Protocol design of non-linear function in secure multi-party computation based on secret sharing

Secure Multi-Party Computation (MPC) enables a group of untrusted parties to collaboratively compute the output of a specified function, while ensuring that each party’s private input remains confidential. Coupled with secret sharing, MPC facilitates privacy-preserving computations, a technique increasingly utilized in diverse fields, such as machine learning. While efficient protocols exist within MPC for linear functions, the evaluation of non-linear functions presents a significant challenge. Existing methods for non-linear functions are often either inefficient or lack the generality for widespread adoption, making them a major impediment in both the design and practical implementation of MPC schemes. In this study, we explore the development of a generic protocol for non-linear function computation in MPC, grounded in secret sharing. We have devised a series of protocols to compute fundamental non-linear functions in a three-party setting under a semi-honest security model, representing secret-shared decimal numbers in fixed-point format. These protocols include Π exp for exponential functions, Π log for logarithmic functions, and Π Inv for inverse proportion functions. By integrating these basic functions, we can formulate protocols for a broad spectrum of non-linear functions. Specifically, we have developed the Π Sigmoid and Π Tanh protocols based on the aforementioned methods. Throughout this paper, unless otherwise specified, comparisons refer exclusively to secret-sharing-based (SS-based) MPC protocols in the three-party, semi-honest setting; constant-round garbled-circuit (GC) approaches are outside our comparison scope due to different cost trade-offs. Within this SS-based literature, our protocols offer the lowest online communication rounds. Furthermore, Π exp and Π inv support an extended range of inputs, and Π log represents the first protocol capable of handling logarithmic functions with fixed-point inputs. This paper provides a thorough analysis of the security and performance of these innovative protocols.