Classes Of Permutation Polynomials Research Articles

Some classes of functions with low c-differential uniformity over finite fields

Multiplicative differential (and the corresponding [Formula: see text]-differential uniformity) was introduced by Ellingsen et al. in [[Formula: see text]-differentials, multiplicative uniformity and (almost) perfect [Formula: see text]-nonlinearity, IEEE Trans. Inf. Theory 66(9) (2020) 5781–5789], which has attracted lots of attention. Functions with low [Formula: see text]-differential uniformity over finite fields, especially the P[Formula: see text]N and AP[Formula: see text]N functions, have been widely investigated due to their applications in cryptography. In this paper, we first compute the [Formula: see text]-differential uniformity of two classes of permutation polynomials. For one of these, we explicitly determine the [Formula: see text]-DDT entries. For the second type of function, we give bounds for its [Formula: see text]-differential uniformity. Besides, several classes of P[Formula: see text]N or AP[Formula: see text]N functions are presented by employing some known functions and the (generalized) AGW criterion.

The boomerang uniformity of three classes of permutation polynomials over 𝔽2n

Permutation polynomials with low boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over [Formula: see text], we determine the boomerang uniformity of these permutation polynomials: (1) [Formula: see text], where [Formula: see text], [Formula: see text] with [Formula: see text]; (2) [Formula: see text], where [Formula: see text], [Formula: see text] with [Formula: see text]; (3) [Formula: see text], where [Formula: see text], [Formula: see text] with [Formula: see text]. The results show that the boomerang uniformity of [Formula: see text], [Formula: see text] and [Formula: see text] can attain [Formula: see text].

Several classes of permutation polynomials over the finite field [formula omitted

Let Fq denote the finite field with q elements. In this paper, based on the AGW criterion and determining the number of solutions to certain equations over the finite field Fp2m, several classes of permutation polynomials with given forms are proposed. Some of them are the permutation polynomials of the form (xpm−x+δ)s+L(x), where L(x) is a linearized polynomial. Others are the permutation polynomials of the form (xpm−x+δ)s1+(xpm−x+δ)s2+x.

The c-differential uniformity and boomerang uniformity of three classes of permutation polynomials over [formula omitted

Permutation polynomials with low c-differential uniformity and boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over F2n, we determine the c-differential uniformity and boomerang uniformity of these permutation polynomials: (1) f1(x)=x+Tr1n(x2k+1+1+x3+x+ux), where n=2k+1, u∈F2n with Tr1n(u)=1; (2) f2(x)=x+Tr1n(x2k+3+(x+1)2k+3), where n=2k+1; (3) f3(x)=x−1+Tr1n((x−1+1)d+x−d), where n is even and d is a positive integer. The results show that the involutions f1(x) and f2(x) are APcN functions for c∈F2n﹨{0,1}. Moreover, the boomerang uniformity of f1(x) and f2(x) can attain 2n. Furthermore, we generalize some previous works and derive the upper bounds on the c-differential uniformity and boomerang uniformity of f3(x).

On the Classification of Low-Degree Ovoids of Q(4,q)

Ovoids of the non-degenerate quadric Q(4,q) of PG(4,q) have been studied since the end of the’ 80s. They are rare objects and, beside the classical example given by an elliptic quadric, only three classes are known for q odd, one class for q even, and a sporadic example for q = 35. It is well known that to any ovoid of Q(4,q) a bivariate polynomial f (x,y) can be associated. In this paper we classify ovoids of Q(4,q) whose corresponding polynomial f (x, y) has “low degree” compared with q, in particular, \(\deg \left( f \right) < {\left( {{1 \over {6.3}}q} \right)^{{3 \over {13}}}} - 1\). Finally, as an application, two classes of permutation polynomials in characteristic 3 are obtained.

