A function from $\Bbb F_{2^n}$ to $\Bbb F_{2^n}$ is said to be {\em $k$th order sum-free} if the sum of its values over each $k$-dimensional $\Bbb F_2$-affine subspace of $\Bbb F_{2^n}$ is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function $f_{\text{\rm inv}}(x)=x^{-1}$ (with $0^{-1}$ defined to be $0$). It is known that $f_{\text{\rm inv}}$ is 2nd order (equivalently, $(n-2)$th order) sum-free if and only if $n$ is odd, and it is conjectured that for $3\le k\le n-3$, $f_{\text{\rm inv}}$ is never $k$th order sum-free. The conjecture has been confirmed for even $n$ but remains open for odd $n$. In the present paper, we show that the conjecture holds under each of the following conditions: (1) $n=13$; (2) $3\mid n$; (3) $5\mid n$; (4) the smallest prime divisor $l$ of $n$ satisfies $(l-1)(l+2)\le (n+1)/2$. We also determine the ``right'' $q$-ary generalization of the binary multiplicative inverse function $f_{\text{\rm inv}}$ in the context of sum-freedom. This $q$-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case.

翻译：若函数从 $\Bbb F_{2^n}$ 到 $\Bbb F_{2^n}$ 在每个 $k$ 维 $\Bbb F_2$-仿射子空间上的函数值之和均非零，则称该函数为{\em $k$阶无和}函数。这一概念由 C. Carlet 最近引入，除其他意义外，它是对 APN 函数的一种推广。这一新课题的核心是关于乘法逆函数 $f_{\text{\rm inv}}(x)=x^{-1}$（其中定义 $0^{-1}=0$）的无和性的一个猜想。已知 $f_{\text{\rm inv}}$ 是 2 阶（等价地，$(n-2)$阶）无和的当且仅当 $n$ 为奇数，并且猜想对于 $3\le k\le n-3$，$f_{\text{\rm inv}}$ 永远不是 $k$ 阶无和的。该猜想在 $n$ 为偶数时已被证实，但在 $n$ 为奇数时仍然悬而未决。在本文中，我们证明了在以下任一条件下该猜想成立：(1) $n=13$；(2) $3\mid n$；(3) $5\mid n$；(4) $n$ 的最小素因子 $l$ 满足 $(l-1)(l+2)\le (n+1)/2$。我们还确定了在无和性背景下，二进制乘法逆函数 $f_{\text{\rm inv}}$ 的“正确” $q$ 元推广。这一 $q$ 元推广不仅保持了其二进制版本的大多数结果，而且展现出一些在二进制情况下未观察到的非凡现象。