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4.1. Hyper-bentness characterizations 111to cite only a few of them.Hence, it has been shown that the zeros of binary Kloosterman sums lead to several familiesof bent, hyper-bent and semi-bent functions. We summarize the known results in Table 4.1 withthe following conventions:• A class of functions is given in terms of a ∈ F ∗ 2 m, b ∈ F∗ 4, c ∈ F 2 n \ F 2 m and r co-prime to2 m + 1; remember that a ∈ F ∗ 2m, but that the corresponding Boolean functions have n = 2minputs.• Unless stated otherwise, the given conditions on a are necessary and sufficient for theBoolean functions to verify the given property.Table 4.1: Families of hyper-bent and semi-bent functions for K m (a) = 0( ) Class of functions Property Conditions ReferencesTr n 1 ax r(2 m −1) hyper-bent K m(a) = 0 [70, 157, 167, 46]( ) )Tr n 1 ax r(2 m −1) + Tr n 1(cx (2m −1) 1 2 +1 semi-bent K m(a) = 0 [198]( ) )Tr n 1 ax r(2 m −1) + Tr n 1(cx (2m −1) 1 2 +1 semi-bent K m(a) = 0 [198])+ Tr n 1(x (2m −1) 1 4 +1 ;Tr n m( (c) = 1, m odd ) )Tr n 1 ax r(2 m −1) + Tr n 1(cx (2m −1) 1 2 +1( ) semi-bent K m(a) = 0 [198]+ Tr n 1 x (2 m −1)3+1 ;Tr n m( (c) = 1 ) )Tr n 1 ax r(2 m −1) + Tr n 1(cx (2m −1) 1 2 +1 semi-bent K m(a) = 0 [198])+ Tr n 1(x (2m −1) 1 6 +1 ;Tr n m ( (c) = 1, m even ) ( )Tr n 1 ax r(2 m −1) + Tr n 1 αx 2 m +1 semi-bent K( ∑2)m(a) = 0 [198]+ Tr n ν−1 −11x (2m −1) i2i=1ν +1;gcd(ν, m) = 1, α ∈ F 2 n, Tr n m (α) = 1Quite surprisingly, all the aforementioned characterizations involve zeros of Kloosterman sumsand it is only in 2009 that Mesnager [195] has shown that another value of such sums — the value4 — also gives rise to bent, hyper-bent and semi-bent functions. We will call this criterion the firstMesnager criterion. Afterwards, other families 2 of (hyper-)bent functions and semi-bent functionswere as well described by Mesnager [197, 198]. The known results about (hyper-)bent functionsare summarized in Table 4.2, those about semi-bent functions in Table 4.3. The conventions arethe same as for Table 4.1.Such characterizations are obviously much more pleasant than the original definition involvingthe Walsh–Hadamard transform. This is also a first step towards an efficient way to explicitlybuild (hyper-)bent functions.4.1.2 Functions with multiple trace termsIn fact, Charpin and Gong devised in the same article [46] a quite more general characterizationof hyper-bentness for a large class of Boolean functions with multiple trace terms. In particular,2 The remark of Footnote 1 applies here as well