A recent survey of permutation trinomials over finite fields

1.   Introduction

Let $F_{q}$ be the finite field with $q = p^{n}$ elements, where $p$ is a prime number and $n$ is a positive integer. A polynomial $f(x)\in F_{q}[x]$ is called a permutation polynomial over $F_{q}$ if it is a bijection of $F_{q}$ into itself. The study of permutation polynomials on finite fields began by Hermite ^[27], Dickson ^[15] and Carlitz ^[12] has since been carried out by many other researchers ^{[1,3,4,5,8,9,35,85]}. The study of permutation polynomials over finite fields has been attracting researcher interest for many years due to their wide applications in cryptography ^{[36,59,61,62]}, coding theory ^[17,18] and combinatorial designs ^[16]. In many cases, compositional inverses of permutation polynomials are necessary. In block ciphers, substitution boxes, which form the confusion layer during encryption, are often designed by using permutation polynomials and their compositional inverses. Very important progress has been achieved in the area of the construction of permutation polynomials and their compositional inverses (see ^{[55,77,78,79]}).

Permutation polynomials with fewer terms are particularly desirable due to their good algebraic structure over finite fields. The three most basic types of polynomials are monomials, binomials and trinomials. Permutation polynomials of the monomial or binomial type have been widely investigated in recent decades. However, much less is known about polynomials with more than two terms, such as trinomials and quadrinomials. The monomial $x^{r}$ is a permutation polynomial over $F_{q}$ if and only if $\gcd(r, q-1) = 1$. It is challenging to ascertain the conditions on $a, b, n, m$ and $q$ under which the binomials $ax^{n}+bx^{m}$ are permutations on $F_{q}$.

In 2015, Hou ^[32] briefly surveyed the known classes of permutation binomials and trinomials, but very few classes were known at that time. The purpose of the present paper is to review some of the recent contributions to the area while providing more details and background. Our primary focus is on the results of permutation trinomials that have appeared in the last decade. In addition, we will present the reader with a selection process for recently developed approaches and methods.

An old and yet very useful result on the theory of permutation polynomials is the following theorem, proved by Hermite ^[27] for prime fields, and Dickson ^[15] in the general case.

Lemma 1.1. (Hermite-Dickson criterion) ^[27] Let $F_{q}$ be a finite field of characteristic $p$. Then, $f(x)\in F_{q}[x]$ is a permutation polynomial of $F_{q}$ if and only if the following two conditions hold:

(1) $f(x)$ has exactly one root in $F_{q}$;

(2) For each integer $t$ with $1\leq t\leq q-2$ and $t\not\equiv0\pmod{p}$, the reduction of $f(x^{t})\pmod {x^{q}-x}$ has a degree $\leq q-2$.

The following Lemma 1.2 shows that polynomials of the form $x^{r}h(x^{(q-1)/d})$ over $F_{q}$ have a close connection with the $d$-order subgroup $\mu_d$ of $F_{q}^{*}$.

Indeed, there have been many construction results that were obtained by using the method of constructing permutation polynomials over $\mu_d$ to obtain permutation polynomials over the original finite field. It was first stated by Wan and Lidl ^[69], and later modified by Wang ^[70] and Zieve ^[85].

Lemma 1.2. ^[69,70,85] Let $d, r > 0$ with $d|(q-1)$ and $h(x)\in F_{q}[x]$. Then, $f(x) = x^{r}h(x^{(q-1)/d})$ permutes $F_{q}$ if and only if

(1) $\gcd(r, (q-1)/d) = 1$,

(2) $x^{r}h(x)^{(q-1)/d}$ permutes $\mu_{d}$.

Theorem 1.1. ^[47] Let $p$ be a prime and $n, r_{1}, r_{2}, \cdots, r_{t}$ be non-negative integers such that

Then,

Further, it follows that $\binom{n}{r_{1}, r_{2}, \ldots, r_{t}}\not\equiv0 \pmod{p}$ if and only if $\sum\limits_{i = 1}^{t}d_{ij} = d_{j}$, $\forall$ $0\leq j\leq s$.

Definition 1.1. ^[74] Two permutation polynomials $f(x)$ and $g(x)$ in $F_{q}[x]$ are called the quasi-multiplicative equivalence if there exists an integer $1\leq d < q-1$ such that $\gcd(d, q-1) = 1$ and $f(x) = ag(cx^{d})$, where $a, b\in F_{q}^{*}$.

Definition 1.2. ^[47] A polynomial $f(x)\in F_{q}[x]$ is said to be a complete permutation polynomial over $F_{q}$ if both $f(x)$ and $f(x)+x$ are permutations of $F_{q}$.

To check the permutation property of any given polynomial over a finite field, we can use Akbary, Ghioca, Wang(AGW) criterion. We make use of the subfield of the finite field and a known polynomial that permutes the subfield. For instance, a permutation polynomial over $F_{q}$ can be used to check the constructed polynomial over $F_{q^{n}}$, regardless of whether it is a permutation polynomial or not.

Lemma 1.3. ^[1] (AGW criterion) Let $K$, $L$ and $\bar{L}$ be finite sets with $|L| = |\bar{L}|$, and let

and $\bar{\omega}$: $K\rightarrow \bar{L}$ be maps as shown in the Figure 1, such that $\bar{\omega} \circ g = \bar{g} \circ \omega$. If both $\omega$ and $\bar{\omega}$ are surjective, then the following statements are equivalent:

(1) $g$ is a bijection from $K$ to $K$;

(2) $\bar{g}$ is a bijection from $L$ to $\bar{L}$ and $g$ is injective on $\omega^{-1}(l)$ for each $l\in L.$

So, if $\bar{g}$ is a bijection on $L$, we can make conclusions about the permutation of $g$ over $K$.

There are several approaches that can be employed to decide whether $f(x)$ is a permutation polynomial. When investigating the permutational properties of a polynomial, a well-established connection with algebraic curves is provided by the following observation. For a given polynomial $f(x)\in F_q[x]$, let us consider the curve $C_{f}$ with the affine equation

Deciding whether a polynomial is a permutation polynomial over any field is based on the investigation of the set of $F_{q^{2}}$-rational point of $C_{f}$. The Hermite-Dickson criteria ^[27] constitute another well-known tool for use in the construction of any type of permutation polynomial, and when using these criteria, one may use the Lucas sequence ^[47] to compute binomial coefficients in the polynomial expressions. Here, the main aim is to compute the sum $\sum\limits_{x\in F_{q}} f(x)^{s}$ for the integer $0\leq s \leq q-1$. However, computing the sum is a little lengthy and the toughest one. To overcome this difficulty, recent discoveries have been made, including Lemma 1.2. The main construction method involves the use of the multivariate method ^[21] and discussions of the number of solutions of special equations $f(x) = d$. Recently, one more method has come into existence for the computation of the permutation binomials and trinomials, i.e., the computation of the fractional polynomials ^[39]. The AGW criterion is another significant technique to determine a polynomial's permutation property (see ^[1]).

In Section 2, we list all of the existing permutation trinomials, and those of the forms $x^{r}h(x^{s})$, $\lambda_{1}x^{a}+\lambda_{2}x^{b}+\lambda_{3}x^{c}$ and $x+x^{s(2^{m}-1)+1} +x^{t(2^{m}-1)+1}$ with Niho-type exponents $s, t$ are discussed in the Sections 3–5, respectively. In addition, in Sections 3–5, we provide a discussion about current conjectures and open problems that have been mentioned in the literature.

2.   Permutation trinomials

There are compelling findings about both the existence and non-existence of permutation binomials, but there is not a comparable instance of non-existence in the case of permutation trinomials. Nonetheless, there have been numerous permutation trinomial discoveries from a variety of backgrounds. The Dickson polynomial of order five, i.e., $D_{5}(x, a) = x^{5}+ax^{3}-a^{2}x$, is a permutation trinomial over $F_{3^{m}}$ with a wide range of uses; it also produces a family of ideal nonlinear functions for cryptography, as well as linear codes for data transmission and storage. Similarly, the Dickson polynomial $D_{7}(x, a) = x^{7}+x^{5}+x$ of degree 7 with $m\not\equiv0\pmod{3}$ comprises permutation trinomials over $F_{3^{m}}$. In 1997, Lee and Park ^[38] investigated permutation trinomials; they assumed that $q\equiv1\pmod{3}$ and there is a fixed primitive root $g$ of $F_{q}$ for $\alpha\in F_{q}^{*}$, and that $\log_g{\alpha}$ is the residue class $k\pmod{ q-1}$ such that $\alpha = g^{k}$; here, $s = \frac{q-1}{3}$ and $\omega = g^{s}$, $h(x) = x^{r}f(x^{s})$, where $f(x) = ax^{2}+bx+c\in F_{q}[x]$. Similar to this, in ^[13,21] both authors used two distinct concepts to investigate the permutation property of the trinomial

over $F_{2^{m}}$ when $m$ is odd. Lee and Park ^[38] established the following theorem, which is the primary method of creating numerous permutation polynomials of the form $h(x) = x^{r}f(x)$.

Theorem 2.1. ^[38] Let $h(x) = x^{r}f(x^{s})\in F_{q}[x]$, where $f(x) = ax^{2}+bx+c\in F_{q}[x]$ and $s = \frac{q-1}{d}$ with $d\mid(q-1)$. Then, $h(x)$ is a permutation polynomial over $F_{q}$ if and only if the following conditions are satisfied:

(1) $(r, s) = 1$,

(2) for $0\leq i\leq3$, $f(\omega^{i})\not = 0$,

(3) $\log_g\frac{f(1)}{f(\omega)}\equiv \log_g \frac{f(\omega)}{f(\omega^{2})}\not\equiv r\pmod{3}$.

3.   Polynomials of the form $x^{r}h(x^{s})$

The permutation behavior of polynomials of the form $f(x) = x^{r}h(x^{(q-1)/d})$ over a finite field $F_{q}$ has drawn the interest of many scholars since the majority of permutation polynomials can be expressed as $x^{r}h(x^{s})$ over a finite field $F_{q}$. Over the past few years, a large number of permutation polynomials of this form have been discovered. Wan and Lidl ^[69] first investigated the permutation behavior of the form $f(x) = x^{r}h(x^{(q-1)/d})$, where $d|q-1$ and $1 < r < \frac{q-1}{d}$ over $F_{q}$. In this section, we list all of the permutation polynomials of the form $x^{r}h(x^{s})$. In his contribution to the field of permutation, Zieve ^[85] introduced a common method to find permutations of the type $f(x) = x^{r}h(x^{(q-1)/d})$ over $F_{q}$ by exploiting the behavior of the permutations $g(x) = x^{r}h(x)^{q-1}$ in $\mu_{q+1}$; as an added feature, and from this fact, Zieve ^[86] constructed several classes of permutation trinomials over $F_{q^{2}}$.

The approach involves rational functions over $F_{q^{2}}$ that map $\mu_{q+1}$ to either $\mu_{q+1}$ or $F_{q}\cup\{\infty\}$. He first constructed a general polynomial of the form

for $n > 0$, $k\geq0$, $\beta, \gamma\in F_{q^{2}}$ with $\beta^{q+1} = 1$ and $\gamma^{(q+1)}\not = 1$, which permutes $F_{q^{2}}$ if and only if $\gcd(n+2k, q-1) = 1$ and $\gcd(n, q+1) = 1$. He also investigated polynomials of the form

with $\delta\in F_{q^{2}}$ but $\delta\not\in F_{q}$; it is a permutation polynomial over $F_{q^{2}}$ if and only if $\gcd(n(n+2k), q-1) = 1$. As a consequence of the above fact, he mentioned some trinomials in the following corollaries for $n = 3$ and certain values of $\beta, \delta, \gamma$.

Corollary 3.1. ^[86] Let $q$ be a prime power and $k\geq0$. The polynomial

permutes $F_{q^{2}}$ if and only if $\gcd(2k+3, q-1) = 1$ and $3\not\mid q$.

For the values $k = q-3$, $k = 1$ and $k = 0$, the above corollary reduces to the following corollary.

Corollary 3.2. ^[86] Let $q$ be a prime power with $3\not\mid q$. Then,

(1) $x^{q}+3x^{2q-1}-x^{q^{2}-q+1}$ permutes $F_{q^{2}}$.

(2) $x^{q+4}+3x^{2q+3}-x^{4q+1}$ permutes $F_{q^{2}}$ if and only if $q\not\equiv1\pmod{5}$.

(3) $x^{3}+3x^{q+2}-x^{3q}$ permutes $F_{q^{2}}$ if and only if $q\equiv2\pmod{3}$.

