here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

4.1. Hyper-bentness characterizations 113that family includes the well-known monomial functions with the Dillon exponent as a specialcase. The characterization involves exponential sums and Dickson polynomials and is given below.Theorem 4.1.1 (Charpin–Gong criterion [46]). Let n = 2m be an even integer. Let S be a setof representatives of the cyclotomic classes modulo 2 m + 1 whose cosets have full size n. Let f abe the function defined on F 2 n by f a (x) = ∑ (r∈R Trn 1 ar x ) r(2m −1), where R ⊆ S and a r ∈ F ∗ 2 m.Let g a be the Boolean function defined on F 2 m by g a (x) = ∑ r∈R Trm 1 (a r D r (x)), where D r (x) isthe Dickson polynomial of degree r. Then f a is hyper-bent if and only if∑x∈F ∗ 2 m χ ( Tr m 1(x−1 ) + g a (x) ) = 2 m − 2 w H (g a ) − 1 .Finally, Mesnager [196] gave a similar characterization of hyper-bentness 3 for another largeclass of hyper-bent functions with multiple trace terms which do not belong to the familyconsidered by Charpin and Gong [46]. We call it the second Mesnager criterion to avoid confusionwith the first Mesnager criterion for binomial functions.Theorem 4.1.2 (Second Mesnager criterion [196]). Let n = 2m be an even integer with m oddand S be a set of representatives of the cyclotomic classes modulo 2 m + 1 whose cosets have fullsize n. Let b ∈ F ∗ 4. Let f a,b be the function defined on F 2 n byf a,b (x) = ∑ r∈RTr n 1(a r x r(2m −1) ) + Tr 2 1) (bx 2n −13, (4.1)where R ⊆ S and all the coefficients a r are in F ∗ 2 m. Let g a be the related function defined on F 2 mby g a (x) = ∑ r∈R Trm 1 (a r D r (x)), where D r (x) is the Dickson polynomial of degree r. Then:1. f a,b is hyper-bent if and only if f a,b is bent.2. If b is a primitive element of F 4 , then the three following assertions are equivalent:(a) f a,b is hyper-bent;∑(b)χ (g a (D 3 (x))) = −2;(c)x∈F ∗ 2 m ,Tr m 1 (x−1 )=1∑x∈F ∗ 2 m χ ( Tr m 13. f a,1 is hyper-bent if and only if2(x−1 ) + g a (D 3 (x)) ) = 2 m − 2 w H (g a ◦ D 3 ) + 3.∑x∈F ∗ 2 m ,Tr m 1 (x−1 )=1χ (g a (D 3 (x))) − 3∑x∈F ∗ 2 m ,Tr m 1 (x−1 )=1χ (g a (x)) = 2 .Once more, these criteria reduce the test of hyper-bentness from the computation of the fullWalsh–Hadamard transform to that of a finite number of exponential sums.3 There was a typo in the theorem given in the original article [196] where the last term in the right hand sideof Condition 2c reads 4 instead of 3. This is an unfortunate consequence of the fact that the summation set usedin the statement of that condition within the theorem is F ∗ 2 m whereas it is F 2m within the proof of the theorem.