8 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 conjecture. The equation has $q^2$ solutions for one value of $b$, $q^2-q$ solutions for $2^n$ values of $b$, and at most $2$ solutions for all remaining points…

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

PcN power-exponent conjecture. For all $0\leq j\leq 6$, the only possible values of $d$ are the elements of

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…

Maximal differential uniformity distribution conjecture. There exists $\varepsilon_m>0$ such that, for all sufficiently large $n$, at least $\varepsilon_m2^{2n}$ pairs…

Blondeau et al.'s conjecture. For each of these two values of $d$, $F_d$ is differentially $8$-uniform, and all values $0,2,4,6,8$ occur in its differential spectrum. The paper pre…

Let $L$ be a finite field of odd characteristic, and let $s$ be an exponent. Write $s\not\approx 1$ for the stated exponent equivalence relation, and call $s$ nice when it has the…