Several classes of permutation polynomials with trace functions over $$\mathbb {F}_{p^n}$$

Several classes of permutation polynomials with trace functions over $$\mathbb {F}_{p^n}$$

The c-Differential Uniformity and Boomerang Uniformity of Two Classes of Permutation Polynomials

The Difference Distribution Table (DDT) and the differential uniformity play a major role for the design of substitution boxes in block ciphers, since they indicate the function’s resistance against differential cryptanalysis. This concept was extended recently to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -DDT and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -differential uniformity, which have the potential of extending differential cryptanalysis. Recently, a new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to quantify the resistance of a block cipher against boomerang-style attacks. Here we concentrate on two classes (introduced recently) of permutation polynomials over finite fields of even characteristic. For one of these, which is an involution used to construct a 4-uniform permutation, we explicitly determine the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -DDT entries and BCT entries. For the second type of function, which is a differentially 4-uniform function, we give bounds for its <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -differential and boomerang uniformities.

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

In this paper, by transforming the permutation problem into the root distribution problem in the unit circle of certain quadratic and cubic equations, we investigate the permutation behavior of the type $f(x)=x+x^{2^{3m}-2^m+1}+x^{2^{4m}-2^{3m}+2^m}$ over $F_{2^{4m}}$ and $f(x)=x+x^{2^m}+x^{2^{m+1}-1}+ax^{2^{2m}-2^m+1}$ over $F_{2^{2m}}$, respectively.

Some specific classes of permutation polynomials over $ {\textbf{F}}_{q^3} $

&lt;abstract&gt;&lt;p&gt;Constructing permutation polynomials is a hot topic in finite fields. Recently, huge kinds of permutation polynomials over $ {\bf F}_{q^2} $ have been studied. In this paper, by using AGW criterion and piecewise method, we construct several classes of permutation polynomials over $ {\bf F}_{q^3} $ of the forms similar to $ (x^{q^2}+x^q+x+\delta)^{\frac{q^{3}-1}{d}+1}+L(x) $, for $ d = 2, 3, 4, 6, $ where $ L(x) $ is a linearized polynomial over $ {\bf F}_{q} $.&lt;/p&gt;&lt;/abstract&gt;

Constructions and Necessities of Some Permutation Polynomials over Finite Fields

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

Necessary and sufficient conditions of two classes of permutation polynomials

In this paper, two types of permutation polynomials, (x2m+x+δ)s+cx and xr(xq−1+a), are studied. To formulate the necessary and sufficient conditions for the polynomials to be permutation polynomials over finite fields, the structures and properties of the field elements are analyzed. Meanwhile, the number of solutions to some equations over finite fields is investigated. For quadratic equations, the necessary and sufficient conditions are utilized thoroughly to study the permutation polynomials. But for cubic equations, only one direction is investigated by the existing literatures. This paper employs the properties of these equations in full strength to study (x2m+x+δ)s+cx over F23m⁎. For the binomials of the form xr(xq−1+a), two different methods are exploited to formulate the necessary and sufficient conditions over Fq3, and some partial results are obtained for Fqe.

Finding Compositional Inverses of Permutations From the AGW Criterion

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria.

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.

Some classes of permutation polynomials with the form (xk + x) s + x 2 n −2

For a prime 2, let n be a positive integer, and F 2 n be the finite field with 2 n elements. In this paper, we propose several classes of permutation polynomials with the form (xk + x) s + x 2 n −2 over finite fields F 2 n which have the highest degree of algebra.

A link between two classes of permutation polynomials

The study of permutation polynomials over finite fields has attracted many scholars' attentions due to their wide applications and there are several different forms of permutations over finite fields. However, there is little literature on the relation between different forms of permutations.In this paper, we find an equivalent relation between permutation polynomials of the form f(x)=xh(x2n−1) over F22n and permutations of the form F(x)=H1(Δ)H1(Δ)+H2(Δ)+x on F2n, where Δ=1x2+x+δ, δ∈F2n with Tr2n(δ)=1 and H1,H2 are the x-coordinate and 1-coordinate of h (see Definition 2.4), respectively. A special case with H1+H2=1, which means F(x)=H1(1x2+x+δ)+x, has been already studied [8,16,21,22]. We summarize the known permutations and then construct 8 new classes of permutations of such form, from which one can derive some permutation polynomials of the form xh(x2n−1) over F22n directly. Finally, we give an asymptotic result about (1x2+x+δ)2k+x to be a permutation over F2n using the Hasse-Weil bound.

