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120 Chapter 4. Efficient characterizations for bentnessProposition 4.3.3 ([129]). Let m ≥ 3 be any positive integer and a ∈ F ∗ 2m. Suppose that thereexists t ∈ F ∗ 2 such that a = m t4 + t 3 .• If m is odd, then K m (a) ≡ 1 (mod 3).• If m is even, then K m (a) ≡ 0 (mod 3) if Tr m 1Tr m 1 (t) = 1.(t) = 0 and K m (a) ≡ −1 (mod 3) ifFurthermore, Charpin, Helleseth and Zinoviev [48] gave additional results about values ofK m (a) modulo 3.Proposition 4.3.4 ([48]). Let m ≥ 3 be any positive integer and a ∈ F ∗ 2m. Then we have:• If m is odd, then K m (a) ≡ 1 (mod 3) if and only if Tr m 1a =b(1+b)for some b ∈ F ∗ 4 2 m.(a1/3 ) = 0. This is equivalent to• If m is even, then K m (a) ≡ 1 (mod 3) if and only if a = b 3 for some b such that Tr m 2 (b) ≠ 0.Further divisibility results exist and could be used to further refine the tests proposed inthis chapter. For example, results up to 64 can be found in a paper of Göloğlu, McGuire andMoloney [124], and results up to 256 in an even more recent paper of Göloğlu, Lisoněk, McGuireand Moloney [123].Most of these results about divisibility were first proved studying the link between exponentialsums and coset weight distribution [129, 48]. However some of them can be proved in a completelydifferent manner as we show in the next subsection.4.3.2 Using torsion of elliptic curvesTheorem 4.2.1 giving the value of K m (a) as the cardinality of an elliptic curve can indeed be usedto deduce divisibility properties of Kloosterman sums from the rich theory of elliptic curves. Werecall that the quadratic twist of the ordinary elliptic curve E a that we denote by Ẽa is given bythe Weierstraß equationẼ a : y 2 + xy = x 3 + bx 2 + a ,where b ∈ F 2 m has absolute trace 1; it has cardinality:#Ẽa = 2 m + 2 − K m (a) .First of all, we recall a proof of the divisibility by 4 stated in Proposition 4.3.1 as it canbe found for example in the preprint of Ahmadi and Granger [3]. For m ≥ 3, K m (a) ≡ #E a(mod 4), so K m (a) ≡ 0 (mod 4) if and only if #E a ≡ 0 (mod 4). This is equivalent to E a havinga non-trivial rational point of 4-torsion. This can also be formulated as both the equation of E aand its 4-division polynomial f 4 (x) = x 6 + ax 2 having a rational solution. It is easily seen thatP = (a 1/4 , a 1/2 ) is always a non-trivial solution to this problem.Lisoněk [180] used similar techniques to give a different proof of Proposition 4.3.2. Indeed, form ≥ 3, K m (a) is divisible by 8 if and only if E a has a non-trivial rational point of 8-torsion. Thisis easily shown to be equivalent to Tr m ( )1 a1/4= Tr m 1 (a) = 0.Finally, it is possible to prove directly that the condition given in Proposition 4.3.3 is notonly sufficient, but also necessary, using torsion of elliptic curves 6 . We use this property inSubsection 4.4.3.6 We were recently made aware that such a result was also proved in a different way also involving ellipticcurves by Garashuck and Lisoněk [108] in the case where m is odd.