76 problems

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…

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…

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

Let $q=p^e$ be an odd prime power and let $k$ be an integer with $1\le k\le q-1$. Define … and … as polynomials in $\mathbb F_q[X]$. Dmytrenko–Lazebnik–Williford's conjecture. Each…

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…

Goresky–Klapper conjecture. Apart from the identity map $(p;A,k)=(p;1,1)$ and the six cases

Let $q$ be even, let $a,b\in\mathbb F_{q^2}^*$, and define … Assume that $f$ is a permutation polynomial of $\mathbb F_{q^2}$. Tu–Zeng–Li–Helleseth's conjecture. The conditions … a…

Let $p$ be a prime, let $\d$ be an integer with $d\ge 2$, and let $f\in\mathbb{F}_p[x]$ be a complete mapping polynomial, meaning that both $f$ and $f(x)+x$ induce permutations of…

Let $f$ be a non-constant monic polynomial in $Q[x]$. A global permutation polynomial (GPP) over $Q$ is a polynomial $P(x)\inQ[x]$ whose reduction modulo $p$ is a permutation of…

Finiteness conjecture. There are only finitely many pairs $(q,a)$ for which $f$ is a permutation polynomial of $\mathbb{F}_{q^2}$.

Performance conjecture. Restricting quadratic interleavers to those that admit a quadratic inverse should not impair their performance when they are applied to turbo codes.

Two-class EA-equivalence conjecture. For every odd prime $p\geq 11$, these permutation polynomials fall into exactly two EA-equivalence classes. One class consists of the two speci…

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…

Jiang–Li–Qu's conjecture. The condition $\gamma=1$ is necessary and sufficient for the polynomial to be a permutation polynomial.

Let $n$ be a positive integer, let $\mathbb{F}_2$ be the field with two elements, and let $\chi_n$ denote the mapping obtained from the coordinate rule $y_i=x_i+(x_{i+1}+1)x_{i+2}$…

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…

Let $q$ be a prime power, and let $f(X)\in\mathbb{F}_q[X]$ be an exceptional polynomial, meaning that $f$ induces a bijection on $\mathbb{F}_{q^k}$ for infinitely many positive int…

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…

Sziklai's conjecture. If

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 $1leq nleq p^2-1$. Write $D_n(1,x)$ for the reversed Dickson polynomial of the first kind. A polynomial that permutes $\mathbb{F}_p$ is ca…