Crooked Maps in Finite Fields

International audience We consider the maps $f:\mathbb{F}_{2^n} →\mathbb{F}_{2^n}$ with the property that the set $\{ f(x+a)+ f(x): x ∈F_{2^n}\}$ is a hyperplane or a complement of hyperplane for every $a ∈\mathbb{F}_{2^n}^*$. The main goal of the talk is to show that almost all maps $f(x) = Σ_{b ∈B}c_b(x+b)^d$, where $B ⊂\mathbb{F}_{2^n}$ and $Σ_{b ∈B}c_b ≠0$, are not of that type. In particular, the only such power maps have exponents $2^i+2^j$ with $gcd(n, i-j)=1$. We give also a geometrical characterization of this maps.