y ∈ (αA) ⎪⎩−1 ,0 otherwise .