On Constructing Two Classes of Permutation Polynomials over Finite Fields

In this paper, we construct two classes of permutation polynomials over finite fields. First, by one well-known lemma of Zieve, we characterize one class permutation polynomials of the finite field, which generalizes the result of Marcos. Second, by using the onto property of functions related to the elementary symmetric polynomial in multivariable and the general trace function, we construct another class permutation polynomials of the finite field. This extends the results of Marcos, Zieve, Qin and Hong to the more general cases. Particularly, the latter result gives a rather more general answer to an open problem raised by Zieve in 2010.

Some new forms of permutation polynomials based on the AGW criterion

In this paper, we propose several classes of permutation polynomials over F22m of the form x+(Trmn(x)k+δ)s and x+(Trmn(x)k1+δ1)s1+(Trmn(x)k2+δ2)s2 based on the AGW criterion.

New constructions of involutions over finite fields

Involutions over finite fields are permutations whose compositional inverses are themselves. Involutions especially over $ \mathbb {F}_{q} $ with q is even have been used in many applications, including cryptography and coding theory. The explicit study of involutions (including their fixed points) has started with the paper (Charpin et al. IEEE Trans. Inf. Theory, 62(4), 2266–2276 2016) for binary fields and since then a lot of attention had been made in this direction following it; see for example, Charpin et al. (2016), Coulter and Mesnager (IEEE Trans. Inf. Theory, 64(4), 2979–2986, 2018), Fu and Feng (2017), Wang (Finite Fields Appl., 45, 422–427, 2017) and Zheng et al. (2019). In this paper, we study constructions of involutions over finite fields by proposing an involutory version of the AGW Criterion. We demonstrate our general construction method by considering polynomials of different forms. First, in the multiplicative case, we present some necessary conditions of f(x) = xrh(xs) over $\mathbb {F}_{q}$ to be involutory on $\mathbb {F}_{q}$, where s∣(q − 1). Based on this, we provide three explicit classes of involutions of the form xrh(xq− 1) over $\mathbb {F}_{q^{2}}$. Recently, Zheng et al. (Finite Fields Appl., 56, 1–16 2019) found an equivalent relationship between permutation polynomials of $g(x)^{q^{i}} - g(x) + cx +(1-c)\delta $ and $g\left (x^{q^{i}} - x + \delta \right ) +c x$. The other part work of this paper is to consider the involutory property of these two classes of permutation polynomials, which fall into the additive case of the AGW criterion. On one hand, we reveal the relationship of being involutory between the form $ g(x)^{q^{i}} - g(x) + cx +(1-c)\delta $ and the form $ g\left (x^{q^{i}} - x + \delta \right ) +c x $ over $ \mathbb {F}_{q^{m}} $ ; on the other hand, the compositional inverses of permutation polynomials of the form $ g\left (x^{q^{i}} - x + \delta \right ) + cx $ over $ \mathbb {F}_{q^{m}} $ are computed, where $ \delta \in \mathbb {F}_{q^{m}} $, $ g(x) \in \mathbb {F}_{q^{m}}[x] $ and integers m, i satisfy 1 ≤ i ≤ m − 1. In addition, a class of involutions of the form $ g\left (x^{q^{i}} - x + \delta \right ) + cx $ is constructed. Finally, we study the fixed points of constructed involutions and compute the number of all involutions with any given number of fixed points over $ \mathbb {F}_{q} $.

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.

Two types of permutation polynomials with special forms

Let q be a power of a prime and Fq be a finite field with q elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form cx−xs+xqs, and investigate the relationship between two types of widely discussed permutation polynomials. Based on this relation, many classes of permutation polynomials having the form (xqk−x+δ)s+cx without restriction on δ are derived.