论文标题
在奇数有限字段及其$ c $ -Differential属性上的一类APN功率功能的差分光谱上
On the differential spectrum of a class of APN power functions over odd characteristic finite fields and their $c$-differential properties
论文作者
论文摘要
在文献中，已经研究了三类几乎完美的非线性（对于简短的APN）功率功能，并确定了它们的差异光谱。功率函数的差异均匀性$ f（x）= x^{\ frac {p^{n} -3} -3} {2}} $上的有限字段$ f_ {p^n} $ $ f_ {p^n} $ $ p^n $（$ p $ p $ p $是奇怪的prime），在1997年$ p pm pm p pm pm pm pm pm pm pm pm pm pm pm pm pm pm off。用$ p^n> 7 $的电源。结果表明，当$ p^n = 27 $，$ 5 $是$ f_ {p^n} $的非Quare时，$ f $是pn，而当$ 3 $ 3 $ - 5 $是$ f_ {p^n} $时，则$ 3 $ - 均匀。在本文中，通过研究某些方程式系统和某些字符总和$ f_ {p^n} $，可以完全确定$ f $的差异光谱。我们专注于功率函数$ x^d $，即使$ d $超过$ f_ {p^n} $（$ p $奇数），我们认为的功率函数$ f $是APN，它具有最低的差异均匀性和非平凡的差异光谱。此外，我们通过调查$ f $的$ c $ differtial属性来研究所谓的$ c $差异均匀性的扩展。具体而言，给出了$ f $的$ c $差异均匀性的上限，并且在$ c = -1 $的情况下，考虑了其$ c $ -Differential Spectrum。最后，我们强调的是，在我们对所考虑的功率函数的差异频谱的整个研究中，我们提供了评估具有与椭圆曲线连接的特定字符总和的方法，并确定了有限场上方程式系统的解决方案的解决方案数量。
Only three classes of Almost Perfect Nonlinear (for short, APN) power functions over odd characteristic finite fields have been investigated in the literature, and their differential spectra were determined. The differential uniformity of the power function $F(x)=x^{\frac{p^{n}-3}{2}}$ over the finite field $F_{p^n}$ of order $p^n$ (where $p$ is an odd prime), was studied by Helleseth and Sandberg in 1997, where $p^n\equiv3\pmod{4}$ is an odd prime power with $p^n>7$. It was shown that $F$ is PN when $p^n=27$, APN when $5$ is a nonsquare in $F_{p^n}$, and differentially $3$-uniform when $5$ is a square in $F_{p^n}$. In this paper, by investigating some equation systems and certain character sums over $F_{p^n}$, the differential spectrum of $F$ is completely determined. We focusing on the power functions $x^d$ with even $d$ over $F_{p^n}$ ($p$ odd), the power functions $F$ we consider are APN which are of the lowest differential uniformity and the nontrivial differential spectrum. Moreover, we examine the extension of the so-called $c$-differential uniformity by investigating the $c$-differential properties of $F$. Specifically, an upper bound of the $c$-differential uniformity of $F$ is given, and its $c$-differential spectrum is considered in the case where $c=-1$. Finally, we emphasize that, throughout our study of the differential spectrum of the considered power functions, we provide methods for evaluating sums of specific characters with connections to elliptic curves and for determining the number of solutions of specific systems of equations over finite fields.
