Efficient Multiparty Protocols Using Generalized Parseval's Identity and the Theta Algebra

We propose a protocol able to show publicly addition and multiplication on secretly shared values. To this aim we developed a protocol based on the use of masks and on the FMPC (Fourier Multi-Party Computation). FMPC is a novel multiparty computation protocol of arithmetic circuits based on secret-sharing, capable to compute addition and multiplication of secrets with no communication. We achieve this task by introducing the first generalisation of Parseval's identity for Fourier series applicable to an arbitrary number of inputs and a new algebra referred to as the "Theta-algebra". FMPC operates in a setting where users wish to compute a function over some secret inputs by submitting the computation to a set of nodes, without revealing them those inputs. FMPC offloads most of the computational complexity to the end users, and includes an online phase that mainly consists of each node locally evaluating specific functions. FMPC paves the way for a new kind of multiparty computation protocols; making it possible to compute addition and multiplication of secrets stepping away from circuit garbling and the traditional algebra introduced by Donald Beaver in 1991. Our protocol is capable to compute addition and multiplication with no communication and its simplicity provides efficiency and ease of implementation.