23 problems

Let $q$ be a power of two, and let $f\in\mathbb{F}_q[X]$ be an exceptional APN polynomial, meaning that $f$ induces an APN function on $\mathbb{F}_{q^r}$ for infinitely many positi…

Let $m\ge 8$ be an even integer, let $b,c\in\mathbb{F}_{2^m}$, and let $r$ satisfy $r\le m/2$ and $\gcd(r,m)=1$. Let $f_{b,c,r}$ denote the function defined in Theorem. Non-APN con…

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

Let $n=2m$. For integers $r<m$ with $\gcd(r,m)=1$, define $s$ by … Set $F_1(x)=x^{s(2^m-1)+1}$ over $\mathbb{F}_{2^{2m}}$. A power function is Niho type locally-APN if it is both o…

Degree bound conjecture. The degree of an APN mapping is less than $m$.

APN distance-one conjecture. If $F\neq G$, then $d(F,G)>1$. It is unknown whether two APN functions can have Hamming distance one; the conjecture is known for several classes, incl…

Non-permutation conjecture. The function $C_u$ is not a permutation and is not APN.

Axis-value conjecture. For every $a\in\mathbb{F}_2^n$ and every nonzero $b\in\mathbb{F}_2^n$,

Uniform-exclude-distribution conjecture. If $d_{\mathcal{G}_F}$ is uniform on $\mathcal{Q}(\mathbb{F}_2^n,F)$, then $\mathcal{G}_F$ is maximal.

Single-point modification conjecture. The function $G$ is not APN.

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…

Recursive APN function conjecture. There exists a recursive APN function in every dimension $n\in\mathbb{N}$ with $n\geq 2$.

Non-APN and non-permutation conjecture. The function $C_u$ is not APN. Further, if $u\neq 0$, then $C_u$ is not a permutation.

Component-degree conjecture. For every nonzero $v\in\mathbb{F}_2^n$,

Degree conjecture. Then

Let $F\colon\mathbb{F}_2^n\rightarrow\mathbb{F}_2^n$ be an APN function, and let $\mathsf{Aut}(F)$ denote its automorphism group. If $F$ is an APN permutation, let…

Let $F:{\mathbb F}_{2^n}\to{\mathbb F}_{2^n}$ be an APN function. Fix $x_0\in{\mathbb F}_{2^n}$ and let $F'$ be its $(x_0,\epsilon)$-modification, meaning that…

Let $n\geq5$ be odd, and let … Let $\phi(n)$ denote Euler's totient function. Kassami-code inequivalence conjecture. The $\frac{\phi(n)}2$ binary linear codes…

Let $n\geq5$ be odd. Let $\mathsf{C}_2(\mathbb{KA}_{n,1})$ denote the binary linear code generated by the block characteristic vectors of $\mathbb{KA}_{n,1}$. Kassami-code paramete…

Let $n\geq5$ be odd, and let … Let $\phi(n)$ denote Euler's totient function, and let $\mathbb{KA}_{n,i}$ be the corresponding Kassami APN $3$-designs. Kassami-design non-isomorphi…

Let $q=2^n$, with $n\geq5$ odd, and let $B_s$ be the associated base block. Two-exponent APN 3-design conjecture. The incidence structure … is a $3$-…

Let $q=2^n$. Let $n\geq5$ be odd, let $s=2^{2i}-2^i+1$, and assume $\gcd(3i,n)=1$. Let $B_s$ be the associated base block and let $\operatorname{GA}_1(q)$ denote the relevant affin…

Let ${\mathbb F}_{2^n}$ be the finite field with $2^n$ elements, and for $2\leq t\leq n-1$ define … A function is APN if its differential uniformity is at most $2$. The classificat…