Budaghyan–Carlet–Helleseth conjecture on the algebraic degree of APN functions

Let $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ be an almost perfect nonlinear (APN) function, meaning that for every $a,b \in \mathbb{F}_2^n$ with $a \ne 0$, the equation $F(x+a)+F(x)=b$ has either zero or two solutions. The algebraic degree of $F$ is the maximum degree of a monomial appearing in its algebraic normal form.

Budaghyan–Carlet–Helleseth conjecture. No APN function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ has algebraic degree $n$ for all $n \geq 3$.

APN functions are important in cryptography because they provide optimal resistance to differential attacks, and their graphs are Sidon sets. The conjecture is stated in the source as a completely open question.