论文标题
一个无限的0-APN单一族的家族，带有两个参数
An infinite family of 0-APN monomials with two parameters
论文作者
论文摘要
我们考虑一个无限的指数家族$ e（l，k）$，带有两个参数，$ l $和$ k $，并在$ e（l，k）$的情况下得出了足够的条件，使得超过$ \ mathbb {f} _ {2^n} $ 0-apn。这些条件使我们能够为每种选择$ l $和$ k $，这是$ n $的无限列表，其中$ x^{e（l，k）} $比一般的0-apn效率要高得多。我们观察到，黄金和反向指数以及黄金指数的倒数可以以$ e（l，k）的形式表示，适合合适的$ l $和$ k $。我们表征了$ e（l，k）$的所有案例，可以与黄金，卡萨米，韦尔奇，NIHO和倒数指数家族的代表相等。我们表征$ e（l，k）$可以与多伯丁指数（不考虑倒置）相同的环形固定位置，并提供计算数据，表明多伯丁倒数永远不会等于$ e（l，k）$。我们计算$ e（l，k）$的apn- $ l $和$ k $ over $ \ mathbb {f} _ {2^n} $ for $ n \ le 100 $，并绘制可以使用当前可用技术执行此类测试的限制。我们得出的结论是，在已知类别之外，测试的功能之间没有APN单元。
We consider an infinite family of exponents $e(l,k)$ with two parameters, $l$ and $k$, and derive sufficient conditions for $e(l,k)$ to be 0-APN over $\mathbb{F}_{2^n}$. These conditions allow us to generate, for each choice of $l$ and $k$, an infinite list of dimensions $n$ where $x^{e(l,k)}$ is 0-APN much more efficiently than in general. We observe that the Gold and Inverse exponents, as well as the inverses of the Gold exponents can be expressed in the form $e(l,k)$ for suitable $l$ and $k$. We characterize all cases in which $e(l,k)$ can be cyclotomic equivalent to a representative from the Gold, Kasami, Welch, Niho, and Inverse families of exponents. We characterize when $e(l,k)$ can lie in the same cyclotomic coset as the Dobbertin exponent (without considering inverses) and provide computational data showing that the Dobbertin inverse is never equivalent to $e(l,k)$. We computationally test the APN-ness of $e(l,k)$ for small values of $l$ and $k$ over $\mathbb{F}_{2^n}$ for $n \le 100$, and sketch the limits to which such tests can be performed using currently available technology. We conclude that there are no APN monomials among the tested functions, outside of known classes.
