论文标题
更多无限类的类似APN的功率功能
More infinite classes of APN-like Power Functions
论文作者
论文摘要
在文献中，有许多类似APN的函数可以推广APN属性或类似于APN函数，例如局部APN函数，0-APN函数或具有Boomerang均匀性的功能2。 在本文中，我们研究了构建类似APN的无限类别但不是APN功率功能的问题。 一方面，我们找到了两种无限类别的局部apn类，但没有apn功率在$ \ gf_ {2^{2m}}} $上，$ m $偶，即，即$ \ mathcal {f} _1 _1（x）= x^x^^= x^（2^M-1）} $ with $ \ gcd（j（2^m-1）} $ $ \ mathcal {f} _2（x）= x^{j（2^m-1）+1} $，$ j = \ frac {2^m+2} {3} $。据作者所知道的，我们无限的本地APN类但不是APN功能是过去11年中仅有的两个。此外，我们还证明了这个无限类$ \ Mathcal {f} _1 $不仅具有最佳的Boomerang均匀性$ 2 $，而且具有有趣的属性，其差异均匀性严格比其Boomerang均匀性大。对于另一件事，使用多元方法，包括上述无限类$ \ Mathcal {f} _1 $，我们构造了七个新的无限类别的0-APN类，但不构造APN功率功能。
In the literature, there are many APN-like functions that generalize the APN properties or are similar to APN functions, e.g. locally-APN functions, 0-APN functions or those with boomerang uniformity 2. In this paper, we study the problem of constructing infinite classes of APN-like but not APN power functions. For one thing, we find two infinite classes of locally-APN but not APN power functions over $\gf_{2^{2m}}$ with $m$ even, i.e., $\mathcal{F}_1(x)=x^{j(2^m-1)}$ with $\gcd(j,2^m+1)=1$ and $\mathcal{F}_2(x)=x^{j(2^m-1)+1}$ with $j = \frac{2^m+2}{3}$. As far as the authors know, our infinite classes of locally-APN but not APN functions are the only two discovered in the last eleven years. Moreover, we also prove that this infinite class $\mathcal{F}_1$ is not only with the optimal boomerang uniformity $2$, but also has an interesting property that its differential uniformity is strictly greater than its boomerang uniformity. For another thing, using the multivariate method, including the above infinite class $\mathcal{F}_1$, we construct seven new infinite classes of 0-APN but not APN power functions.
