16 problems

Let $f(x)=ax^d$ be a monomial that is an exceptional APN polynomial, meaning that it is APN over infinitely many extensions of its base field. The Gold numbers are $2^k+1$, and the…

Let $q=p^n$ be an odd prime power, let $\mathbb{F}_{q}$ be the finite field of $q$ elements, and define … A function on $\mathbb{F}_{q}$ is almost perfect nonlinear (APN) when its…

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…

Let $\gamma\in\mathbb F_{3^n}\setminus\{0,1\}$, and let $f=Inv\circ(1,\gamma)$ be the swapped inverse function. Define … Let $\nabla_f$ denote the second-order zero differential un…

Let $p$ be an odd prime, let $\mathbb F_{p^n}$ be a finite field, and let $Inv$ denote the inverse function. For a parameter $\gamma$, let $Inv\circ(1,\gamma)$ be the swapped inver…

Let $n>1$ be odd, let $\mathbb F_{2^n}$ be the finite field with $2^n$ elements, and let $\gamma\in\mathbb F_{2^n}\setminus\{0,1\}$. Define the sets $S_{\gamma,9}$ and…

McGuire–Rodier conjecture. Up to CCZ equivalence, these monomials are the only exceptional APN polynomials. This conjecture proposes a classification extending the resolved monomia…

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…

Attainability conjecture. The bound ${_c}\delta_F^{(2)}=6$ is attainable for every $n\geq 8$.

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…

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…