Related papers: Two low differentially uniform power permutations …
It was shown by Boukerrou et al.~[IACR Trans. Symmetric Cryptol. 1 (2020), 331--362] that the $F$-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear…
This paper considers permutation polynomials over the finite field $F_{q^2}$ in even characteristic by utilizing low-degree permutation rational functions over $F_q$. As a result, we obtain two classes of permutation binomials and six…
Let $n$ be an odd positive integer, $p$ be a prime with $p\equiv3\pmod4$, $d_{1} = {{p^{n}-1}\over {2}} -1 $ and $d_{2} =p^{n}-2$. The function defined by $f_u(x)=ux^{d_{1}}+x^{d_{2}}$ is called the generalized Ness-Helleseth function over…
Let $q = p^m$, where $p$ is an odd prime number and $m$ is a positive integer. In this paper, we examine the finite field $\mathbb{F}_{q^2}$, which consists of $q^2$ elements. We first present an alternative method to determine the…
Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$ over…
Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication…
The characterization of permutations over finite fields is an important topic in number theory with a long-standing history. This paper presents a systematic investigation of low-degree bivariate polynomial systems $F=(f_1(x,y),f_2(x,y))$…
Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature…
Functions with low differential uniformity can be used as the s-boxes of symmetric cryptosystems as they have good resistance to differential attacks. The AES (Advanced Encryption Standard) uses a differentially-4 uniform function called…
Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However APN…
Let $m$ be a positive integer and $p$ a prime. In this paper, we investigate the differential properties of the power mapping $x^{p^m+2}$ over $\mathbb{F}_{p^n}$, where $n=2m$ or $n=2m-1$. For the case $n=2m$, by transforming the derivative…
Recent studies on binomials of the form $F_r(x) = x^r(1 + \chi(x))$ over $\mathbb{F}_{p^n}$ have shown that these functions can exhibit very low boomerang uniformity. In this paper, we focus on the specific behavior of such binomials in…
Recently, a new concept called multiplicative differential was introduced by Ellingsen et al. Inspired by this pioneering work, power functions with low c-differential uniformity were constructed. Wang et al. defined the c-differential…
We consider rational functions of the form $V(x)/U(x)$, where both $V(x)$ and $U(x)$ are polynomials over the finite field $\mathbb{F}_q$. Polynomials that permute the elements of a field, called {\it permutation polynomials ($PPs$)}, have…
Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low $c$-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of…
Recently, a new concept called the $c$-differential uniformity was proposed by Ellingsen et al. (2020), which allows to simplify some types of differential cryptanalysis. Since then, finding functions having low $c$-differential uniformity…
Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation…
Let $F$ be a finite field, let $f$ be a function from $F$ to $F$, and let $a$ be a nonzero element of $F$. The discrete derivative of $f$ in direction $a$ is $\Delta_a f \colon F \to F$ with $(\Delta_a f)(x)=f(x+a)-f(x)$. The differential…
Very recently, a new concept called multiplicative differential and the corresponding $c$-differential uniformity were introduced by Ellingsen et al. A function $F(x)$ over finite field $\mathrm{GF}(p^n)$ to itself is called…
We investigate permutation polynomials F over finite fields F_{p^n} whose generalized derivative maps x -> F(x + a) - cF(x) are themselves permutations for all nonzero shifts a. This property, termed perfect c-nonlinearity (PcN), represents…