Secure Network Function Computation for Linear Functions—Part II: Target-Function Security
Yang Bai, Xuan Guang, Raymond W. Yeung · IEEE Transactions on Information Theory · 2026
In Part I of this two-part paper, we put forward the problem of secure network function computation.We investigated securely computing linear target functions with a wiretapper who can eavesdrop any edge subset up to a certain sizer, referred to as the security level, where the security function is the identity function, i.e., we need to prevent the source messages from being leaked to the wiretapper. The notion of this security is called source security. In the current Part II of the two-part paper, we consider another interesting model which is the same as the above one except that the security function is identical to the target function, i.e., we need to prevent the target function from being leaked to the wiretapper. The notion of this security is calledtarget-function security. We first prove a non-trivial upper bound on the secure computing capacity defined as the maximum average number of times that the target function can be securely computed with zero error at the sink node for one use of the network. The upper bound obtained is applicable to arbitrary network topologies and arbitrary security levels. In particular, when the security levelris equal to 0, the upper bound reduces to the computing capacity without security consideration. This upper bound is always not less than the upper bound obtained in Part I for source security, which accords with the relation between the secure computing capacities for target-function security and source security. We further obtain a bound on the gap between the two upper bounds, and show that the gap between the two upper bounds can be unbounded. Furthermore, we present an algebraic framework for linear (function-computing) secure codes for the target-function-security model by proving two equivalent conditions for computability and target-function security, respectively. With this framework, we develop a construction of linear secure codes for the target-function-security model and thus obtain a lower bound on the secure computing capacity. In the same spirit, we also prove an equivalent algebraic condition for source security, and so obtain an algebraic framework for linear secure codes for the source-security model. With this, we not only generalize the code construction developed in Part I for the source-security model but also can present it in a more succinct and understandable way. In addition, motivated by the fact that a source-secure code is target-function-secure but not vice versa, we compare our code construction for the target-function-security model and the generalized code construction for the source-security model, and show that the codes constructed by the generalized code construction for source security are only a very special subclass of the codes constructed by the code construction for target-function security. Further, a considerably smaller field size can be used for our code construction for target-function security compared with the generalized code construction for source security.