Regarding the case of $q = 2^{2m+1}$, the first two parts of the above corollary were conjectured by Tu et al. ^[64]. Later, Zieve ^[86] provided the alternative proof for [28,Theorem 1] in [86,Theorem 5.1]. Using this, he further determined the permutation binomial for $h(x) = x^{d}+\beta^{-1}$ in the following corollary. The proof encompasses demonstrating that $h(x)$ has roots in $\mu_{q+1}$ if and only if $-\beta^{-1}$ is in $(\mu_{q+1})^{d}$, which equals $\mu_{(q+1)}/\gcd(q+1, d)$.

Corollary 3.3. ^[86] Let $q$ be a prime power, let $r$ and $d$ be positive integers and let $\beta$ be the $(q+1)^{th}$ root of unity in $F_{q^{2}}$. Then, $x^{r+d(q-1)}+\beta^{-1}x^{r}$ permutes $F_{q^{2}}$ if and only if all of the following conditions hold:

(1) $\gcd(r, q-1) = 1$,

(2) $\gcd(r-d, q+1) = 1$,

(3) $(-\beta)^{(q+1)/\gcd(q+1, d)}\not = 1$.

There are numerous classes of permutation trinomials over finite fields in the literature, some of which have profound connections to other areas. Recently, a complete determination of the permutation trinomials of the type

over $F_{q^{2}}$ was demonstrated in ^[29,30,31]. Trinomials of the form

were taken into consideration by Hou ^[29]. The main theme is dependent on the fact that

so $f(x)$ is a permutation polynomial of $F_{q}$ if and only if $(a+b+1)\not = 0$.

Motivated by the work of Fernando et al. ^[22] i.e., a study of permutation polynomial over finite fields defined by functional equations, which also detailed the discovery of a class of permutation binomials in ^[28] over $F_{q^{2}}$, in 2013, Hou ^[29] determined a class of permutation trinomials of the form

over $F_{q^{2}}$ when $q > 2$. The following theorem is similar to the theorems mentioned in ^[28], but the method presented is different from that of ^[28].

Theorem 3.1. ^[29] Let $q > 2$ be a prime power and $f(x) = -x+bx^{q}+x^{2q-1}\in F_{q}[x]$, where $b\in F_{q}^{*}$. Then, $f(x)$ is a permutation polynomial of $F_{q^{2}}$ if and only if one of the following conditions applies:

(1) $q$ is even and $Tr_{q/2}\left(\frac{1}{b}\right) = 0$,

(2) $q\equiv1\pmod{8}$ and $b^{2} = -2$.

Regarding sufficiency, the author applied the existence of the solution $x\in F_{q^{2}}$ of the equation $f(x) = y$ for every $y\in F_{q^{2}}$, and, regarding the necessity

for $1\leq s\leq q-2$. The sum can be expressed as a double sum in terms of the binomial coefficients; it can be solved for $s = (q-1)q$ and $s = 1+(q-2)q$. When $s = (q-1)q$, the sum

gives $1+4b^{-2}$ as a square in $F_{q}^{*}$, and when $s = 1+(q-2)q$, the simplified sum implies that $b^{2} = -2$.

Hou ^[30] used another approach that involves the use of $Tr(x)$ and $N(x)$. The idea behind this approach is that the polynomial $f(x)\in F_{q}[x]$ permutes $F_{q^{2}}$ if and only if $f(x) = c$ has a unique solution in $F_{q^{2}}$ for any $c\in F_{q^{2}}$. This can be achieved by proving that $f(F_{q^{2}}\backslash F_{q})\subseteq F_{q^{2}}\backslash F_{q}$, followed by $c\in F_{q^{2}}\backslash F_{q}$. Then, it is sufficient to prove that $(Tr(x), N(x))$ is uniquely determined by $c$, since $(Tr(x), N(x))$ can be uniquely determined by the set $\{x, x^{q}\}$ and $f(x)\not = f(x^{q}) = f(x)^{q}$. Here, Hou ^[30] considered two cases on $q$ and stated the following theorem.

Theorem 3.2. ^[30] Let $f(x) = ax+bx^{q}+x^{2q-1}\in F_{q^{2}}[x]$. Then, $f(x)$ is a permutation polynomial of $F_{q^{2}}$ if and only if one of the following conditions is satisfied when $q$ is odd:

(1) $a(a-1)$ is a square in $F_{q}^{*}$, and $b^{2} = a^{2}+3a$,

(2) $a = 1$ and $b^{2}-4$ is a square in $F_{q}^{*}$,

(3) $a = 3$, $b = 0$ and $q\equiv -1\pmod{6}$,

(4) $a = b = 0$ and $q\equiv1, 3\pmod{6}$.

Alternatively, it is a permutation polynomial if and only if either condition is satisfied when $q$ is even and $q > 2$:

(1) $a\not = 1$, $Tr_{q/2}\left(\frac{1}{a+1}\right) = 0$ and $b^{2} = a^{2}+a$,

(2) $a = 1$, $b\not = 0$ and $Tr_{q/2}\left(\frac{1}{b}\right) = 0$.

After the computation of permutation trinomials ^[30] of the form $f(x) = ax+bx^{q}+x^{2q-1}\in F_{q}[x]$ for all primes with $q > 2$, Hou ^[31] came up with certain new conditions to determine general permutation trinomials of the form $ax+bx^{q}+x^{2q-1}$ over $F_{q^{2}}$. The results cover two cases on $q$, odd and even, and the methodology is based on the method described in ^[30].

Theorem 3.3. ^[31] Let $f(x) = ax+bx^{q}+x^{2q-1}\in F_{q^{2}}[x]$ and $q$ be a prime power. Then, $f(x)$ is a permutation polynomial of $F_{q^{2}}$ if and only if one of the following conditions is satisfied:

(1) When $q$ is odd:

(a) $a = b = 0$ and $q\equiv1, 3\pmod{6}$,

(b) $(-a)^{(q+1)/2} = -1$ or 3 and $b = 0$,

(c) $ab\not = 0$, $a = b^{1-q}$ and $1-\frac{4a}{b^{2}}$ is a square of $F_{q}^{*}$,

(d) $ab(a-b^{1-q})\not = 0$, $1-\frac{4a}{b^{2}}$ is a square of $F_{q}^{*}$ and $b^{2}-a^{2}b^{q-1}-3a = 0$.

(2) When $q$ is even:

(a) $a = b = 0$ and $q = 2^{2k}$,

(b) $ab\not = 0$, $a = b^{1-q}$ and $Tr_{q/2}(b^{-1-q}) = 0$,

(c) $ab(a-b^{1-q})\not = 0$, $\frac{a}{b^{2}}\in F_{q}$, $Tr_{q/2}\left(\frac{a}{b^{2}}\right) = 0$ and $b^{2}+a^{2}b^{q-1}+a = 0$.

Remark 3.1. In ^[30,31], Hou considered the general trinomial

over $F_{q^{2}}$. The construction discussed in ^[31] included more general conditions on coefficients, including the conditions mentioned in ^[30].

Using the criterion proposed by Akbary et al. ^[1], Tu et al. ^[65] determined the permutation trinomials over a finite field of even characteristics. They investigated whether, for a positive integer $m$ and $v\in F_{2^{m}}^{*}$, the trinomial

is a permutation polynomial over $F_{2^{3m}}$. Later, Wu and Lin ^[73] noticed that

can be written as

using this, they derived complete permutation polynomials of the form

over $F_{2^{nm}}$ for any $v\in F_{2^{m}}\backslash\{0, 1\}$.

By combining different techniques with the multivariate method proposed by Dobbertin ^[21], Ding et al. ^[19] constructed different types of permutation polynomials over $F_{2^{m}}$ with non-zero trivial coefficients. Two classes of permutation polynomials have been constructed under the condition that $m$ is odd. Both classes of trinomials are described in the following theorem.

Theorem 3.4. ^[19] For any odd integer $m > 1$, the following trinomials are permutation polynomials over $F_{2^{m}}$:

(1) $f(x) = x+x^{2^{(m+1)/2}-1}+x^{2^{m}-2^{(m+1)/2}+1}$,

(2) $f(x) = x+x^{3}+x^{2^{m}-2^{(m+3)/2}+2}$.

And, using the fact that, for a positive integer $m$, the equation $x^{2}+ux+v = 0$, where $u, v\in F_{2^{m}}$ and $u\not = 0$ has roots in $F_{2^{m}}$ if and only if $Tr_{v/ u^{2}} = 0$, they determined that the trinomial

is a permutation polynomial over $F_{2^{m}}$ for any even $m\geq2$. They observed that one of these above classes works for any finite field $F_{q^{m}}$ such that $q\not\equiv0\pmod{3}$. Using the Dobbertin ^[21] multivariate methods, they obtained the general class of permutation trinomials listed below.

Theorem 3.5. ^[19] Let $k$ be a positive integer, $q$ be a prime power with $q\not\equiv0\pmod{3}$ and $m$ be an even positive integer. Then,

is a permutation polynomial of $F_{q^{m}}$ if and only if one of the following three conditions holds:

(1) $m\equiv0\pmod{4}$,

(2) $q\equiv1\pmod{4}$,

(3) $m\equiv2\pmod{4}$, $q\equiv2\pmod{3}$ and $exp_{3}(k)\geq exp_{3}(q^{m/2}+1)$, where $exp_{i}$ denotes the exponent of three in the canonical factorization of $i$.

Normally, it is hard to determine an explicit expression of the compositional inverse of a permutation polynomial. However, they obtained the compositional inverse of the permutation trinomial by substituting $k = 2$ and $q = 2$ in the above theorem. Similarly, for $q = 2$ and $k = 1$.

Motivated by the work of Ding et al. ^[19], Li et al. ^[40] constructed four classes of permutation trinomials over $F_{2^{m}}$. In the following theorems, they considered two classes of trinomials with non-zero trivial coefficients.

Theorem 3.6. ^[40] Let $q = 2^{2k}$ and $k$ be a positive integer. Then, $f(x) = x+x^{2^{k}}+x^{2^{2k-1}-2^{k-1}+1}$ is a permutation trinomial over $F_{q}$ if and only if $k\not\equiv0\pmod{3}$.

They only mentioned a sufficient condition to be a permutation trinomial in the preceding Theorem 3.6. Later, Gupta and Sharma ^[26] observed that, as a consequence of Theorem 3.10, case 1 the above-mentioned Theorem 3.6 reduces to

which is a permutation trinomial over $F_{2^{2k}}$ if and only if $\gcd(m, 3) = 1$.

Theorem 3.7. ^[40] Let $q = 2^{2k}$ and $k > 0$ be an odd integer. Then, $f(x) = x+x^{2^{k}+2}+x^{2^{2k-1}+2^{k-1}+1}$ is a permutation trinomial over $F_{q}$.

Nevertheless, Gupta and Sharma ^[26] enhanced Theorem 3.7 by asserting that

is a permutation trinomial over $F_{q}$ if and only if $k$ is odd.

By constructing the fractional polynomial $g(x)$, Li et al. ^[39] observed that Theorem 3.7 can be improved for $q = 2^{k}$ and $\gcd(3, k) = 1$; then,

for $r = 2$ and $h(x) = 1+x^{2}+x^{-1}$; the fractional polynomial becomes

which permutes $\mu_{q+1}$ when $k\not\equiv0\pmod{3}$. In the next two theorems, they introduced two new classes of permutation trinomials of the form $f(x) = x+ax^{\alpha}+bx^{\beta}$ over $F_{q}$, where $a, b\in F_{q}^{*}$.

Theorem 3.8. ^[40] Let $q = 2^{2k}$ and $k > 0$ be an integer:

where $a\in F_{q}$ and the order of $a$ is $2^{k}+1$. Then, $f(x)$ is a permutation trinomial over $F_{q}$.

Theorem 3.9. ^[40] Let

where $a\in F_{q}$. Then, $f(x)$ is a permutation trinomial over $F_{q}$.

Later, Gupta and Sharma ^[26] presented four new classes of permutation trinomials of the form $x^{r}h(x^{2^{m}-1})$ over $F_{2^{2m}}$. Among those four classes of permutation trinomials, two classes give necessity conditions for Theorem 3.6 and Theorem 3.7.

Theorem 3.10. ^[26]

(1) The polynomial $f(x) = x^{4}+x^{2^{m}+3}+x^{3\cdot2^{m}+1}\in F_{2^{2m}}[x]$ is a permutation polynomial over $F_{2^{2m}}$ if and only if $\gcd(m, 3) = 1$.

(2) The polynomial $f(x) = x^{2}+x^{2\cdot2^{m}}+x^{3\cdot2^{m}-1}\in F_{2^{2m}}[x]$ is a permutation polynomial over $F_{2^{2m}}$ if and only if $\gcd(m, 3) = 1$.

(3) The polynomial $f(x) = x^{5}+x^{2^{m}+4}+x^{4\cdot2^{m}+1}\in F_{2^{2m}}[x]$ is a permutation polynomial over $F_{2^{2m}}$ if and only if $m$ is odd.

(4) The polynomial $f(x) = x^{3}+x^{3\cdot2^{m}}+x^{2^{m+2}-1}\in F_{2^{2m}}[x]$ is a permutation polynomial over $F_{2^{2m}}$ if and only if $m$ is odd.

In addition to the above four classes of permutation trinomials, they also stated the following two conjectures on permutation trinomials.

Conjecture 3.1. ^[26] The polynomial

is a permutation trinomial over $F_{2^{2m}}$ if and only if $m\equiv2\pmod{4}$.

Conjecture 3.2. ^[26] The polynomial

is a permutation trinomial over $F_{2^{2m}}$ if and only if $m\equiv2\pmod{4}$.

Wu et al. ^[74] observed that the above Conjecture 3.2 can be written as

which permutes $F_{2^{2m}}$ if and only if $\gcd(5+2k, 2^{m}-1) = 1$ and $2/m$ for $m > 0, k > 0$. Using this, the authors determined some of the similar permutation trinomials. Later, Zha et al. ^[80] also proved the above two conjectures. Based on the work done by Gupta and Sharma ^[26], Zha et al. ^[80] investigated permutation trinomials of the form $x^{r}h(x^{2^{m}-1})$ over $F_{2^{2m}}$ by applying a bijection over the unit circle of $F_{2^{2m}}$ with order $2^{m}+1$. And, from Conjecture 3.1, they derived new permutation trinomials.

Theorem 3.11. ^[80] The trinomial

is a permutation trinomial over $F_{2^{2m}}$ if and only if $m\equiv2\pmod{4}$.

Subsequently, they demonstrated that Conjecture 3.2 was correct, and they obtained the following classes of permutation trinomials from Conjecture 3.2.

Theorem 3.12. ^[80]

(1) $x^{3}+x^{2^{m+1}+1}+x^{3\cdot2^{m}}\in F_{2^{2m}}$ permutes $F_{2^{2m}}$ if and only if $m$ is odd.

(2) $x^{3}+x^{2^{m}+2}+x^{3\cdot2^{m}}\in F_{2^{2m}}$ permutes $F_{2^{2m}}$ if and only if $m$ is odd.

(3) $x^{5}+x^{4\cdot2^{m}+1}+x^{5\cdot2^{m}}\in F_{2^{2m}}$ permutes $F_{2^{2m}}$ if and only if $m\equiv2\pmod{4}$.

(4) The mapping $g_{5}(x) = x^{5}(1+x+x^{5})^{2^{m}-1}$ permutes $\mu_{2^{m}+1}$ if and only if $m$ is even.

They established a connection between two families of permutation polynomials over $F_{2^{2m}}$ in the following theorem.

Theorem 3.13. ^[80] Let $r, l$ be integers and $m$ be even with $\gcd(r, 2^{m}-1) = 1$. Assume the following:

The polynomial

permutes $F_{2^{2m}}$ if and only if $\gcd(r+2l, 2^{m}-1) = 1$ and the polynomial $f(x) = x^{r}h(x^{2^{m}-1})$ permutes $F_{2^{2m}}$.

Fernando ^[23] extracted permutation trinomials from reversed Dickson polynomials of the $(k+1)^{th}$ kind when $n = p^{l}+2$ and $p > 3$, where $l\in N$.

Theorem 3.14. ^[23] Let $p > 3$ be an odd prime and $q = p^{e}$, where $e$ is a non-negative integer. Let $k$ be an integer such that $k\not = 0, 2, 4$ and $0\leq k\leq p-1$. Let

Then, $f(x)$ is a permutation polynomial of $F_{q}$ if and only if $l = 0$ and $k\not = 3$.

Through the study of the number of solutions of special equations, Ma et al. ^[52] constructed one class of complete permutation trinomials:

for any odd prime $p$ over $F_{p^{3m}}$. They also proved that

is a permutation polynomial over $F_{p^{3m}}$ if and only if

applies over $F_{p^{3m}}$. Using the multivariate method introduced by Dobbertin ^[21], they constructed the two classes of trinomial permutations over $F_{2^{m}}$ listed below.

Theorem 3.15. ^[52] Let $m > 1$ be an odd integer, and write $k = \frac{m+1}{2}$. Then, for each $u\in F_{2^{m}}^{*}$, the following trinomials are permutation polynomials over $F_{2^{m}}:$

(1) $f(x) = x+u^{2^{k-1}-1}x^{2^{k}-1}+u^{2^{k-1}}x^{2^{k}+1}$,

(2) $f(x) = x+ux^{2^{k}-1}+u^{2^{k}}x^{2^{m}-2^{k+1}+2}$.

For the field of characteristic 2, let $q = 2^{k}$ and $f(x) = x^{r}h(x^{q-1})\in F_{q^{2}}$, where

then, $f(x)$ permutes $F_{q^{2}}$ if and only if $\gcd (r, q-1) = 1$ and $g(x) = x^{r}h(x)^{q-1}$ permutes $\mu_{q+1}$. The fractional polynomial of $g(x)$ is

using this specified equation, Li et al. ^[39] constructed several classes of permutation trinomials of the form $x^{r}h(x^{(p^{m}-1)/d})$ with $m = 2k$ and $d = p^{k}+1$ for $p = 2, 3$, and they proposed Conjectures 3.3 and 3.4.

By construction of the fractional polynomial and Lemma 1.2 the next theorem was developed and it is a generalization of Corollary 3.2. For the next theorem, $r = 3+(q+1)l$ in the first case and $r = 2+(q+1)l$ in the second case were considered.

Theorem 3.16. ^[39] Let $q = 2^{k}$; then, $f(x) = x^{r}h(x^{q-1}) = x^{a}+x^{b}+x^{c}$ is a permutation trinomial of $F_{q^{2}}$ in the following cases:

(1) $a = lq+l+3$, $b = (l+4)q+l-1$ and $c = (l-1)q+l+4,$ when $k$ is even and $\gcd(2l+3, q-1) = 1$,

(2) $a = lq+l+2$, $b = (l+2)q+l$ and $c = (l-1)q+l+3,$ when $k\not\equiv0\pmod{3}$ and $\gcd(l+1, q-1) = 1$.

In the following theorem, they considered $f(x) = x^{r}h(x^{q-1})$, where $h(x) = 1+x^{4}+x^{-1}$ and $r = 2+(q+1)l$, and they obtained new class of permutation trinomial.

Theorem 3.17. ^[39] Let $q = 2^{k}$. Then,

is a permutation trinomial over $F_{q^{2}}$ when $k\equiv2, 4\pmod{6}$ and $\gcd(l+1, q-1) = 1$.

In the following theorem, they considered $f(x) = x^{r}h(x^{q-1})$, where $h(x) = 1+x^{3}+x^{-1}$ and $r = 3+(q+1)l$ in the first case and $r = 1+(q+1)l$ in the second case.

Theorem 3.18. ^[39] Let $q = 2^{k}$; then, $f(x) = x^{r}h(x^{q-1}) = x^{a}+x^{b}+x^{c}$ is a permutation trinomial over $F_{q^{2}}$ in the following cases:

(1) $k\not\equiv2\pmod{4}$, $a = lq+l+3$, $b = (l+3)q+l$, $c = (l-1)q+l+4$ and $\gcd(2l+3, q-1) = 1$.

(2) $k$ is even, $a = lq+l+1$, $b = (l+3)q+l-2$, $c = (l-1)q+l+2$, $l\geq0$ is an integer and $\gcd(2l+1, q-1) = 1$.

Theorem 3.19. ^[39] Let $q = 2^{k}$, $r = 1+(q+1)l$, where $\gcd(2l+1, q-1) = 1$, and $h(x) = 1+x^{4}+x^{-2}$. Then,

is a permutation trinomial over $F_{q^{2}}$ if $k\not\equiv0\pmod{3}$.

Following the method described by Hou ^[30], Li et al. ^[39] obtained two permutation trinomials over $F_{3^{2k}}$ with fixed exponents by applying $Tr(x)$ and $N(x)$. Then, they generalized these trinomials to two classes of permutation trinomials with one parameter in each class.

They obtained the following two classes of trinomials over $F_{3^{2k}}$, of the form $f(x) = x^{r}h(x^{q-1})$, by using the fractional polynomial method. In the following theorem, the first case was considered for $h(x) = 1-x^{2}+x^{-2}$; they later computed the fractional polynomial $g(x)$, which permutes $\mu_{q+1}$ when $k\not\equiv0\pmod{4}$, as does $f(x)$. Similarly, in the second case of $r = 1$, $h(x) = 1+x^{3}+x^{-1}$, and the fractional polynomial $g(x)$ permutes $\mu_{q+1}$ when $k$ is odd; this is also true for $f(x)$.

Theorem 3.20. ^[39] Let $q = 3^{k}$. Then,

(1) when $k\not\equiv0\pmod{4}$, $f(x) = x-x^{2q-1}+x^{q^{2}-2q+2}$ is a permutation trinomial over $F_{q^{2}}$,

(2) when $k$ is odd, $f(x) = x-x^{3q-2}+x^{q^{2}-q+1}$ is a permutation trinomial over $F_{q^{2}}$.

Together with the above classes of permutation trinomials over $F_{q^{2}}$, they also proposed the following two conjectures. They verified Conjecture 3.4 by using MAGMA for $1\leq k\leq6$, and they observed that, by using Lemma 1.2, one can prove Conjecture 3.4.

Conjecture 3.3. ^[39]

(1) Let $q = 3^{k}$, $k$ be even and $f(x) = x^{lq+l+5}+x^{(l+5)q+l}-x^{(l-1)q+l+6}$, where $\gcd(5+2l, q-1) = 1$. Then, $f(x)$ is a permutation polynomial over $F_{q^{2}}$.

(2) Let $q = 3^{k}$ and $f(x) = x^{lq+l+1}-x^{(l+4)q+l-3}-x^{(l-2)q+l+3}$, where $\gcd(1+2l, q-1) = 1$. Then, $f(x)$ is a permutation polynomial over $F_{q^{2}}$.

(3) Let $q = 3^{k}$ and $f(x) = x^{lq+l+1}+x^{(l+2)q+l-1}-x^{(l-2)q+l+3}$, where $\gcd(1+2l, q-1) = 1$. Then, $f(x)$ is a permutation polynomial over $F_{q^{2}}$ if $k\not\equiv2\pmod{4}$.

Conjecture 3.4. ^[39]

(1) Let $q = 3^{k}$, $k$ be even and $g(x) = \frac{-x^{7}+x^{3}+x}{x^{6}+x^{4}-1}$. Then, $g(x)$ permutes $\mu_{q+1}$.

(2) Let $q = 3^{k}$ and $g(x) = \frac{x^{6}+x^{4}-1}{-x^{7}+x^{3}+x}$. Then, $g(x)$ permutes $\mu_{q+1}$.

(3) Let $q = 3^{k}$ and $g(x) = \frac{-x^{5}+x^{3}+x}{x^{4}+x^{2}-1}$. Then, $g(x)$ permutes $\mu_{q+1}$ if $k\not\equiv2\pmod{4}$.

Remark 3.2. According to our observation, all three parts of Conjecture 3.4 have been resolved. In a way, Conjecture 3.3 can be considered to be resolved as well since it is equivalent to Conjecture 3.4 based on Lemma 1.2, although no direct proof is available in the literature.

Using the approach involving the resultant of two polynomials and rational points on curves, Bartoli and Giulietti ^[5] proved the first part of Conjecture 3.4. Later, by analyzing the quadratic factors of the corresponding fifth and seventh-degree equations over $F_{3^{2k}}$, Li ^[41] proved the last two cases of Conjecture 3.4. He used the theory of the existence of a unique solution in $\mu_{q+1}$ for the equation $g(x) = t$ for any $t\in\mu_{q+1}$. Also, using the technique introduced in ^[39], Liu and Sun ^[49] partially settled parts (2) and (3) of Conjecture 3.4. To prove these two parts, the authors first determined two classes of permutation trinomials of the form $f(x) = xh(x)^{q-1}$ over $F_{q^{2}}$, which are listed below. They considered $f(x) = xh(x)^{q-1};$ in the first case of the below theorem, $f(x)$ can be written as $x(1-x^{4}+x^{-2})^{q-1}$, and the corresponding fractional polynomial is

which permutes $\mu_{q+1}$ if $m\not\equiv0\pmod{3}$. Similarly, in the second case of the below theorem, $f(x)$ can be written as $x(1-x^{2}+x^{-2})^{q-1};$ the corresponding fractional polynomial is

which permutes $\mu_{q+1}$ if $m$ is odd.

Theorem 3.21. ^[49] Let $q = 3^{m}$. Then, the following holds:

(1) The polynomial $f(x) = x-x^{4q-3}+x^{3-2q}$ is a permutation trinomial over $F_{q^{2}}$ if $m\not\equiv0\pmod{3}$.

(2) The polynomial $f(x) = x-x^{2q-1}+x^{3-2q}$ is a permutation trinomial over $F_{q^{2}}$ if $m$ is odd.

Bhattacharya and Sarkar ^[10] constructed permutation binomials and trinomials. They extracted permutation trinomials of the form

where $s$ and $t$ are positive integers from permutation binomials of the form

They used Hermite-Dickson criteria, the Lucas theorem and Wan-Lidl criteria to get the following trinomial.

Theorem 3.22. ^[10] Let $s$ and $t$ be positive integers. Then, the polynomial

is a permutation polynomial over $F_{2^{t}}$ if and only if $t\alpha = 1$, $s = \{1, 2\}$ and $t$ is odd.

Wu et al. ^[74] explicitly determined all permutation trinomials over $F_{2^{m}}$ from the results of Zieve's paper ^[86]; they also proved the conjecture proposed in ^[26]. All of the explicit trinomials were obtained from

by simplifying the coefficients of $x^{(n-k)(q-1)}$ and $x^{k(q-1)}$ from the polynomial $(\gamma x^{q-1}-\beta)^{n}-\gamma(x^{q-1}-\gamma^{q}\beta)^{n}$. To construct the permutation polynomial $f_1$, they considered

using the Lucas formula, they came to know that there are $\prod\limits_{i = 0}^{l}(n_{i}+1)$ integers $k$ with $0\leq k\leq n$ and $\binom{n}{k}\not\equiv0\pmod{p}$. Then, by simplifying and considering all cases on $\gamma$ in

they ended up with conditions which are mentioned in the following theorem. Following a similar procedure for the polynomial

they obtained $f_{2}$.

Theorem 3.23. ^[74] Let $m > 0$ and $k$ be an integer, $q = 2^{m}$. Then, the polynomial

permutes $F_{q^{2}}^{*}$ if and only if $\gcd(3+2k, q-1) = 1$. The polynomial

permutes $F_{q^{2}}^{*}$ if and only if $\gcd(6+2k, q-1) = 1$.

Using Conjecture 3.2, Wu et al. determined some similar permutation trinomials in the following theorems.

Theorem 3.24. ^[74] Let $m > 0, k > 0$ and $q = 2^{m}$. Then, the polynomials of the form

permute $F_{2^{2m}}$ if and only if $\gcd(n+2k, 2^{m}-1) = 1$, when

(1) $l_{1} = 4$, $l_{2} = 5$ and $2\mid m$,

(2) $l_{1} = 2$, $l_{2} = 3$ and $3\not\mid m$,

(3) $l_{1} = 1$, $l_{2} = 3$ and $3\not\mid m$,

for $n = 5, 4, 2,$ respectively.

Li et al. ^[43] constructed four classes of permutation trinomials over $F_{q^{2}}$ using Lemma 1.2 and the fraction polynomial method. In the first case of the following theorem, they considered polynomials of the form

where $h(x) = 1+x^{6}+x^{-2}$, and, in the second case, they considered $f(x) = xh(x^{q-1})$, where

Later, Zheng et al. ^[81] listed all of the necessary values of $s$ for these classes of permutation trinomials.

Theorem 3.25. ^[43] Let $q = 2^{k}$. Then the following holds:

(1) If $k\geq1$, $l$ is an integer and $f(x) = x^{lq+l+3}+x^{(l+6)q+l-3}+x^{(l-2)q+l+5}$, then $f(x)$ is a permutation trinomial over $F_{q^{2}}$ if and only if $\gcd(3+2l, q-1) = 1$ and $k\not\equiv0\pmod{4}$.

(2) If $k$ is odd and $f(x) = x+x^{\frac{q^{2}-3q+5}{3}}+x^{\frac{2q^{2}-3q+4}{3}}$, then $f(x)$ is a permutation trinomial over $F_{q^{2}}$.

(3) If $k\not\equiv1\pmod{3}$, then $f(x) = x+x^{{q^{2}-q+q}}+x^{{q^{3}-q^{2}+q}}$ is a permutation trinomial over $F_{q^{3}}$.

(4) If $k\not\equiv1\pmod{3}$, then $f(x) = x+x^{{q^{2}}}+x^{{q^{3}-q^{2}+q}}$ is a permutation trinomial over $F_{q^{3}}$.

Using the Niho exponents, Bai and Xia ^[2] investigated permutation trinomials of the form

for $p = 3, 5$ and a positive integer $k$. They verified that $(p-1)q+1$, $pq$ and $q+(p-1)$ are Niho exponents over $F_{p^{2k}}$. They also considered general polynomials of the form

where $l$ is a non-negative integer and $\gcd(2l+p, q-1) = 1$. The authors concluded that the same two polynomials may not be permutation polynomials when $p > 5$. They proved the following theorems by using an idea that originated from ^[30] and was used in ^[39].

Theorem 3.26. ^[2] Let $q = p^{k}$ and

be the trinomial. Then, for $p = 3$ or $p = 5$, $f(x)$ is a permutation polynomial of $F_{q^{2}}$ if and only if $k$ is even.

Theorem 3.27. ^[2] Let $q = p^{k}$ with $p\in\{3, 5\}$ and $l$ be a non-negative integer satisfying $\gcd(2l+p, q-1) = 1$. Then,

is a permutation polynomial of $F_{q^{2}}$ if and only if $k$ is even.

Bartoli and Zini ^[5] constructed permutation trinomials of the form $x^{2p^{s}+r}+x^{p^{s}+r}+\lambda x^{r}$ over $F_{p^{t}}$, for the case that $2p^{s}+r < p^{t}$, and it was an extension of Bhattacharya and Sarkar's ^[10] work when $p = 2$ and $r = 1$. The authors characterized certain classes of permutation trinomials when $s$ and $r$ are non-negative integers given that $\lambda \in F_{p^{t}}$ and $f_{\lambda}(x) = x^{2p^{s}+r}+x^{p^{s}+r}+\lambda x^{r}$ in $F_{p^{t}}[x]$. If $r = 0$, then $d = 0$. If $r\not = 0$, write $r = p^{u}v$ with $u\geq0$ and $p\not\mid v$; then, $d = 2p^{s-u}+v$ if $u\leq s$ and $d = 2+p^{u-s}v$ if $u > s$, that is,

Theorem 3.28. ^[5] Assume that $d^{4} < p^{t}$. Then, $f_{\lambda}(x)$ is a permutation polynomial of $F_{p^{t}}$ if and only if one of the following cases holds:

(1) $p = 2$, $t$ is odd and $f_{\lambda}(x) = x^{3}+x^{2}+x$ or $f_{\lambda}(x) = x^{5}+x^{3}+x$,

(2) $p\equiv2\pmod{3}$, $t$ is odd and $f_{\lambda}(x) = x^{3}+x^{2}+\frac{1}{3}x$.

There are some papers that make considerable use of the methodology based on the relationship between permutation polynomials and algebraic curves; however, the most challenging aspect of such a methodology is the investigation of the singular points of the curves. However, difficulty can be minimized by considering the algebraic curve $C$: $F(X, Y) = 0$, defined over $F_{q}$, and demonstrating that $C$ has no completely irreducible components over $F_{q}$. Using this technique, Bartoli and Timpanella ^[7] determined permutation trinomials of the form

over $F_{q^{2}}$, where $q = 2^{2s+1}$ and $n$ and $m$ are odd.

To show that $C$ has no absolutely irreducible components over $F_{q}$, Bartoli and Timpanella investigated singular points of $C$, and they proved that $C$ splits into two components that share no common irreducible components whose degrees are close enough. By finding the lower bound on $(degA)(degB)$, later, by using Bezout's theorem ^[24,57], they arrived at the final conclusion by contradiction.

Theorem 3.29. ^[7] Let $(m, q+1) = 1$. Suppose that $F_{A, B, m, n}(X)$ is a permutation polynomial of $F_{q^{2}}$. Let $n\equiv i\pmod{8}$ and $m\equiv j\pmod{8}$. Then, one of the following conditions is satisfied:

(1) $2(n+m)\geq d < \sqrt[4]{q}$,

(2) $B\in F_{q}$,

(3) $A^{2}+B^{q+1} \not = 0$, $B\not\in F_{q}$ and $n < 5m$,

(4) $A^{2}+B^{q+1} = 0$, $B\not\in F_{q}$, $(i, j)\in\{(1, 5), (3, 7), (5, 1), (7, 3)\}$ and $n < 38m$,

(5) $A^{2}+B^{q+1} = 0$, $B\not\in F_{q}$ and $(i, j)\not\in\{(1, 5), (3, 7), (5, 1), (7, 3)\}$.

Kyureghyan and Zieve ^[37] determined permutation polynomials of the form $x+\gamma Tr_{q^{2}/q}(x^{(q^{2}+1)/4})$ over $F_{q^{2}}$, where $\gamma\in F_{q^{2}}$ satisfies $(2\gamma)^{(q+1)/2} = 1$, which is equivalent to

Similarly, following Theorem 4.3, Qin and Yan ^[58] constructed permutation trinomials with the index $q+1$ over $F_{q^{2}}$ by using monomials of $\mu_{(q+1)/2}$ and $-\mu_{(q+1)/2}$ to study the permutation property of $x^{r}h(x)^{q-1}$ on $\mu_{q+1}$. In the following theorem, they characterized more generalized permutation trinomials over $F_{q^{2}}$.

Theorem 3.30. ^[58] Let $c\in F_{q^{2}}$ satisfy that $(c/2)^{(q+1)/2} = 1$, and let $k$ be an integer. Let $\gcd(r, q^{2}-1) = 1$ and $\gcd(2r-2k-1, (q+1)/2) = 1$. Then, the polynomial $x^{r}(c+x^{s}+x^{qs-1})$ is a permutation trinomial in the following cases:

(1) $s = \frac{q^{2}+2q-3+4k(q-1)}{4}$,

(a) $q\equiv1\pmod{4}$,

(b) $q\equiv1\pmod{8}$ and $k$ is an odd integer.

(2) $s = \frac{3q^{2}+2q-5+4k(q-1)}{4}$,

(a) $q\equiv1\pmod{4}$,

(b) $q\equiv5(\pmod{8})$ and $k$ is an odd integer.

Furthermore, motivated by the idea of Li et al. ^[43], Qin and Yan ^[58] determined several classes of permutation trinomials of the form $x^{r}(1+x^{s_{1}(q-1)}+x^{s_{2}(q-1)})$ over $F_{q^{2}}$ with the index $q+1$, where $q = 2^{k}$ and $k$ is odd. In addition, they considered another kind of permutation trinomial of the form $x^{r}(c-x^{s_{1}(q-1)}+x^{s_{2}(q-1)})$ over $F_{q^{2}}$, which is listed below.

Theorem 3.31. ^[58] Let $c\in F_{q^{2}}$ and $k$ be an odd integer. Let $\gcd(r, q^{2}-1) = 1$ and $\gcd(2r-2k-1, (q+1)/2) = 1$. Then, the polynomial $x^{r}(c-x^{s_{1}(q-1)}+x^{s_{2}(q-1)})$ is a permutation trinomial in the following cases:

(1) $s_{1} = \left(\frac{q+3}{4}+k\right)$, $s_{2} = \left(\frac{q^{2}+3q}{4}+k+1\right)$ and $(c/2)^{(q+1)/2} = 1$, $q\equiv5\pmod{8}$,

(2) $s_{1} = \left(\frac{3q+5}{4}+k\right)$, $s_{2} = \left(\frac{3q^{2}+5q}{4}+k+1\right)$ and $(c/2)^{(q+1)/2} = 1$, $q\equiv5\pmod{8}$,

(3) $s_{1} = \left(\frac{3q+5}{4}+k\right)$, $s_{2} = \left(\frac{q^{2}+3q}{4}+k+1\right)$ and $(-c/2)^{(q+1)/2} = 1$, $q\equiv1\pmod{8}$,

(4) $s_{1} = \left(\frac{3q+5}{4}+k\right)$, $s_{2} = \left(\frac{3q^{2}+5q}{4}+k+1\right)$ and $(-c/2)^{(q+1)/2} = 1$, $q\equiv1\pmod{8}$.

Theorem 3.32. ^[58] Let $c\in F_{q^{2}}^{*}$ and $k$ be an integer. Let $\gcd(r, q^{2}-1) = 1$. Then, the polynomial $x^{r}(c-x^{s_{1}(q-1)}+x^{s_{2}(q-1)})$ is a permutation trinomial in the following cases:

(1) $s_{1} = \left(\frac{q+3}{4}+k\right)$, $s_{2} = \left(\frac{3q^{2}+5q}{4}+k+1\right)$ and $q\equiv1\pmod{4}$,

(2) $s_{1} = \left(\frac{3q+5}{4}+k\right)$, $s_{2} = \left(\frac{q^{2}+3q}{4}+k+1\right)$ and $q\equiv1\pmod{4}$.

4.   Polynomials of the form $\lambda_{1}x^{a}+\lambda_{2}x^{b}+\lambda_{3}x^{c}$

The permutation polynomials of the form $\lambda_{1}x^{a}+\lambda_{2}x^{b}+\lambda_{3}x^{c}$ are included in this section. One can see that the majority of polynomials of any form can be expressed in the form of $x^{r}h(x^{s})$. In this case, there are also a lot of polynomials of the form $\lambda_{1}x^{a}+\lambda_{2}x^{b}+\lambda_{3}x^{c}$ that can be expressed in the form of $x^{r}h(x^{s})$, but we are separately listing this form for the benefit of future research so that it will be easier to obtain new classes by referring to this article.

Using the multivariate method introduced by Dobbertin ^[21], Wang et al. ^[71] constructed six new classes of permutation trinomials over $F_{2^{n}}$. In the following theorem, all five classes of permutation trinomials over $F_{2^{n}}$ when $n\equiv0, 1, 2\pmod{3}$ are listed.

Theorem 4.1. ^[71] Let $k$ be a positive integer. Then, $f(x) = x^{a}+x^{b}+x$ is a permutation polynomial over $F_{2^{n}}$ if, for the case that $n\equiv0\pmod{3}$,

(1) $k\not\equiv2\pmod{3}$ for $a = 2^{2k}+2^{k}-1$, $b = 2^{2k}$,

(2) $k\not\equiv2\pmod{3}$ for $a = 2^{2k}+2^{k}-1$, $b = 2^{2k}+2^{k}-1;$

for the case that $n\equiv1\pmod{3}$,

$\ \ \ a = 2^{2k+1}+2^{k+1}+1$, $b = 2^{k+1}+1;$ for the case that $n\equiv2\pmod{3}$ and $n = 3k-1$,

(1) $a = 2^{3k-1}-2^{2k}+2^{k}$, $b = 2^{k}-1$,

(2) $a = 2^{2k}+2^{k}+1$, $b = 2^{2k}+1$.

In the following theorem, the last class of a permutation trinomial over $F_{2^{n}}$ with $n\equiv0\pmod{4}$ is stated.

Theorem 4.2. ^[71] Let $k$ be an odd integer, and let $m, n, d$ be positive integers satisfying

Then, $f(x) = x^{d}+x^{2^{2m}}+x$ is a permutation polynomial over $F_{2^{n}}$.

Yuan ^[78,79] determined many classes of permutation trinomials of the form $cx-x^{s}+x^{qs}$ by using the AGW criterion. The obtained permutation polynomials were derived from a linear bijection between subsets $S$ and $\bar{S}$ of a finite field, as well as their permutation behavior, which was related to $\delta$. Using this construction as their reference, Zheng et al. ^[81] looked into the relationship between permutation trinomials of the form $cx-x^{s}+x^{qs}$ and permutation polynomials of the form $(x^{q}-x+\delta)^{s}+cx$, restricting them on $\delta$ over $F_{q^{2}}$ (see ^[81]HY__HY, Proposition 3]). There is no inclusion of subsets or their bijections in the proposed relationship between these two varieties of permutation polynomials. Furthermore, they stated that a variety of classes of permutation polynomials of the form $(x^{q}-x+\delta)^{s}+cx$ without a restriction on $\delta$ can be constructed on the basis of this relation. In addition, they constructed the four classes of permutation trinomials of the form $cx-x^{s}+x^{qs}$ over $F_{q^{2}}$ that are listed below.

Theorem 4.3. ^[81] Let $c\in F_{q^{2}}^{*}$. The polynomial $cx-x^{s}+x^{qs}$ permutes $F_{q^{2}}$ in each of the following cases:

(1) $s = \frac{3q^{2}+2q-1}{4}$,

(a) $q\equiv1\pmod{8}$ and $\left(-\frac{2}{c}\right)^{\frac{q+1}{2}} = 1$,

(b) $q\equiv5\pmod{8}$ and $\left(\frac{2}{c}\right)^{\frac{q+1}{2}} = 1$.

(2) $s = \frac{(q+1)^{2}}{4}$,

(a) $q\equiv5\pmod{8}$ and $\left(-\frac{2}{c}\right)^{\frac{q+1}{2}} = 1$,

(b) $q\equiv1\pmod{8}$ and $\left(\frac{2}{c}\right)^{\frac{q+1}{2}} = 1$.

(3) $s = \frac{q^{2}+q+1}{3}$, $c = 1$ and $q\equiv1\pmod{3}$.

Theorem 4.4. ^[81] Let $q$ be power of an odd prime and $s = q^{3}+q^{2}-q$. Then, the polynomial $f(x) = x-x^{s}+x^{q^{2}s}$ permutes $F_{q^{4}}$.

Li et al. ^[44] proposed several classes of complete permutation binomials over a finite field based on certain polynomials over its subfields or subsets. In addition, a class of complete permutation trinomials with Niho exponents was studied, and the number of these complete permutation trinomials was also determined. They found that certain polynomials over $F_{2^{n}}$ can have the same properties those of $ax^{k}+bx$ over $F_{2^{m}}$ for a positive divisor $m$ of $n$ such that $\frac{n}{m}$ is odd. Later, they presented complete permutation trinomials of the form $ax^{p^{2m}-p^{m}+1}+a^{p^{m}}x^{p^{m}}+(a^{p^{m}+1}+1)x$ with Niho-type exponents. Liu ^[48] modified the conditions and generalized the results based on the results presented in ^[44]; they also showed that the relationship exists between two permutation polynomials proposed by Zheng et al. ^[81].

The following corollary is a generalization of the result stated in ^[81].

Corollary 4.1. ^[48] For a positive integer $m$ and $c\in F_{2^{m}}^{*}$, the polynomial

is a permutation over $F_{2^{3m}}$.

In the next proposition, Liu ^[48] considers an arbitrary value for $c$ and generalizes the result, which was constructed with the coefficient 1 in Theorem 4.3 case 3 by Zheng et al. ^[81].

Proposition 4.1. ^[48] Let $q = 2^{m}$ be even with $s = \frac{q^{2}+q+1}{3}$, and let $q\equiv1\pmod{3}$. Then, the polynomial $cx+x^{s}+c^{q}x^{qs}$ permutes $F_{q^{2}}$ for $c\in F_{q^{2}}^{*}$ satisfying $Tr_{1}^{m}(c^{q+1}) = 0$.

The binomial $ax^{k}+bx$ over $F_{2^{m}}$, which was proved for the case of an odd value of $m$ in ^[44], Liu considered the same binomial and proved it to be a permutation trinomial for all positive values of $m$ by using different methods.

When we go through the permutation polynomials' application part, involve the concept of differential uniformity, which is crucial to the S-box. On the way to determining the permutation polynomial's differential uniformity, Peng et al. ^[56] constructed permutation trinomials of the form $x^{2^{k+1}+3}+ax^{2^{k}+2}+bx$ over $F_{2^{2k}}$ and established its differential uniformity by using MAGMA. They observed that $x^{2^{k+1}+3}+ax^{2^{k}+2}+bx\in F_{2^{2k}}$ permutes $F_{2^{2k}}$ if and only if

permutes $F_{2^{k}}$. Furthermore, $g(x)$ permutes $F_{2^{k}}$ if and only if $g(x) = x^{5}$ for $k\equiv2\pmod{4}$ or $g(x) = x^{5}+ax^{3}+a^{2}x$, $a\in F_{2^{k}}$ for odd values of $k$. Later, Peng et al.^[56] conjectured that the differential uniformity of the permutation trinomials is 12 and 10 for both cases when $k\geq5$.

Sharma and Gupta ^[60] constructed permutation trinomials of the form $ax+bx^{q+2}+x^{2q+3}$ and $ax^{2}+bx^{q+3}+x^{2q+4}$. The polynomial

can be written as $xh(x^{q+1})$, where $h(x) = x^{2}+bx+a$. For any $\alpha\in F_{q},$

it can be further simplified as

Using this simplification, they proved that $f(x)$ permutes $F_{q^{2}}$ if and only if $g(x)$ permutes $\mu_{q+1}$ if and only if $g(x)$ permutes $F_{q}$, which is subject to the condition of if and only if $G(x)$ permutes $F_{q}$. Following certain assumptions on coefficients $a$ and $b$, they proved the following theorems.

Theorem 4.5. ^[60] Let $q = 2^{m}$. The polynomial $f(x) = ax+bx^{q+2}+x^{2q+3}$ permutes $F_{q^{2}}$ if and only if one of the following conditions is satisfied:

(1) $m = 1$, $a\not = b+1$, $a_{2} = 0$ and either $a_{3} = 1$ or $a_{1} = a_{4}$,

(2) $m\geq3$, $m\not\equiv0\pmod{4}$, $a_{2} = a_{3} = 0$ and $a_{1}^{4} = a_{4}$,

(3) $m\geq3$ is odd, $a_{1}a_{2}+a_{3} = 0$ and $a_{2}^{2} = a_{1}^{4}+a_{1}^{2}a_{2}+a_{4}$.

In the above theorem, under the condition that $a\in F_{q}^{*}$, $f(x)$ permutes $F_{q^{2}}$ if and only if either (1) $m = 1$ and $a\not = b+1$ or (2) $m\geq3$ is odd, either $b^{q-1} = 1$ and $a = b^{2}$, or $a = b^{q+1}$ and $b^{2(q-1)}+b^{q-1}+1 = 0$. Similarly, under the condition that $b\in F_{q}^{*}$, $f(x)$ permutes $F_{q^{2}}$ if and only if either (1) $m = 1$ and $a\not = b+1$; (2) $m = 1$, $a\not = b+1$ and $a+a^{4} = b^{2}$ or (3) $m\geq3$ is odd, $a^{q-1} = 1$ and $a = b^{2}$.

Similarly, Sharma and Gupta ^[60] determined the necessary and sufficient conditions for the coefficients when $q = 3^{m}$, $q = 5^{m}$ and $q = p^{m}$, $p > 5$ for the permutation polynomial of the form $f(x) = ax+bx^{q+2}+x^{2q+3}$ over $F_{q^{2}}$. And, they determined conditions on $q$ and coefficients $a, b\in F_{q}$ for the polynomial of the form $ax^{2}+bx^{q+3}+x^{2q+4}$ over $F_{q^{2}}$. The polynomial

can be written as

where

By Lemma 1.2, $f(x)$ permutes $F_{q^{2}}$ if and only if

permutes $\mu_{q-1} = F_{q}^{*}$. Here, $g(x)$ permutes $F_{q}^{*}$ if and only if $g(x)$ permutes $F_{q}$, which is subject to the condition of if and only if

permutes $F_{q}$; so, by taking different conditions on $q$ and coefficients $a, b\in F_{q}$ over $F_{q^{2}}$, this can be achieved.

A permutation polynomial $f(x)\in F_{q}[x]$ of degree $n$ is said to be normalized if $f(x)$ is monic and $f(0) = 0$, and the coefficient of $x^{n-1}$ equals 0 if $p\not\mid n$, where $p$ is a characteristic of $F_{q}$. By knowing some of the normalized permutation polynomials of degree 5 over $F_{2^{n}}$, Liu et al. ^[50] determined some necessary and sufficient conditions for the coefficients $(b_{1}, b_{2})\in F_{2^{n}}^{2}$ such that

is a permutation polynomial, where $\bar{x}$ is a conjugation of any $x\in F_{2^{2m}}$, which is denoted as $\bar{x} = x^{2^{m}}$. Permutation polynomials that have been determined are presented in the next theorem.

Theorem 4.6. ^[50] For the two positive integers $m, n$ with $n = 2m$, let $(b_{1}, b_{2})\in F_{2^{n}}$, which are not both zero. Then, the polynomial

is a permutation polynomial over $F_{2^{m}}$ under the following condition:

(1) if and only if $m$ is even with $m\equiv2\pmod{4}$, $b_{1} = \theta b_{2}$ and $b_{2}$ as a root of $x^{2}+\theta^{-2}\omega x+\theta^{-4}\omega = 0$, where $\theta\in F_{2^{m}}^{*}$ and $\omega\in F_{2^{m}}^{*}$ is a primitive third root of unity.

It is also true if condition (1) is not satisfied but one of the following two conditions are satisfied when $m$ is odd:

(2) $b_{1} = 0$, $b_{2} = \theta\omega$ or $b_{2} = \theta\omega^{2}$, where $\theta\in F_{2^{m}}^{*}$ and $\omega\in F_{2^{n}}$ is a primitive third root of unity.

(3) $b_{1} = \theta\omega$ or $b_{1} = \theta\omega^{2}$ and $b_{2} = \theta^{2}+\theta\eta\omega^{2}$ or $b_{2} = \theta^{2}+\theta\eta\omega$, where $\theta\in F_{2^{m}}^{*}$, $\eta\in F_{2^{m}}^{*}$ and $\omega\in F_{2^{n}}$ is a primitive third root of unity.

Guo et al. ^[25] observed that, by reversing the method introduced by Tu et al. ^[64], so many new classes of permutation trinomials with a coefficient of 1 can be determined. On that note, they determined the permutation polynomial $x^{d_{1}}+x^{d_{2}}+x^{d_{3}}$ and its compositional inverse for $i = j+m-1$, which is listed below.

Theorem 4.7. ^[25] Let $n = 2m$, $m > 0$, $i, j\in N^{+}$, $I\not = j$, $u\in Z$, $J = 2^{j}$ and $I = 2^{i}$. Suppose that $\gcd(d_{1}, 2^{2m}-1) = 1$, $\gcd(2^{i}-2^{j}, 2^{m}+1) = 1$ and

where $\frac{1}{J-I} \pmod{2^{m}+1}\equiv t$, $0\leq t\leq2^{m}+1$. Then, $x^{d_{1}}+x^{d_{2}}+x^{d_{3}}$ is a permutation polynomial over $F_{2^{n}}$.

Considering the work of Guo et al. ^[25], Zieve ^[87] made remarkable observations and provided another and simpler proof, which demonstrates the general method for producing the permutation polynomials that were introduced in ^[86]. Moreover, Zieve observed that the newly constructed permutation trinomials ^[25] are multiplicatively equivalent to many previous results.

Recently, Xie et al. ^[76] proposed two classes of permutation trinomials over $F_{q^{3}}$ for an arbitrary odd characteristic based on the multivariate method and some suitable manipulation; this was done to solve equations with low degrees over finite fields. In the following theorem, two classes of permutation polynomials are listed. They stated that sufficient conditions in both classes are also necessary, but they were kept as an open problem.

Theorem 4.8. ^[76] Let $q$ be an odd prime power and $a, b\in F_{q^{3}}^{*}$. Then,

(1) $f(x) = ax^{q(q^{2}-q+1)}+bx^{q^{2}-q+1}+2x$ is a permutation polynomial of $F_{q^{3}}$ if one of the following conditions is satisfied:

(a) $ab = 1$ and $a^{q^{2}+q+1}\not = -1$,

(b) $ab\in F_{q}^{*}\backslash\{1\}$ and $a^{q^{2}+q+1}+2ab+b^{q^{2}+q+1} = 0.$

(2) $f(x) = x^{q^{2}-q+1}+ax^{q^{2}}+bx$ is a permutation polynomial of $F_{q^{3}}$ if one of the following conditions is satisfied:

(a) $a^{q}b = 1$ and $a^{q^{2}+q+1}\not = -1$,

(b) $a^{q}b\in F_{q}^{*}\backslash\{1\}$ and $a^{q^{2}+q+1}-2a^{q}b+b^{q^{2}+q+1} = 0$.

Inspired by the work of Wang et al. ^[71], Zheng et al. ^[82] investigated infinite classes of permutation polynomials with the form $f(x) = x+ax^{2^{m}}+bx^{d}$, defined on $F_{2^{n}}$, under the condition that $m$ is any positive integer, $d = \sum\limits_{i = 0}^{\frac{n}{2}}2^{ik}$, with $k$ being an odd positive integer satisfying $\gcd(k, m) = 1$, and $a, b\in F_{2^{m}}^{*}$. And, following the case considered in ^[45],

where $k$ is a positive integer satisfying $1\leq k\leq m-1$ and $\gcd(2^{k}-1, 2^{m}+1) = 1$; also, assuming that $a, b\in F_{2^{m}}^{*}$, they obtained some sufficient conditions for $a$ and $b$ such that

is a permutation polynomial of $F_{2^{n}}$. Also, they observed that

is a permutation polynomial of $F_{q^{2}}$, where $ab\not = 0$; then, $a+b+1\not = 0$. In the following theorem, they proved that $f(x) = x+ax^{2^{m}}+bx^{d}$ permutes $F_{2^{n}}$ when $m$ is any positive integer, $d = \sum\limits_{i = 0}^{\frac{n}{2}}2^{ik}$, with $k$ being an odd positive integer satisfying $\gcd(k, m) = 1$, and $a, b\in F_{2^{m}}^{*}$.

Theorem 4.9. ^[82] Let $k$ be an odd positive integer and $m, n$ and $d$ be positive integers such that

Let $a, b\in F_{2^{m}}^{*}$, with $a+b+1\not = 0$. Then, $f(x) = x+ax^{2^{m}}+bx^{d}$ permutes $F_{2^{n}}$ if one of the following items is satisfied, where $B\in F_{2^{n}}$ such that $B^{2^{k}-1} = a$:

(1) $b = B^{2^{k}}\left(\frac{1}{B}+\frac{1}{B^{2}}+\cdots+\frac{1}{B^{2^{k-1}}}\right)^{2}$ and $Tr_{1}^{m}\left(\frac{b}{a+b+1}\right) = 0$,

(2) $b = B^{2^{k}}\left(\frac{1}{B}+\frac{1}{B^{2}}+\cdots+\frac{1}{B^{2^{k-1}}}\right)^{2}+B^{2^{k}}$ and $Tr_{1}^{m}\left(\frac{b}{a+b+1}\right) = 0$.

Later, they mentioned an open problem based on the above theorem with the same conditions as mentioned above; as with $a\in F_{2^{m}}^{*}$ and $b\in F_{2^{n}}^{*}$, $f(x)$ will be a permutation polynomial for any $B\in F_{2^{m}}$. Recently, Ding and Zieve ^[20] analyzed these open problems by using geometric techniques. In the following theorem, another class of permutation trinomial is described.

Theorem 4.10. ^[82] Let $n = 2m$ and $\gcd(2^{k}-1, 2^{m}+1) = 1$, where $m, k$ are positive integers with $k < m$. Let $d = ord_{2}(\gcd(k, m))$ and $a, b\in F_{2^{m}}^{*}$, with $a+b+1\not = 0$. If

and

then the trinomial

permutes $F_{2^{n}}$, where

In the above theorem, when $k = 1$ and $a\in F_{2^{m}}^{*}$ sufficient conditions are also necessary when $m$ is even, but it is not true when $m$ is odd.

By transforming the permutation problem into a root distribution problem in the unit circle of certain quadratic and cubic equations, Liu ^[51] investigated the permutation behavior of the trinomials over $F_{2^{4m}}$ and quadrinomials over $F_{2^{2m}}$. In the following theorem, he investigated the permutation trinomial of the form

over $F_{2^{4m}}$.

Theorem 4.11. ^[51] Let $m$ be a positive integer; then,

is a permutation trinomial over $F_{2^{4m}}$.

Together with trinomials, he investigated permutation quadrinomials of the form

over $F_{2^{2m}}$, which is mentioned below.

Theorem 4.12. ^[51] Let $n = 2m$ be an even integer, with $m$ being even and $m > 2$. Assume that $a, \mu\in F_{2^{m}}$ satisfy

and

Then, the quadrinomial

is a permutation polynomial over $F_{2^{n}}$.

Meanwhile, Zheng et al. ^[84] constructed permutation quadrinomials of the form

over $F_{2^{2m}}$ for the cases of

5.   Polynomials of the form $f(x)=x+x^{s(q^{m}-1)+1} +x^{t(q^{m}-1)+1}$ with Niho exponents

The Niho exponents were introduced by Niho ^[54] for the case $n = 2m;$ $q = 2$ and $1\leq s, t\leq2^{m};$ a positive integer $d$ is called a Niho exponent with respect to the finite field $F_{2^{n}}$ if $d\equiv2^{j}\pmod{2^{m}-1}$ for some non-negative integer $j$. When $j = 0$, the integer $d$ is called a normalized Niho exponent. The inverse of the normalized Niho exponent is also the normalized Niho exponent if it exists. The product of two normalized Niho exponents is also a normalized Niho exponent. So far, the permutation trinomials of the form

have drawn a lot of interest. The primary objective for this type of polynomial is to identify a pair $(s, t)$ such that

is a permutation polynomial over any given finite field with Niho exponents. For a complete source of information, Li and Zeng ^[46] surveyed some recent advances in the field of Niho exponents.

Li and Helleseth ^[42,45] determined several new classes of permutation trinomials over $F_{2^{n}}$ by finding the pair $(s, t)$ of the polynomial $f(x) = x+x^{s(2^{m}-1)+1} +x^{t(2^{m}-1)+1}$. In the following theorem, they listed a pair $(s, t)$ such that the trinomial $f(x)$ is a permutation over $F_{2^{2m}}$.

Theorem 5.1. ^[42] The known pairs $(s, t)$ that exist such that the trinomials

are permutations over $F_{2^{2m}}$ are as follows:

(1) $(s, t) = (k, -k)$, where $k$ is a positive integer, either $m$ is even or $m$ is odd and $exp_{3}(x)\geq exp_{3}(2^{m}+1)$,

(2) $(s, t) = (2, 2^{m}) = (2, -1)$ for every positive integer $m$,

(3) $(s, t) = (1, 2^{m-1}) = (1, -1/2)$, where $m\not\equiv0\pmod{3}$.

Using Lemma 1.2 and Theorem 5.1, they noticed some of the pairs to be

if $\gcd(2^k-1, 2^{m}+1) = 1$, and

if $\gcd(2^k+1, 2^{m}+1) = 1$, for which

is a permutation trinomial. For both pairs, the first condition of Theorem 5.1 is met. Similarly, they verified Theorem 5.1 for $(s, t) = (1, 1/3)$ and $(1, 2/3)$ when $\gcd(3, 2^{m}+1) = 1$, and $(s, t) = (1, 3/2)$ and $(1/4, 3/4)$ when $m\not\equiv0\pmod{3}$. Furthermore, they constructed some more permutation trinomials over $F_{2^{n}}$ by using Lemma 1.2 and some techniques for solving equations with lower degrees over a finite field.

Theorem 5.2. ^[42] Let $n = 2m$ for even integers $m$. The trinomial

is a permutation of $F_{2^{2m}}$ for the following $(s, t)$ pairs:

(1) $(-1/3, 4/3) = \left(\frac{2^{m+1}+1}{3}, \frac{2^{m}+5}{3}\right)$,

(2) $(3, -1) = (3, 2^{m})$,

(3) $(-2/3, 5/3) = \left(\frac{2^{m}-1}{3}, \frac{2^{m+1}+7}{3}\right)$,

(4) $(1/5, 4/5)$, where $n = 2m$ satisfies that $\gcd(5, 2^{m}+1) = 1$.

In particular, Theorems 5.1 and 5.2 produce many more permutation trinomials over $F_{2^{n}}$, which are described in the following corollary.

Corollary 5.1. ^[42] Let $n = 2m$, $q = 2^{m}$ and $k$ be a positive integer with $\gcd(2k+1, q-1) = 1$. Then, the following trinomials are permutations over $F_{2^{n}}$:

(1) $x^{(q+1)k+1}+x^{(q+1)k+(2q^{2}-q+2)/3}+x^{(q+1)k+(q^{2}+4q-2)/3}$ if $m$ is even,

(2) $x^{(q+1)k+1}+x^{(q+1)k+3q-2}+x^{(q+1)k-q+2}$ if $m$ is even,

(3) $x^{(q+1)k+1}+x^{(q+1)k+1+(q-1)^{2}/3}+x^{(q+1)k+(2q^{2}+5q-4)/3}$ if $m$ is even,

(4) $x^{(q+1)k+1}+x^{(q+1)k+1+l(q-1)}+x^{(q+1)k+1+4l(q-1)}$, where $5l\equiv1\pmod {q+1}$.

By observing the above pairs, they stated the following two conjectures for the pairs $(s, t)$.

Conjecture 5.1. ^[42] Determine the conditions on $s$ such that

is a permutation polynomial over $F_{2^{2m}}$ for $s+t = 1$.

Conjecture 5.1. ^[42] Determine the conditions on the integer $k$ such that

is a permutation polynomial over $F_{2^{2m}}$ for $(s, t) = (2k, -k)$.

Remark 5.1. Conjectures 5.1 and 5.2 are not settled yet; one can try to resolve these two conjectures.

In ^[45], the authors identified even more pairs of $(s, t)$ such that

is a permutation trinomial over $F_{2^{2m}}$. We list those new pairs of $(s, t)$ in the following theorem. For the proof, the authors used the fractional polynomial method and Lemma 1.2.

Theorem 5.3. ^[45] Let $n = 2m$, and $m, k$ be positive integers; then, the trinomial

is a permutation over $F_{2^{2m}}$ if the following holds:

(1) $(s, t) = \left(\frac{2^{k}}{2^{k}-1}, \frac{-1}{2^{k}-1}\right)$, where $\gcd(2^{k}-1, 2^{m}+1) = 1$ and $k < m$,

(2) $(s, t) = \left(\frac{1}{2^{k}+1}, \frac{2^{k}}{2^{k}+1}\right)$, where $\gcd(2^{k}+1, 2^{m}+1) = 1$.

If $k = 1$, then Theorem 5.3 (1) generalizes Theorem 3.2 (see ^[19]). If $k = 2$, then Theorem 5.3 (2) generalizes Theorem 5.2 (1). Similarly, if $k = 2$, then Theorem 5.3 (2) generalizes Theorem 5.2 (4).

As we have seen, there are numerous classes of permutation trinomials over $F_{q^{k}}$ for any $k$ that can be computed from Niho exponents. Wu and Li ^[75] also obtained permutation trinomials over $F_{5^{n}}$ by using Niho exponents.

Particularly, they considered polynomials of the form

where $n = 2k$, $1\leq s, t\leq5^{k}$ and $c_{1}, c_{2}\in \{1, -1\}$. Their construction of permutation trinomials is conditional on taking the fractional polynomial that permutes $(5^{k}+1)^{th}$ roots of unity in $F_{5^{2k}}$, which was the utilization of Lemma 1.2. In the following theorem, they listed all of the trinomials for $k$, both even and odd cases.

Theorem 5.4. ^[75] Let $q = 5^{k}$ and $k$ be a positive integer such that, for the following pairs of $(s, t)$,

is a permutation trinomial over $F_{q^{2}}$:

(1) $(c_{1}, c_{2}) = (1, -1)$ and $(s, t) = \left(\frac{q+3}{4}, \frac{q+3}{2}\right)$,

(2) $k$ is odd, $(c_{1}, c_{2}) = (1, -1)$ and $(s, t) = \left(\frac{q-1}{2}, \frac{q+3}{2}\right)$,

(3) when $k$ is odd, $(c_{1}, c_{2}) = (-1, 1)$ and $(s, t) = \left(\frac{q+3}{2}, q\right), \left(\frac{q+1}{2}, 2\right), \left(\frac{q+3}{2}, \frac{q+5}{2}\right), \left(\frac{q+1}{2}, \frac{q-1}{2}\right)$,

(4) when $k$ is even, $(c_{1}, c_{2}) = (-1, 1)$, $(s, t) = \left(2, \frac{q+3}{2}\right), \left(1, \frac{q+5}{2}\right)$,

(5) when $k$ is even, $(c_{1}, c_{2}) = (1, -1)$, $(s, t) = \left(1, \frac{q-1}{2}\right), \left(\frac{q+3}{2}, \frac{q+5}{2}\right), \left(\frac{q+3}{2}, q\right)$.

In addition, Wu and Li ^[75] defined two sets $\lambda_{-} = \{x^{2}|x\in \mu_{q+1}\}$ and $\lambda_{+} = \{-x^{2}|x\in \mu_{q+1}\}$ treating the squares and non-squares separately to construct another kind of fractional permutation polynomial over $\mu_{q+1}$.

Lemma 5.1. ^[75] Let $q = 5^{k}$, where $k$ is a positive integer. Let

where $s = \frac{q+3}{4}$. Then, $g_{1}(x)$ permutes $\mu_{q+1}$.

Before publishing this article, Wu and Li ^[75] proposed the following two conjectures in the online preprint, and those conjectures were addressed by Ma and Ge ^[53]. Later, Bartoli and Giulietti ^[5] provided alternative and shorter proofs for these two conjectures, which are presented in ^[53] and proposed in ^[75].

Conjecture 5.3. ^[75] The polynomial $f(x) = x\left(\frac{x^{2}-x+2}{x^{2}+x+2}\right)^{2}$ is a permutation polynomial over $F_{5^{k}}$ for odd values of $k$.

Conjecture 5.4. ^[75] Let $q = 5^{k}$ and $k$ be an even integer. Then,

permutes $\mu_{q+1}$.

The permutation trinomials of the form $x+a_{1}x^{s_{1}(q-1)+1}+a_{2}x^{s_{2}(q-1)+1},$ for $a_{1}, a_{2}$ equal to one, have been the subject of extensive research to date. However, choosing general coefficients other than one will be a challenging task. On this note, Tu et al. ^[66] defined permutation trinomials of the form $x+a_{1}x^{s_{1}(q-1)+1}+a_{2}x^{s_{2}(q-1)+1}$ for the general coefficients $a_{1}, a_{2}$ by defining two subsets $\Gamma_{1}$ and $\Gamma_{2}$ of $F_{2^{n}}^{*}\times F_{2^{n}}^{*}$ as follows:

and

By analysis of the solutions of the equation $f(x) = b$ for any $b\in F_{2^{n}}$, for all coefficients $(a_{1}, a_{2})$ in the set $\Gamma_{1}\cup\Gamma_{2}$, they proved that $f(x)$ is a permutation trinomial over $F_{2^{n}}$. This can be easily achieved by determining the number of solutions of some low-degree equations in the unit circle of $F_{q^{2}}$. In the following theorem, they assumed that $n = 2m$ and $(s_{1}, s_{2}) = (q, 2)$, with $q = 2^{m}$.

Theorem 5.5. ^[66] For any $(a_{1}, a_{2})$ in the set $\Gamma_{1}\cup\Gamma_{2}$, the trinomial $f(x) = x+a_{1}x^{q(q-1)+1}+a_{2}x^{2(q-1)+1}$ permutes $F_{2^{n}}$.

Tu et al. ^[66] confirms that, via numerical experiments with $n = 6, 8, 10, 12, 14$, the above Theorem 5.5 covers permutation trinomials of the form

for all possible coefficients $a_{1}, a_{2}\in F_{2^{n}}^{*}$. Nonetheless, they suggested the following open problem in order to use effective approaches to demonstrate this truth.

Open problem 1. ^[66] The trinomial

where $q = 2^{m}$ and $a_{1}a_{2}\not = 0$, is a permutation of $F_{q^{2}}$ if and only if the coefficients $a_{1}, a_{2}$ satisfy one of the following two conditions:

(1) $Tr_{1}^{m}\left(1+\frac{1}{a_{1}\bar{a_{1}}}\right) = 0$ if $a_{2} = \frac{\bar{a_{1}}}{a_{1}}$,

(2) $a_{2}(1+a_{1}\bar{a_{1}}+a_{2}\bar{a_{2}})+\bar{a_{1}}^{2} = 0$ and $Tr_{1}^{m}\left(\frac{a_{2}\bar{a_{2}}}{a_{1}\bar{a_{1}}}\right) = 0$ if $a_{2}\bar{a_{2}}\not = 1$.

Dealing with technique involves the connection between the permutation polynomial and algebraic curve $C_{f}$, which does not have $F_{q}-$rational points $(a, b), a\not = b\in F_{q}$, is actually a hard problem. If the degree $d$ of $C$ is small with respect to size $q$ of $F_{q}$, i.e., $d < \sqrt[4]{q}$, then there exists an absolutely irreducible $F_{q}-$rational component in $C$ that is distinct from $x = y$, i.e., the existence of $F_{q}-$rational points off the line $x = y$. By the Hasse-Weil theorem ^[63] if the curve $C_{f}$ has absolutely irreducible components defined over $F_{q}$, then it contains at least $q-2\sqrt{q}-7$ affine $F_{q}-$rational points off the line $x = y$. So, if $q\geq16$ and $f(x)$ is a permutation polynomial of $F_{q^{2}},$ then $C_{f}$ has no affine rational points off the line $x = y;$ therefore, $C_{f}$ splits completely into absolutely irreducible components that are not defined over $F_{q}$. Utilizing this approach, Bartoli ^[4] provided a solution to Open problem 1.

Based on the sufficient conditions for $(a_{1}, a_{2})$ that would result in the polynomial $x+a_{1}x^{s_{1}(q-1)+1}+a_{2}x^{s_{2}(q-1)+1}$ being a permutation over $F_{2^{2m}}$ that were proposed in ^[66], Tu and Zeng ^[68] generalized those results by using the same techniques for odd characteristics. As with $\Gamma_{1}$ and $\Gamma_{2}$ in ^[66] here, the Tu and Zeng ^[68] defined new sets $\Gamma_{1}$ and $\Gamma_{2}$ for $p\geq3$. Tu and Zeng ^[68] considered $p\geq3$ as an odd prime, $n = 2m$ for a positive integer $m$ and $(a_{1}, a_{2})\in F_{p^{n}}^{*}\times F_{p^{n}}^{*};$ they later considered $v_{1} = a_{1}\bar{a_{1}}$ and $v_{2} = a_{2}\bar{a_{2}}$ and defined following sets:

and

Observe that $\Gamma_{2} = \emptyset$ for $p = 3$ and $\Gamma_{1} = \emptyset$, when $p = 3$ and $m = 1$. So, Tu and Zeng ^[68] considered $m\geq2$, when $p = 3$ and defined the following trinomial for $(s_{1}, s_{2}) = (p^{m}, 2)$ and odd primes $p$.

Theorem 5.6. ^[68] Let $n = 2m$ for a positive integer $m$ and $p$ be an odd prime. Then, the polynomial

permutes $F_{p^{2m}}$ if $(a_{1}, a_{2})\in\Gamma_{1}\cup\Gamma_{2}$.

The condition mentioned in ^[66,68] to construct permutation trinomials of the form

for $p = 2$ and $p = 3$, respectively, was just a sufficient condition. On this note, Bartoli ^[4] proved that the condition mentioned in ^[66] for $p = 2$ was also a necessary condition. And, Hou et al. ^[33] proved that the conditions mentioned in ^[68] for $p = 3$ is necessary for $f(x)$ to be a permutation polynomial over $F_{q^{2}}$. The conclusion of the proof was that $f(x) = xh(x^{q-1})$ permutes $F_{q^{2}}$ if and only if

permutes $\mu_{q+1}$, which is subject to the condition of if and only if $bx^{3}+x+a$ has no root in $\mu_{q+1}$ and

permutes $\mu_{q+1}$.

Later, Tu and Zeng ^[67] constructed two classes of permutation trinomials of the form

for the pair $(s_{1}, s_{2}) = (-1/2, 1/2), (3/4, 1/4)$, with general coefficients $a_{1}, a_{2}$. In the following theorem, they mentioned both classes of permutation trinomials over $F_{q^{2}}$ when $q = 2^{m}$.

Theorem 5.7. ^[67]

(1) Let $n = 2m$ be a positive integer with $m\geq3$ and $a_{1}, a_{2} \in F_{2^{n}}^{*}$. Then, the trinomial

permutes $F_{2^{n}}$ if and only if $a_{1} = \bar{a_{2}}$ and $Tr_{1}^{m}(a_{1}\bar{a_{1}}) = 0$.

(2) Let $n = 2m$ be a positive integer. Assume that $(a_{1}, a_{2}) \in F_{2^{n}}^{*}\times F_{2^{n}}^{*}$ satisfying that $a_{1} = \frac{a_{2}^{2}}{a_{2}}$ and $x^{3}+x+\frac{1}{a_{2}\bar{a_{2}}} = 0$ has no solution in $F_{2^{m}}$. Then,

permutes $F_{2^{n}}$.

Since the conditions given in Theorem 5.7 case 1 are both sufficient and necessary, the conditions mentioned in Theorem 5.7 case 2 are just sufficient; as a result, Tu and Zeng ^[67] mentioned it as a conjecture to establish the necessary part.

Hou ^[34] made the observation that

is a permutation polynomial over $F_{q^{2}}$ if $a = b$ and $x^{3}+x+a^{-1}$ has no root in $F_{q}$ by applying an appropriate substitution of $x\rightarrow ux$ and $a\in F_{q}^{*}$, $b\in F_{q^{2}}^{*}$ in order to address this conjecture.

Tu and Zeng ^[68] considered polynomials of the form

where $a, b\in F_{q^{2}}^{*}$. This type of polynomial belongs to a more general family of permutation polynomials of $F_{q^{2}}$. This family has been investigated in several papers, but only a few have been determined as necessary and sufficient conditions. On this note, Bartoli and Timpanella ^[8] characterized this general family of permutation trinomials, which was considered in ^[66,67,68] for the case of $p > 3$, by using the connections with algebraic curves over finite fields. They proved that the condition mentioned in Theorem 5.6 was not only sufficient, but that it was necessary too.

Following the work on permutation polynomials discussed in ^[42,45], Deng and Zheng ^[14] further studied the polynomials of the form

by using the technique provided in ^[30], and they presented new classes of permutation trinomials over $F_{3^{2m}}$. In the following theorem, they discussed permutation trinomials for the pairs $(s, t) = (2/7, 8/7), (-2/7, 8/7)$ over $F_{2^{2m}}$.

Theorem 5.8. ^[14] Let $q = 2^{m}$. The trinomial

is a permutation over $F_{q^{2}}$ if the following holds:

(1) $\gcd(m, 2) = 1$ and $(s, t) = (2/7, 8/7)$,

(2) $m\equiv2, 4\pmod{6}$ and $(s, t) = (-2/7, 8/7)$.

