The $c$-differential properties of a class of power functions

Let $\mathbb{F}_{p^{n}}$ be the finite field with $p^{n}$ elements, where $p$ is a prime and $n$ is a positive integer. The multiplicative cyclic group of the finite field $\mathbb{F}_{p^{n}}$ is denoted by $\mathbb{F}_{p^{n}}^{*}$. Let $\mathbb{F}_{p^{n}}[x]$ denote the polynomial ring over $\mathbb{F}_{p^{n}}$. Any function $f:\mathbb{F}_{p^{n}}\rightarrow\mathbb{F}_{p^{n}}$ can be uniquely represented as a univariate polynomial of degree less than $p^{n}$. Therefore, $f$ can always be seen as a polynomial in $\mathbb{F}_{p^{n}}[x]$. A polynomial $f\in\mathbb{F}_{p^{n}}[x]$ is called a permutation polynomial of $\mathbb{F}_{p^{n}}$ if the induced mapping $x\mapsto f(x)$ is a permutation over $\mathbb{F}_{p^{n}}$. Permutation polynomials are a very important class of polynomials as they have applications in coding theory and cryptography. Nowadays, designing infinite classes of permutation polynomials over finite fields with good cryptographic properties remains an interesting research topic.

Substitution boxes (S-boxes for short) are important nonlinear building blocks in block ciphers, which can be seen as permutation functions over finite fields. To quantify the ability of S-boxes to resist the differential attack[1], one of the most powerful attacks on block ciphers, Nyberg[8] introduced the notion of differential uniformity. More importantly, to estimate the resistance against some variants of differential cryptanalysis, the differential spectrums[2] of S-boxes are also shown to be of great interest.

In [3], Borisov et al. proposed a new differential on block ciphers, called a multiplicative differential, it is an extension of differential cryptanalysis, which helps to identify the weakness of IDEA ciphers. Motivated by the concept of multiplicative differential, Ellingsen et al. in [4] first introduced the (multiplicative) $c$-derivative.

In the same paper they presented a generalized notion of differential uniformity, the so-called $c$-differential uniformity.

If ${}_{c}\Delta_{f}=\delta$, then $f$ is called a differentially $(c,\delta)$-uniform function. Especially, $f$ is called a perfect $c$-nonlinear (PcN) function if ${}_{c}\Delta_{f}=1$, and an almost perfect $c$-nonlinear (APcN) function if ${}_{c}\Delta_{f}=2$. It is clear that if $c=1$ and $a\neq 0$, the $c$-differential uniformity becomes the usual differential uniformity and one use the corresponding notation by omit the symbol $c$. The smaller the value ${}_{c}\Delta_{f}$ is, the better $f$ resists against multiplicative differential attacks. Thus, the research on cryptographic functions with low $c$-differential uniformity has been a hot issue in recent years, the readers can refer to [4, 7, 6, 9, 10, 15, 13, 12, 16, 20, 19, 17, 14] and the references therein.

For a power function $f(x)=x^{d}$ with a positive integer $d$, one can easily see that ${}_{c}\delta_{f}(a,b)={}_{c}\delta_{f}(1,b/a^{d})$, for all $a\in\mathbb{F}_{p^{n}}^{*}$ and $b\in\mathbb{F}_{p^{n}}$. For simplicity, denoted ${}_{c}\delta_{f}(1,b)$ by ${}_{c}\delta_{f}(b)$ with $b\in\mathbb{F}_{p^{n}}$. More precisely, it was proved in [7, Lemma 1] that the $c$-differential uniformity of $f(x)=x^{d}$ is given by

Moreover, as a generalization of the differential spectrum, the $c$-differential spectrum of a power function is defined as follows.

Generally speaking, it is difficult to determine the $c$-differential spectrum of power functions. To the best of our knowledge, only a few classes of power functions over odd characteristic finite fields have a nontrivial $c$-differential spectrum. The known results on power functions $f$ over $\mathbb{F}_{p^{n}}$, for which $c$-differential spectrum has been determined are summarized in Table 1, where Tr${}_{n}(\cdot)$ is the absolute trace mapping from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2}$.

