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116 Chapter 4. Efficient characterizations for bentnessProposition 4.2.5. Let f : F 2 m → F 2 m be a function, g = Tr m 1 (f) be its composition with theabsolute trace and H f be the (affine) curve defined over F 2 m byH f : y 2 + xy = x + x 2 f(x) ,Then∑x∈F ∗ 2 m χ ( Tr m 1(x−1 ) + g(x) ) = −2 m + #H f .Proof. The proof is similar as that of Theorem 4.2.1 and is summarized by the following equalities:∑χ ( Tr m (1 x−1 ) + g(x) ) = ∑(1 − 2(Tr m (1 x−1 ) + g(x)))x∈F ∗ 2 m x∈F ∗ 2 m= 2 m − 1 − 2# { x ∈ F ∗ 2 | ( m Trm 1 x−1 ) + g(x) = 1 }= −2 m + 1 + 2# { x ∈ F ∗ 2 | ( m Trm 1 x−1 ) + g(x) = 0 }= −2 m + 1 + 2# { x ∈ F ∗ 2 | ∃t ∈ F m 2 m, t2 + t = x −1 + f(x) }= −2 m + 1+ 2# { x ∈ F ∗ 2 m | ∃t ∈ F 2 m, (t/x)2 + (t/x) = x −1 + f(x) }= −2 m + 1 + 2# { x ∈ F ∗ 2 m | ∃t ∈ F 2 m, t2 + xt = x + x 2 f(x) }= −2 m + 1 + #H f − # {P ∈ H f | x = 0}= −2 m + #H f .We can now easily deduce the reformulation of the Charpin–Gong criterion given by Lisoněk.Theorem 4.2.6 (Reformulation of the Charpin–Gong criterion [182, 181]). The notation is asin Theorem 4.1.1. Moreover, let H a and G a be the (affine) curves defined over F 2 m byG a : y 2 + y = ∑ r∈Ra r D r (x) ,H a : y 2 + xy = x + x 2 ∑ r∈Ra r D r (x) .Then f a is hyper-bent if and only if#H a − #G a = −1 .Proof. According to Proposition 4.2.5, the left hand side of the Charpin–Gong criterion satisfies∑χ ( Tr m (1 x−1 ) + g a (x) ) = −2 m + #H a ;x∈F ∗ 2 mand, according to Proposition 4.2.4, the right hand side of the Charpin–Gong criterion satisfies∑x∈F ∗ 2 m χ (g a (x)) = −2 m − 1 + #G a .Let us now fix a subset of indices E ⊆ R and denote by r max the maximal index. We cansuppose r max to be odd and will do so for two reasons: