Classes Of Permutation Trinomials Research Articles

On Some Permutation Trinomials in Characteristic Three

In this paper, we determine the permutation properties of the polynomial x^3+x^q+2−x^4q−1 over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of the form x^4q−1 +x^2q+1 ±x^3. In particular, we first show that x^3 +x^q+2 −x^4q−1 permutes F_q^2 with q = 3^m if and only if m is odd. This enables us to show that the sufficient condition in [35, Theorem 4] is also necessary. Next, we prove that x^4q−1 + x^2q+1 − x^3 permutes F_q^2 with q = 3^m if and only if m is congruent to 0 modulo 4. Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial x^4q−1 +x^2q+1 +x^3 and show that it is never a permutation polynomial of F_q^2 in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.

Two classes of permutation trinomials over [formula omitted] in characteristic two

Let q=2m and Fq3 be the finite field with q3 elements. In this paper, based on the multivariate method, resultant elimination, and transforming into dealing with some equations over finite fields, we propose two classes of permutation trinomials of Fq3. We illustrate that these two classes of permutation trinomials are QM-inequivalent to all known permutation polynomials over Fq3. Some well-known results can be covered by our theorems.

Two new classes of permutation trinomials over [formula omitted] with odd characteristic

Let q be an odd prime power and Fq3 be the finite field with q3 elements. In this paper, we propose two classes of permutation trinomials of Fq3 for an arbitrary odd characteristic based on the multivariate method and some subtle manipulation of solving equations with low degrees over finite fields. Moreover, we demonstrate that these two classes of permutation trinomials are QM inequivalent to all known permutation polynomials over Fq3. To the best of our knowledge, this paper is the first to study the construction of nonlinearized permutation trinomials of Fq3 with at least one coefficient lying in Fq3﹨Fq.

Further results on a class of permutation trinomials

Further results on a class of permutation trinomials

A recent survey of permutation trinomials over finite fields

<abstract><p>Constructing permutation polynomials is a hot topic in the area of finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials were constructed. In 2015, Hou surveyed the achievements of permutation polynomials and novel methods. But, very few were known at that time. Recently, many permutation binomials and trinomials have been constructed. Here we survey the significant contribution made to the construction of permutation trinomials over finite fields in recent years. Emphasis is placed on significant results and novel methods. The covered material is split into three aspects: the existence of permutation trinomials of the respective forms $ x^{r}h(x^{s}) $, $ \lambda_{1}x^{a}+\lambda_{2}x^{b}+\lambda_{3}x^{c} $ and $ x+x^{s(q^{m}-1)+1} +x^{t(q^{m}-1)+1} $, with Niho-type exponents $ s, t $.</p></abstract>

New classes of permutation trinomials over [formula omitted

In this paper, we propose three new classes of permutation trinomials over Fq3. The first two classes are of the form x+L(xq2+q−1), where L(x)∈{x+Axq,x+Axq2}, q even, and the third class is x+Axq2−q+1+A2xq2, q odd. In fact we prove a conjecture proposed by Gong, Gao and Liu [8] about permutation trinomials of the form xq+1+L(x)∈Fq3[x] as a particular case of our results. Moreover, we determine necessary and sufficient conditions for the permutation trinomials studied. We also show that the permutation trinomials proposed in this paper are new in the sense that they are not quasi-multiplicative equivalent to permutation trinomials over Fq3 known so far.

Permutation polynomials of the form xd + L(xs) over [formula omitted

In this paper, permutation polynomials of the form xd+L(xs) over finite fields Fq3 are investigated, where q=2m and m is a positive integer. By means of the iterative method, the multivariate method and the resultant elimination, six classes of permutation trinomials are obtained from certain integers d, s and linearized polynomials L(x). It is also showed that the permutations proposed in this paper are quasi-multiplicative inequivalent to all known permutation trinomials over Fq3.

Two classes of permutation trinomials with Niho exponents

In this paper, we consider two classes of permutation trinomials with Niho-type exponents over the finite field F22m, where m is a positive integer. We transform the problem into investigating on some quartic equations (2k-th degree equations) over the subfield F2m in the first class (second class, respectively). We show that these equations have no solutions in F2m. Some sufficient conditions are established to characterize the coefficients in the two classes of permutation polynomials. The numerical result suggests that the sufficient conditions on the coefficients for the case of m odd in the first class, and in the second class, are also necessary.

Two classes of permutation trinomials with Niho exponents over finite fields with even characteristic

In this paper, we consider two classes of permutation trinomials with Niho-type exponents over the finite field F22m. Some sufficient conditions are obtained to characterize the coefficients of the permutation trinomials. Our numerical result suggests that those sufficient conditions for one class are also necessary.

