Secure Network Function Computation for General Target and Security Functions

TL;DR

This paper establishes a general information-theoretic upper bound on secure network function computation capacity, applicable to arbitrary networks and functions, and provides specific bounds and constructions for linear functions.

Contribution

It introduces a universal upper bound for secure network function computation and develops efficient algorithms and constructions for linear target and security functions.

Findings

Derived a nontrivial upper bound on secure computing capacity for arbitrary networks and functions.

Proposed an efficient linear-time algorithm to compute the upper bound for vector-linear target functions.

Characterized conditions and constructed schemes for linear secure network codes, including bounds on finite field size.

Abstract

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 {\zeta} 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.