On sum-free functions

A function from $F_{2^n}$ to $F_{2^n}$ is said to be kth order sum-free if the sum of its values over each k-dimensional $F_2$-affine subspace of $F_{2^n}$ 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 $f_{\text{inv}}\left(x\right)=x^{-1}$ (with $0^{-1}$ defined to be 0). It is known that $f_{\text{inv}}$ is 2nd order (equivalently, $(n-2)$th order) sum-free if and only if n is odd, and it is conjectured that for $3\le k\le n-3$, $f_{\text{inv}}$ is never kth 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) $3|n$; (3) $5|n$; (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 $f_{\text{inv}}$ 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.