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.

翻译：考虑一类具有两个参数$l$和$k$的指数函数$e(l,k)$的无限族，并推导出$e(l,k)$在$\mathbb{F}_{2^n}$上为0-APN的充分条件。这些条件允许我们针对每一组$l$和$k$的选取，高效生成使得$x^{e(l,k)}$为0-APN的维度$n$的无限列表（效率远超一般情况）。观察到Gold和Inverse指数以及Gold指数的逆均可通过适当选取$l$和$k$表示为$e(l,k)$形式。我们刻画了$e(l,k)$可循环等价于Gold、Kasami、Welch、Niho和Inverse指数族中代表元的所有情形，并确定了$e(l,k)$可与Dobbertin指数（不考虑逆元）位于同一循环陪集中的条件，同时提供计算数据表明Dobbertin逆元从不与$e(l,k)$等价。通过对小参数$l$和$k$在$n \le 100$的$\mathbb{F}_{2^n}$上进行APN性计算测试，并勾勒出当前技术条件下此类测试的极限边界。结论表明，在已测试函数中除已知类之外不存在APN单项式。