2 YVES AUBRY AND FRANC ¸OIS RODIER Henceforth we fix q = 2m. In order to simplify our study of such functions, let us recall the following elementary results on differential uniformity the proofs are straightforward: Proposition 1. (i) Adding a polynomial whose monomials are of degree 0 or a power of 2 to a function f does not change δ(f). (ii) For all a, b and c in Fq, such that a = 0 and c = 0 we have δ(cf(ax + b)) = δ(f). (iii) One has δ(f 2) = δ(f). Hence, without loss of generality, from now on we can assume that f is a polynomial mapping from Fq to itself which has neither terms of degree a power of 2 nor a constant term, and which has at least one term of odd degree. To any function f : Fq −→ Fq, we associate the polynomial f(x) + f(y) + f(z) + f(x + y + z). Since this polynomial is clearly divisible by (x + y)(x + z)(y + z), we can consider the polynomial Pf (x, y, z) := f(x) + f(y) + f(z) + f(x + y + z) (x + y)(x + z)(y + z) which has degree deg(f) − 3 if deg(f) is not a power of 2. 2. A characterization of functions with δ ≤ 4 We will give, as in [17], a geometric criterion for a function to have δ ≤ 4. We consider in this section the algebraic set X defined by the elements (x, y, z, t) in the aﬃne space A4(Fq) such that Pf (x, y, z) = Pf (x, y, t) = 0. We set also V the hypersurface of the aﬃne space A4(Fq) defined by (1) (x + y)(x + z)(x + t)(y + z)(y + t)(z + t)(x + y + z + t) = 0. The hypersurface V is the union of the seven hyperplanes H1, . . . , H7 defined respectively by the equations x + y = 0, . . . , x + y + z + t = 0. We begin with a simple lemma: Lemma 2. The following two properties are equivalent: (i) there exist 6 distinct elements x0, x1, x2, x3, x4, x5 in Fq such that ⎧ ⎪x0 ⎨ ⎪ ⎩ + x1 = α, f(x0) + f(x1) = β x2 + x3 = α, f(x2) + f(x3) = β x4 + x5 = α, f(x4) + f(x5) = β (ii) there exist 4 distinct elements x0, x1, x2, x4 in Fq such that x0 + x1 + x2 + x4 = 0 and such that f(x0) + f(x1) + f(x2) + f(x0 + x1 + x2) = 0 f(x0) + f(x1) + f(x4) + f(x0 + x1 + x4) = 0. 2

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