36 problems

Let $q$ be an odd prime power and let $n\geq 3$. A quadratic system is a map … where each $f_i\in\mathbb{F}_q[x_1,\dots,x_n]$ has degree at most $2$. Classification conjecture. Suc…

Nonexistence conjecture. The polynomial $f(x)$ is not a permutation polynomial over $\mathbb{F}_{q^2}$.

Let $q=p^m$ and $Q=p^e$. For integers $n$ and $\ell\geq 1$, let $g_{n,q}^{(\ell)}$ be the multivariate polynomial obtained from $g_{n,q}$ by composing with…

Additive–multiplicative index dichotomy conjecture. Apart from monomials of the form $x\mapsto ax^{p^j}$, which have additive and multiplicative indices equal to $1$, any function…

Let $q$ be a prime power. A polynomial $f(X)\in\mathbb{F}_q[X]$ is exceptional if it induces a bijection on $\mathbb{F}_{q^k}$ for infinitely many positive integers $k$. Carlitz co…

Permutation criterion. The polynomials $f_1$ and $f_2$ are permutation polynomials over $\mathbb{F}_{q^3}$ if and only if

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

Gupta–Rai conjecture. The trinomial $f(X)$ is a permutation polynomial over $\mathbb F_{q^2}$ if and only if $\alpha=-1$ and $k=2$. This conjecture concerns the permutation behavio…

Let $p>7$ be a prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. A permutation polynomial (PP) is a polynomial that permutes $\mathbb{F}_p$. The…

Let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind, and let a permutation polynomial (PP) be a polynomial that permutes the field. The $p=5$ reversed Dickson…

Let $p>3$ be a prime, and let $n$ satisfy $1\leq nleq p^2-1$. Write $D_n(1,x)$ for the reversed Dickson polynomial of the first kind. A polynomial that permutes $\mathbb{F}_p$ is ca…

Let $m$ be a positive integer and define … A polynomial over a finite field is a permutation polynomial if it induces a one-to-one map from the field to itself. Garg–Sharma's conje…

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 $q=2^n$ with $n$ odd and $\gcd(i,n)=1$. For … set … Let $\Gamma$ be the parameter set defined in equation. Butterfly permutation parameter conjecture. If $V_i$ is a permutation…

Let $n=2m$ be a positive integer, and let $f(x)$ be the trinomial defined by equation (1) in the source over ${\mathbb F}_{2^{2m}}$. A pair $(s,t)$ specifies the exponents in this…

Let $\mathbb F_q$ be the finite field with $q$ elements, let $p=\operatorname{char}\mathbb F_q$, and for $a\geq 1$ define … A polynomial over $\mathbb F_{q^e}$ is a permutation pol…

Let $n$ be a positive integer, let $p$ be a prime with $p>2n$, and partition the reduced residues modulo $p$ into residue classes $I_i$ modulo $n$. Let $f(x)=Ax^k\pmod p$ be a powe…

Tu–Zeng–Li–Helleseth conjecture. If $f_{\alpha,\beta}(x)$ permutes $\mathbb{F}_{q^2}$, then condition (1) or condition (2) holds.

Let $p$ be an odd prime, let $n+1=p$, and set … Let $a\in\mathbb{F}_{p^{nk}}^*$ satisfy $a^{p^k-1}=-1$. Complete permutation monomial conjecture. The monomial $a^{-1}x^d$ is a comp…

Existence conjecture. There exists $a\in\mathbb{F}_{p^{nk}}^*$ such that $h_a(x)$ is a Dickson polynomial of degree $n+1$ over $\mathbb{F}_{p^k}$. This conjecture was subsequently…

Let $k$ be an even integer, set $q=5^k$, and let … Define $g(x)=-x\big(\frac{x^2-2}{x^2+2}\big)^2$. Wu–Li's conjecture. The map $g$ permutates $\mu_{q+1}$. This is the second of tw…

Let $k$ be a positive odd integer and let $f(x)=x\big(\frac{x^2-x+2}{x^2+x+2}\big)^2$ over the finite field $\mathbb{F}_{5^k}$. Wu–Li's conjecture. The polynomial $f$ is a permutat…

Let $k$ be an even integer and set $q=5^k$. Let … be the group of $(q+1)$-st roots of unity in ${\mathbb F}_{q^2}$. Permutation conjecture. The rational function … permutes…

Let $k$ be a positive integer, set $q=3^k$, and let ${\mathbb F}_{q^2}$ be the finite field with $q^2$ elements. A polynomial is a permutation trinomial over ${\mathbb F}_{q^2}$ if…

Permutation trinomial conjectures. (1) If $k$ is even and $\mathrm{gcd}(5+2l,q-1)=1$, then $f_1(x)$ is a permutation trinomial over $\mathbb{F}_{q^2}$. (2) If…