Secure network function computation is a critical research direction in network coding, which aims to ensure that the target function is correctly computed at the sink node while preventing the wiretapper from obtaining any information about the security function. In this paper, we focus on the general secure network function computation model, where the target function f and the security function ζ are arbitrary, and the wiretapper can eavesdrop on any subset of edges with size at most a given security level. Using information-theoretic techniques, we establish a nontrivial upper bound on the secure computing capacity, which is applicable to arbitrary networks, arbitrary target and security functions, and arbitrary security levels. This upper bound is shown to degenerate to the existing bounds in the literature when the target and security functions are specific forms. Furthermore, we consider two specific models: one where the target function is vector-linear and the security function is the identity function, and another where both functions are vector-linear. For the former, we derive a simplified form of the upper bound on the secure computing capacity via order-theoretic methods and propose an efficient algorithm to compute this bound with linear time complexity in the number of network edges. For the latter, we characterize the equivalent conditions for the computability and security of linear secure network codes, develop two constructive schemes for such codes, and derive an upper bound on the minimal finite field size required for the constructions, thereby obtaining a nontrivial lower bound on the secure computing capacity.

翻译：安全网络函数计算是网络编码中的一个关键研究方向，旨在确保目标函数在宿节点被正确计算，同时阻止窃听者获取任何与安全函数相关的信息。本文聚焦于通用安全网络函数计算模型，其中目标函数f和安全函数ζ是任意的，且窃听者最多能窃听给定安全等级所限定的任意边子集。利用信息论技术，我们建立了一个关于安全计算容量的非平凡上界，该上界适用于任意网络、任意目标与安全函数以及任意安全等级。当目标与安全函数具有特定形式时，该上界可退化为文献中已有的界。此外，我们考虑了两类特例模型：一类中目标函数为向量线性函数且安全函数为恒等函数，另一类中两者均为向量线性函数。针对前者，我们通过序理论方法导出了安全计算容量上界的简化形式，并提出了一种高效算法，其时间复杂度与网络边数呈线性关系。针对后者，我们刻画了线性安全网络码的可计算性与安全性的等价条件，为此类码设计了两类构造方案，推导了构造所需最小有限域大小的上界，并由此获得了安全计算容量的非平凡下界。