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1.2. Families of Boolean functions with good cryptographic properties 17This generalized conjecture obviously includes all the previous ones.The construction of their functions is as follows.Definition 1.2.11 (Construction{of Jin et al. [143]). } Let n = 2k ≥ 4 be an even integer, αa primitive element of F 2 n, A = 1, α, . . . , α 2k−1 −1and g : F 2 k → F 2 Boolean function in kvariables 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( )f(x, y) = g xy 2k −1−u.They proved that such a function f is1. of algebraic degree between n/2 and n − 2 depending on the value of u,2. of optimal algebraic immunity n/2 if Conjecture 1.2.10 is true,3. of nonlinearity at least2 n−1 − 2 π ln 4(2n/2 − 1)2 n/2 − 1 ≈ 2 n−1 − ln 2ππ n2n/2 .The proof of the optimality of the algebraic immunity is once again similar to the previous ones.It should be noted that resistance to fast algebraic attacks is not studied by Jin et al. [143].Modifying these functions as before, Jin et al. obtained balanced functions with high algebraicdegree and nonlinearity. They proved that for n = 2k ≥ 4, these modified functions are1. balanced,2. of optimal algebraic degree n − 1,3. of optimal algebraic immunity n/2 if Conjecture 1.2.10 is true,4. of nonlinearity at least2 n−1 − 2 π ln 4(2n/2 − 1)π2 n/2 − 2 π ln 4(2n/2 − 1)2 n/4 − 2 ≈ 2 n−1 − ln 2ππ n2n/2 − ln 2π n2n/4 .Jin et al. [142] applied a similar generalization to the 1-resilient Boolean function of Tu andDeng [262] and obtained a family of functions which are1. 1-resilient,2. of optimal algebraic degree n − 2,3. of optimal algebraic immunity n/2 up to Conjecture 1.2.10 and an additional assumption,4. of nonlinearity at least2 n−1 − 2 π ln 4(2n/2 − 1)π≈ 2 n−1 − ln 2π (n + 1)2n/2 − 2 ln 2π n2n/4 .2 n/2 − 2 n/2−1 − 4 π ln 4(2n/2 − 1)2 n/4 − 3πFinally, for specific values of the parameter u — in particular for the family of Tang, Carlet andTang [259] presented in Subsection 1.2.4 —, the conjecture, and so the optimality of the algebraicdegree, can be proved using the results of the next chapter.