8 problems

Blondeau et al.'s conjecture. The power mapping $F$ over $\mathbb{F}_{2^n}$ is differentially $8$-uniform. This conjecture concerns the second-order differential behavior of a powe…

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…

Let $n\geq 4$ and let $f:\mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be a cubic power function, meaning $f(x)=x^d$. Two such functions are considered equivalent when they differ by the af…

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 classification conjecture. All PcN functions in $\mathbb{F}_{2^n}$ are linear functions, Gold functions, or inverses of Gold functions.

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…

Dobbertin–Mills–Müller–Pott–Willems conjecture. The function $x\mapsto x^d$ is APN over $\mathbb{F}_{3^n}$.