Function Computation Without Secure Links: Information and Leakage Rates

Consider <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> users, who each hold private data, and one fusion center who must compute a function of the private data of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> users. To accomplish this task, each user may utilize a public and noiseless broadcast channel in a non-interactive manner. In this setting, and in the absence of any additional resources such as secure links, we study the optimal communication rates and minimum information leakages on the private user data that are achievable. Specifically, we study the information leakage of the user data at the fusion center (beyond the knowledge of the function output), as well as at predefined groups of colluding users who eavesdrop one another. We derive the capacity region when the user data is independent, and inner and outer regions for the capacity region when the user data is correlated.