Inspired by the proof of the previous theorem, they presented the following new permutation polynomial for integers $n$ and $k_{i}$ satisfying that $n\geq2$, $k_{0} = 0$ and $k_{i}+k_{n-i} = k_{n}$ for $0\leq i\leq n$, $q = 2^{m}$, and that $R(x) = \sum\limits_{i = 0}^{n}x^{k_{i}}$ is a polynomial over $F_{q^{2}}$.

Theorem 5.9. ^[14] Let $r$ and $l$ be positive integers. Let $h(x)$ be a polynomial over $F_{q^{2}}$. Assume that $R(x)\not = 0$ for $x\in\mu_{q+1}$; then, the polynomial

permutes $F_{q^{2}}$ if and only if $\gcd(r+k_{n}l, q-1) = 1$ and the polynomial $g(x) = x^{r}h(x)^{q-1}$ permutes $\mu_{q+1}$.

In the following theorem, they constructed permutation trinomials of the form $f(x) = x^{4(q-1)+1}+x^{(q-1)^{2}+1}-x$ over $F_{q^{2}}$, where $q = 3^{m}$.

Theorem 5.10. ^[14] Let $m$ be a positive integer with $m\not\equiv0\pmod{6}$. Let $q = 3^{m}$. The polynomial

is a permutation trinomial over $F_{q^{2}}$.

During the study, they observed that the polynomial

is a permutation trinomial for the pairs $(s, t) = (4/11, 10/11)$, and $m$ with $\gcd(m, 5) = 1$, over $F_{2^{2m}}$ and proposed as a conjecture.

As it was noted in ^[75] that their approach cannot be applied to a general characteristic $p > 5$, Cao et al. ^[11] attempted to extend this inquiry and mentioned another proof of Tu and Zeng's ^[67] theorem.

They constructed the permutation trinomials of the form

for the pair $(s, t)$. In the following theorem, they constructed a permutation trinomial $f(x)$ for the pair

Theorem 5.11. ^[11] Let $k$ be a positive odd integer, $q = p^{k}$, where $p\equiv5\pmod{8}$ is a prime number. Let

where $\lambda_{1}, \lambda_{2}\in F_{q^{2}}$ and $r$ is a positive integer with $\gcd(r, q-1) = 1$. Then, $f(x)$ permutes $F_{q^{2}}$ in the following cases:

(1) $(\lambda_{1}, \lambda_{2}) = (a+b, ab)$, where $a, b\in F_{q^{2}}$, $a\not = b$, $a\bar{b} = 1$ and $\gcd(r-2s, q+1) = 1$,

(2) $(\lambda_{1}, \lambda_{2}) = (a+1, a)$, where $a\in F_{q}$, $a^{2} = -1$ and $r = 1$.

The following theorem covers a general proof for the existence of necessary and sufficient conditions for $\lambda_{1} = 2a$, with $r = 1$, $t = 2s$ and $\lambda_{2} = -1$ required for a polynomial to be a permutation of $F_{q^{2}}$ with Niho exponents.

Theorem 5.12. ^[11] Let $q = p^{k}$, where $q\equiv1\pmod{4}$. Let $a\in F_{q}$ and $s = \frac{q+3}{4}$. Then, the trinomial

permutes $F_{q^{2}}$ if and only if $a^{4} = 1$.

In the next theorem, they considered

with $(\lambda_{1}, \lambda_{2}) = (a, a)$ and presented two infinite families of permutation trinomials with odd characteristics.

Theorem 5.13. ^[11] Let $q = p^{k}$, where $p$ is odd prime. Assume that $a\in F_{q}$ is such that $1-4a^{2}$ is square in $F_{q}$. Then,

permutes $F_{q^{2}}$ if and only if one of the following conditions is satisfied:

(1) $a\not = -1/2$ when $q\equiv1\pmod{4}$,

(2) $a\not = \pm1/2$ when $q\equiv3\pmod{4}$.

For the same pair

they presented another class of permutation trinomial of the form

for the case when $p = 5$, $e\geq4$ is even and $a = 2$. They also conjectured that this polynomial is not always a permutation polynomial if $a = 1$. For

they presented another class of permutation trinomial of the form

for the case $\gcd(x^{2}\pm cx+1, x^{q+1}-1) = 1$. Furthermore, they constructed four classes of permutation trinomials:

for some values of $\lambda_{1}, \lambda_{2}$, $(s, t)$ and $q = 5^{k}$. All four classes of these permutation trinomials are listed below.

Theorem 5.14. ^[11] Let $q = 5^{k}$. Then,

permutes $F_{q^{2}}$ if one of the following conditions is satisfied:

(1) $(s, t) = \left(\frac{q+3}{2}, q\right)$, $(s, t) = \left(\frac{q+3}{2}, \frac{q+5}{2}\right)$,

(a) $(\lambda_{1}, \lambda_{2}) = (-1, -1)$, $k$ odd,

(b) $(\lambda_{1}, \lambda_{2}) = (1, 1), (1, -1)$ $k$ even;

(2) $(s, t) = \left(2, \frac{q+3}{2}\right)$,

(a)$(\lambda_{1}, \lambda_{2}) = (-1, -1)$, $k$ odd,

(b) $(\lambda_{1}, \lambda_{2}) = (1, 1), (1, -1)$, $k$ even,

(c) $(\lambda_{1}, \lambda_{2}) = (1, 2)$.

Transforming the problem into an investigation into some quartic equations over the subfield $F_{2^{m}}$ and showing that these equations have no solutions in $F_{2^{m}}$, Zheng et al. ^[83] determined two classes of permutation trinomials over $F_{2^{m}}$. The following theorem is the first class considered by Zheng et al. when $m$ is odd or $m\equiv2\pmod{4}$ and $(s, t) = (2/7, 8/7)$.

Theorem 5.15. ^[83] Let $n = 2m$ be a positive integer and $U$ be a unit circle of $F_{2^{2m}}$. Then,

is a permutation of $F_{2^{2m}}$, if $a, b$ and $m$ satisfy the following conditions:

(1) $m$ is odd, $b\in U$ and $a = b^{\frac{1}{4}}\xi$, where $\xi\in F_{2^{m}}$ satisfies that $Tr_{1}^{m}(1/\xi^{\frac{1}{3}}) = 1$,

(2) $m\equiv 2\pmod{4}$, $b\in U$ and $a = b^{\frac{1}{4}}\xi$, where $\xi\in F_{2^{e}}$ satisfies that $Tr_{1}^{e}(1/\xi^{\frac{1}{3}}) = 1$, and $e = \frac{m}{2}$.

The following class was constructed when $m$ and $k$ are two positive integers satisfying that $k < m$, $\gcd(2^{k}+1, 2^{m}+1) = 1$ and $(s, t) = \left(\frac{1}{2^{k}+1}, \frac{2^{k}}{2^{k}+1}\right)$.

Theorem 5.16. ^[83] Let $n = 2m$ and $d = ord_{2}(\gcd(m, k))$. Let $a, b\in F_{2^{2m}}^{*}$. Then,

is a permutation of $F_{2^{2m}}$ if $a$ and $b$ satisfy

and

They have expressed the open problem based on Lemma 1.2 as, under the identical conditions as listed above,

is a permutation of $F_{2^{2m}}$.

Inspired by the idea proposed in ^[21], Wang et al. ^[72] constructed six classes of permutation trinomials of the form

for $\lambda_{1}, \lambda_{2}, \lambda_{3}\in\{1, -1\}$ and $s_{1}, s_{2}\in\{0, 1, q, q^{2}\}$ over $F_{q^{3}}$, where $q = 3^{k}$. Wang et al. ^[72] used a multivariate method and resultant elimination method to prove that $f(x)$ is a permutation polynomial over $F_{q^{3}}$. They characterized three classes of permutation polynomials for the case when $k\not\equiv1\pmod{3}$, and three classes for the case when $k\not\equiv2\pmod{3}$. The following theorem included all six classes of the permutation trinomials for all values of $\lambda_{1}, \lambda_{2}, \lambda_{3}$ and $s_{1}, s_{2}$.

Theorem 5.17. ^[72] Let $q = 3^{k}$, where $k$ is a positive integer. Then, $f(x)$ is a permutation trinomial over $F_{q^{3}}$ for the following cases when $k\not\equiv1\pmod{3}$:

(1) $(\lambda_{1}, \lambda_{2}, \lambda_{3}) = (1, 1, -1)$, $(s_{1}, s_{2}) = (q, 1)$,

(2) $(\lambda_{1}, \lambda_{2}, \lambda_{3}) = (1, -1, 1)$, $(s_{1}, s_{2}) = (q, q)$,

(3) $(\lambda_{1}, \lambda_{2}, \lambda_{3}) = (-1, 1, 1)$, $(s_{1}, s_{2}) = (0, q)$,

and the following cases when $k\not\equiv2\pmod{3}$:

(1) $(\lambda_{1}, \lambda_{2}, \lambda_{3}) = (1, 1, -1)$, $(s_{1}, s_{2}) = (q, 1)$,

(2) $(\lambda_{1}, \lambda_{2}, \lambda_{3}) = (1, -1, 1)$, $(s_{1}, s_{2}) = (0, q)$,

(3) $(\lambda_{1}, \lambda_{2}, \lambda_{3}) = (-1, 1, 1)$, $(s_{1}, s_{2}) = (q, q)$.

In the above theorem, Wang et al. ^[72] stated all six classes of permutation trinomials for $q = p^{k}$, where $p = 3$. Later, Bartoli ^[6] considered the same form for more general values of $q$, such as $q = p^{h}$ and $p > 3$. He mentioned sufficient conditions for the pairs $(\lambda_{1}, \lambda_{2})$ for which these polynomials permute $F_{q^{3}}$. He also determined the lower bound on the number of these pairs by using the techniques based on function field theory, which provides exact estimates of the number of $F_{q}-$rational solutions of a particular system of equations. The following theorems contained all four classes of the permutation polynomials and lower bounds on the number of pairs $(\lambda_{1}, \lambda_{2})$ for which these polynomials permute $F_{q^{3}}$.

Theorem 5.18. ^[6] Let $\lambda_{1}, \lambda_{2}\in F_{q}$ be such that $\lambda_{1}^{3}+\lambda_{2}^{2}-\lambda_{2}+1 = 0$, $\lambda_{2}\not = 0, 1$. Suppose that

has no roots in $\mu_{q^{2}+q+1}$. Then, the following two polynomials are permutation polynomials over $F_{q^{3}}$:

(1) $f(x) = x^{q(q+1)-1}+\lambda_{1}x^{q(q-1)+1}+\lambda_{2}x$,

(2) $f(x) = x^{q(q+1)-1}+\lambda_{1}x^{q^{2}(q-1)+q}+\lambda_{2}x$.

There are at least $\frac{q-72\sqrt{q}-95}{6}$ pairs of $(\lambda_{1}, \lambda_{2})$ that satisfy the condition that $f(x)$ are permutation trinomials in both of the above cases.

Theorem 5.19. ^[6] Let $q\equiv1\pmod{3}$. If $\lambda_{1}, \lambda_{2}\in F_{q}$, $\lambda_{2}^{2}+\lambda_{2}+1 = 0$, $h_{1}(\lambda_{1})\not = 0$ and $\lambda_{1}^{3}\not = -1$, then the polynomial

is a permutation polynomial over $F_{q^{3}}$.

Theorem 5.20. ^[6] If $\lambda_{1}, \lambda_{2}\in F_{q}$, $\lambda_{2}^{2}+\lambda_{2}+1 = 0$ and $\lambda_{1}^{3}\not = -1$ are such that $\lambda_{2}T^{3}+\lambda_{1}^{2}T^{2}+(-\lambda_{1}\lambda_{2}+\lambda_{1})T-\lambda_{2}$ has no roots in $\mu_{q^{2}+q+1}$, then the polynomial

is a permutation polynomial over $F_{q^{3}}$. Also, there are at least $\frac{q-8\sqrt{q}-50}{3}$ values of $\lambda_{1}$ such that $f(x)$ is a permutation trinomial.

One can find pairs of $(s, t)$ such that

is a permutation polynomial in literature. Nevertheless, a complete value of $(s, t)$ is not known in the literature.

Open problem 2. Find $(s, t)$ such that

is a permutation polynomial over any given finite field with Niho exponents.

6.   Conclusions

In this paper, we surveyed all existing classes of permutation trinomials with all mentioned methodologies. Furthermore, similar methods can be used to generate many new permutation trinomials. We have concluded with remarks and open problems based on recent results on permutation trinomials.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions, which improved the quality of this article. The corresponding author acknowledges the Manipal Institute of Technology (MIT), Manipal Academy of Higher Education, India for their kind encouragement. The first author is grateful to the UGC-CSIR Grant Commission for their support through grant(No. 201610069412), as well as Manipal Academy of Higher Education for their kind encouragement and support.

Conflict of interest

All authors declare no conflicts of interest that could influence the publication of this paper.