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.
翻译:在文献中,许多类似于APN的功能一般化 APN 属性或类似于 APN 函数的 APN 函数, 例如本地 APN 函数、 0- APN 函数或具有 浮标统一性的函数 。 在本文中, 我们研究建造无限类类似 APN 而非 APN 的 APN 动力函数的问题。 首先, 我们发现两个无限的本地- APN 功能类别, 超过$gf ⁇ 2 ⁇ 2 ⁇ 2 ⁇ 2 ⁇ 2 美元。 据作者所知, 过去十一年仅发现两个无限的本地- APN1 (x) =x ⁇ j (2 ⁇ 1) = $gcd (j, 2 ⁇ m+1) 和 $gmccal{F=xj (2 ⁇ m-1) +1} 。 我们发现, 本地- 而非 APN 的 无限级功能, 而不是两个在最后十一年中发现的。 此外, 我们还证明这个无限级的 ASN ASNA 和另一个最高级的 ASUPI ASyal ASloom 。