In this paper, we study the problem of securely computing a function over a network, where both the target function and the security function are vector linear. The network is modeled as a directed acyclic graph. A sink node wishes to compute a function of messages generated by multiple distributed sources, while an eavesdropper can access exactly one wiretap set from a given collection. The eavesdropper must be prevented from obtaining any information about a specified security function of the source messages. The secure computing capacity is the maximum average number of times that the target function can be securely computed with zero error at the sink node with the given collection of wiretap sets and security function for one use of the network. We establish two upper bounds on this capacity, which hold for arbitrary network topologies and for any vector linear target and security functions. These bounds generalize existing results and also lead to a new upper bound when the target function is the sum over a finite field. For the lower bound, when the target function is the sum, we extend an existing method, which transforms a non-secure network code into a secure one, to the case where the security function is vector linear. Furthermore, for a particular class of networks and a vector linear target function, we characterize the required properties of the global encoding matrix to construct a secure vector linear network code.

翻译：本文研究在网络上安全计算函数的问题，其中目标函数与安全函数均为向量线性函数。网络建模为有向无环图。汇聚节点期望计算多个分布式源节点生成消息的某个函数，而窃听者能够从给定集合中恰好访问一个窃听线路集。必须防止窃听者获取关于源消息的指定安全函数的任何信息。安全计算容量是指在给定窃听线路集集合与安全函数条件下，单次使用网络时，汇聚节点能够以零误差安全计算目标函数的最大平均次数。我们建立了该容量的两个上界，这两个上界适用于任意的网络拓扑以及任意的向量线性目标函数与安全函数。这些上界推广了现有结果，并在目标函数为有限域上的求和时导出了一个新的上界。在下界方面，当目标函数为求和时，我们将一种将非安全网络编码转化为安全网络编码的现有方法推广到安全函数为向量线性的情形。此外，针对特定类别的网络与向量线性目标函数，我们刻画了构建安全向量线性网络编码所需的全局编码矩阵的性质。