Private Sum Computation: Trade-Off Between Shared Randomness and Privacy

Consider a scenario involving multiple users and a fusion center. Each user possesses a sequence of bits and can communicate with the fusion center through a one-way public channel. The fusion center's task is to compute the sum of all the sequences under the privacy requirement that a set of colluding users, along with the fusion center, cannot gain more than a predetermined amount
$\delta$
of information, measured through mutual information, about the sequences of other users. Our first contribution is to characterize the minimum amount of necessary communication between the users and the fusion center, as well as the minimum amount of necessary shared randomness at the users. Our second contribution is to establish a connection between secure summation and secret sharing by showing that secret sharing is necessary to generate the local randomness needed for private summation, and prove that it holds true for any
$\delta\geqslant 0$
.