发表机构
University of South Florida（南佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广二元和$q$元函数和自由性的已有结果，给出Welch函数二阶和自由性的新证明，并研究三阶和自由性，提出有数值支持的猜想。
AI 中文摘要
二元函数的和自由性概念最近由C. Carlet引入，作为密码学中使用的APN函数的推广；该概念的$q$元版本是自然扩展。对于满足$0\le k\le n$的每个整数$k$，在$\Bbb F_{2^n}$上存在一个$k$阶和自由函数。已知当$k/n$不接近0或1时，$\Bbb F_{2^n}$上的乘法逆函数不是$k$阶和自由的。我们将这两个结果推广到$q$元函数。APN函数具有编码理论特征。我们将该特征推广到任意有限域上任意阶的和自由函数。众所周知，Welch函数是2阶和自由的。我们给出该结果的另一种证明，这引出一个更一般的代数问题。我们还研究了Welch函数和代数次数为3的幂函数的3阶和自由性。我们提出了关于Welch函数3阶和自由性的猜想，该猜想得到强数值证据的支持。
英文摘要
The notion of sum-freedom of binary functions was introduced recently by C. Carlet as a generalization of the APN functions used in cryptography; the $q$-ary version of the notion is a natural extension. For each integer $k$ with $0\le k\le n$, there is a $k$th order sum-free function on $\Bbb F_{2^n}$. It is also known that when $k/n$ is not close to 0 or 1, the multiplicative inverse function on $\Bbb F_{2^n}$ is not $k$th order sum-free. We generalize these two results to $q$-ary functions. APN functions have a coding theoretic characterization. We generalize the characterization to sum-free functions of arbitrary order over any finite field. It is well known that the Welch functions is 2nd order sum-free. We give an alternative proof for this result which leads to a more general algebraic question. We also investigate that the 3rd order sum-freedom of the Welch function and power functions of algebraic degree 3. We formulate a conjecture about the 3rd order sum-freedom of the Welch function which is supported by strong numerical evidence.
Comments21 pages
