4 problems

Let $n\geq 4$, let $f:\mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be the power function $f(x)=x^d$, and identify exponents under the affine-equivalence transformation $d\mapsto 2^i d$ mod…

Budaghyan et al.'s differential spectrum conjecture. The equation has $q^2$ solutions for one value of $b$, has $q^2-q$ solutions for $q$ values of $b$, and has at most $2$ solutio…

PcN power-function conjecture. $F(x)=x^d$ is a PcN function if and only if one of the following conditions holds:

Bartoli–Timpanella conjecture. For $c=-1$, the power function $F$ is P$c$N over ${\mathrm{GF}}(p^n)$. This conjecture concerns the existence of power functions with perfect $c$-non…