More permutation polynomials with Niho exponents which permute [formula omitted

Constructions of permutation polynomials over finite fields have attracted much interests in recent years, especially those with few terms, such as trinomials, due to their simple form and additional properties. In this paper, we construct several classes of permutation trinomials over Fp2k with Niho exponents of the form f(x)=x+λ1xs(pk−1)+1+λ2xt(pk−1)+1; some necessary and sufficient conditions for the polynomial f(x) to permute Fp2k are provided. Specifically, for p=5, new permutation trinomials are presented. We also give recursive constructions of permutation polynomials using self-reciprocal polynomials.

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

Further results on permutation trinomials with Niho exponents

In this paper, we prove a conjecture proposed by Deng and Zheng about a class of permutation trinomials over finite fields ${\mathbb {F}}_{2^{2m}}$ . In addition, we also construct four classes of permutation trinomials with Niho exponents over ${\mathbb {F}}_{3^{2m}}$ .

On a class of permutation trinomials in characteristic 2

Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form $f(X)=X+aX^{q(q-1)+ 1}+bX^{2(q-1)+ 1}\in \mathbb {F}_{q^{2}}[X]$, where q is even and $a,b\in \mathbb {F}_{q^{2}}^{*}$. They found sufficient conditions on a, b for f to be a permutation polynomial (PP) of $\mathbb {F}_{q^{2}}$ and they conjectured that the sufficient conditions are also necessary. The conjecture has been confirmed by Bartoli using the Hasse-Weil bound. In this paper, we give an alternative solution to the question. We also use the Hasse-Weil bound, but in a different way. Moreover, the necessity and sufficiency of the conditions are proved by the same approach.

Two classes of permutation trinomials with Niho exponents

In this paper, two classes of permutation trinomials with Niho-type exponents are proposed. The coefficients in the first class are completely determined, and a sufficient condition is established to characterize the coefficients in the second class.

A class of permutation trinomials over finite fields of odd characteristic

This paper studies the permutation behavior of the polynomial $f(x)=x+a_{1}x^{q(q-1)+ 1}+a_{2}x^{2(q-1)+ 1}$ in $\mathbb {F}_{q^{2}}[x]$ for odd q, and finds a set of the coefficient pairs $(a_{1}, a_{2})$ that leads $f(x)$ to be a permutation of $\mathbb {F}_{q^{2}}$ . We transform the problem of proving that f is a permutation into determining the number of solutions to some low-degree equations in the unit circle of $\mathbb {F}_{q^{2}}$ .

On a conjecture about a class of permutation trinomials

We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials fα,β(x)=x+αxq(q−1)+1+βx2(q−1)+1∈Fq2[x], αβ≠0, q even, characterizing all the pairs (α,β)∈Fq22 for which fα,β(x) is a permutation of Fq2.

Several classes of permutation trinomials over F 5 n $\mathbb {F}_{5^{n}}$ from Niho exponents

The construction of permutation trinomials over finite fields attracts people’s interest recently due to their simple form and some additional properties. Motivated by some results on the construction of permutation trinomials with Niho exponents, in this paper, by constructing some new fractional polynomials that permute the set of the (q + 1)-th roots of unity in $\mathbb {F}_{q^{2}}$ , we present several classes of permutation trinomials with Niho exponents over $\mathbb {F}_{q^{2}}$ , where q = 5k.

Six new classes of permutation trinomials over $$\mathbb {F}_{3^{3k}}$$ F 3 3 k

Permutation trinomials over finite fields are of great interest for their simple algebraic forms and important applications in many areas of mathematics and engineering. In this paper, six new classes of permutation trinomials over $$\mathbb {F}_{3^{3k}}$$ are presented based on the multivariate method. Their permutation properties are proved by using the resultant elimination method.

More classes of permutation trinomials with Niho exponents

This paper presents two classes of permutation trinomials with the form $x^{s(2^{m}-1)+ 1}+x^{t(2^{m}-1)+ 1}+x$ over the finite field $ \mathbb {F}_{2^{2m}}$ as a supplement of the recent works of Li and Helleseth, and a class of permutation trinomials like this form over $ \mathbb {F}_{3^{2m}}$ . Moreover, we give a method to construct permutation polynomials from known ones.

New permutation trinomials constructed from fractional polynomials

Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six new classes of permutation trinomials over finite fields of even characteristic are constructed from six fractional polynomials. Further, three classes of permutation trinomials over finite fields of characteristic three are raised. Distinct from most of the known permutation trinomials which are with fixed exponents, our results are some general classes of permutation trinomials with one parameter in the exponents. Finally, we propose a few conjectures.