Throughout this paper, let $f(x)=x^{\frac{p^{n}+3}{2}}\in\mathbb{F}_{p^{n}}[x]$, where $p$ is an odd prime and $n$ is a positive integer. We should mention that when $p=3$, $f$ is P$c$N if $n\geq 3$ is odd, and is AP$c$N if $n\geq 2$ is even [7] with its $(-1)$-differential spectrum being given in [19, Theorem 9]. When $p>3$, the $(-1)$-differential uniformity of $f$ was discussed by Mesnager et al. in [7, Theorem 11] and they proved that ${}_{-1}\Delta_{f}\leq 4$ if $p^{n}\equiv 1\pmod{4}$ and ${}_{-1}\Delta_{f}\leq 3$ if $p^{n}\equiv 3\pmod{4}$.

Inspired by the works above, in this paper, we mainly study the $(-1)$-differential spectrum of $f(x)=x^{\frac{p^{n}+3}{2}}$ for all positive integer $n$ when $p\neq 3$. More precisely, we prove that ${}_{-1}\Delta_{f}=3$ if $p^{n}\equiv 3\pmod{4}$ except for $p\in\{7,19,23\}$ where $f$ is an APcN function, and ${}_{-1}\Delta_{f}=4$ if $p^{n}\equiv 1\pmod{4}$, that is, the upper bound of the $(-1)$-differential uniformity of $f$ given by Mesnager et al. can be achieved.

The rest of this paper is organized as follows. In Section 2, we introduce some basic notation about quadratic character and results on quadratic character sums, which will be employed in the sequel. In Section 3, we first determine the $(-1)$-differential spectrum of $f$, and some examples are also presented. In Section 4, we give an upper bound of the $c$-differential uniformity of $f$. Section 5 concludes this paper.

In this section, we mainly introduce some basic results on quadratic character sums over $\mathbb{F}_{p^{n}}$. Let $\eta$ be the quadratic character over $\mathbb{F}_{p^{n}}$, i.e., for any $x\in\mathbb{F}_{p^{n}}$,

It is well-known that $\sum\limits_{x\in\mathbb{F}_{p^{n}}}\eta(x)=0$, and $\eta(-1)=1$ (resp. $-1$) if $p^{n}\equiv 1~{}(\bmod~{}4)$ (resp. $p^{n}\equiv 3~{}(\bmod~{}4)$), which are used extensively in the calculations of character sums.

We consider now sums involving the quadratic character of the form

with $f(x)\in\mathbb{F}_{p^{n}}[x]$. Recall that for $\deg(f(x))=1$, the sums above is trivial, and for $\deg(f(x))=2$, the following explicit formula was established in [5] and [6], respectively.

For $\deg(f(x))\geq 3$, it is challenging to obtain an explicit formula for the character sum $\sum\limits_{x\in\mathbb{F}_{p^{n}}}\eta(f(x))$. However, when $\deg(f(x))=3$, such a sum can be computed by considering $\mathbb{F}_{p^{n}}$-rational points of elliptic curves over $\mathbb{F}_{p}$. More specifically, we denote $\lambda_{p,n}$ as

To calculate $\lambda_{p,n}$, we will use some elementary concepts from the theory of elliptic curves [11]. Let $E/\mathbb{F}_{p}$ be the elliptic curve over $\mathbb{F}_{p}$

and $N_{p,n}$ denote the number of $\mathbb{F}_{p^{n}}$-rational points (remember the extra point at infinity) on the curve $E/\mathbb{F}_{p}$. From the results in [11, P.139, P.142], we have, for all $n\geq 1$,

with

where $\alpha$ and $\beta$ are the two conjugate complex roots of the polynomial $T^{2}+\lambda_{p,1}T+p$. We obtain an explicit and efficient formula of $\lambda_{p,n}$.

Define the following two specific character sums

and

which play a significant role in our main results. As we will see later, the determination of the $(-1)$-differential spectrum of $f(x)=x^{\frac{p^{n}+3}{2}}$ over $\mathbb{F}_{p^{n}}$ heavily depend on the computations of $\lambda^{(1)}_{p,n}$ and $\lambda^{(2)}_{p,n}$.

In the following examples, we will give the exact values of $\lambda^{(1)}_{p,n}$ and $\lambda^{(2)}_{p,n}$, respectively, for specific values of $p$.

In addition, we have the following bound on $\lambda^{(i)}_{p,n}$ for $i=1,2$.

In the end of this section, we present the following results concerning the exact values specific character sums used in Section 3.

The proof of following lemma is very similar to that of [18, Lemma 5], we will no longer prove that.

In this section, we are about to determine the $(-1)$-differential spectrum of $f$ explicitly.

We first introduce some properties of the $c$-differential spectrum of power function presented by Yan and Zhang in [19], which will be used to examine the $(-1)$-differential spectrum of $f$.