On Sum-Free Functions

TL;DR

The work develops and analyzes $k$th order sum-free functions on finite fields, focusing on the binary inverse $f_{\text{inv}}(x)=x^{-1}$ and establishing new confirmations of a central conjecture under several conditions, including the critical case $n=13$. It leverages tools such as the Moore determinant and Lang-Weil bounds to derive criteria and prove non-sum-freeness for many $k$, while also extending the theory to a natural $q$-ary generalization $g_{q-1}(x)=1/x^{q-1}$ that preserves many binary results and exhibits novel phenomena (notably for small $q$). The paper provides explicit constructions for small odd $n$ and smallest prime divisors, and develops general inequalities (e.g., involving the smallest prime divisor $l$ of $n$) that certify the conjecture in broad families. Together, these results deepen understanding of sum-free properties in both binary and $q$-ary settings and connect algebraic, combinatorial, and number-theoretic techniques to cryptographic relevance.

Abstract

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.

Theorems & Definitions (52)

• Conjecture 1.1
• Theorem 2.1
• proof
• Lemma 2.2: Lidl-Niederreiter-FF-1997
• Corollary 2.3
• Theorem 2.4: Carlet-pre, Carlet-Hou
• Theorem 3.1
• proof
• Lemma 3.2
• proof : Proof of Lemma \ref{['lemUV']}