44 problems

A function $f$ over finite fields of characteristic $2$ is exceptional APN if it is APN over infinitely many extensions of the base field. The Gold functions are $f(x)=x^{2^k+1}$,…

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…

APN power-function classification conjecture. All APN functions of the form $F(x)=x^d$, defined over $\mathbb{F}_{2^n}$, are known up to CCZ-equivalence. This conjecture asserts co…

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…

Let $f$ and $g$ be distinct APN functions, with distance measured by the minimum distance notion under discussion for functions in the relevant function space. Minimum-distance con…

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 $m$.

Let $f$ be a hexanomial of the form studied in the paper, and call it APN if it has optimal differential uniformity over the relevant finite field. Dillon's conjecture. Hexanomials…

Kaleyski's conjecture. These two values are both equal to

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.

Maximality conjecture. The graph $\mathcal{G}_F$ is maximal for every APN function $F$.

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

Superpolynomial-growth conjecture. $N(Q)$ is not bounded from above by a polynomial.

Let $F_0$ be the APN permutation in the first row of Table 1, with odd $n=3k$, $\gcd(k,3)=1$, and $\gcd(s,3k)=1$. Budaghyan–Carlet–Leander conjecture. The algebraic degree of the i…

APN monomial classification conjecture. No other APN monomials exist over ${\mathbb F}_{2^n}$ up to cyclotomic equivalence.

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…

A quadratic APN function is an APN function on $\mathbb{F}_2^8$ whose coordinate functions have algebraic degree at most $2$; two such functions are considered equivalent under EA-…

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

Carlet's conjecture. The map $C_u$ is not APN if $m>3$.

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.

Zero-set nonlinearity conjecture. The set $\{x\in\mathbb{F}_2^n\mid v\cdot\Phi_F(x)=0\}$ is not a linear subspace.

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