Paper 2026/765

MBU: Scalable and Constant-Round Evaluation of Non-linear Functions in Standard MPC Setting

Dongcan Guo

Zihang Zhou

Jinxuan Du

Qingshu Meng

Abstract

After more than four decades of research, multi-party computation (MPC) has achieved remarkable success in handling 2-variable multiplication and comparison-based functions (e.g., ReLU) with practical efficiency. However, for general non-linear functions—such as multiplication of many variables, power, exponential, trigonometric functions, sigmoid, softmax, and GeLU—no native MPC algorithm exists as Beaver-based mutiplication that is constant-round, scalable and exact. Existing solutions rely on either polynomial approximations (trading precision for efficiency), iterative multi-round protocols like Multiplication-to-Addition (M2A) conversion (requiring \(\log_2 k\) rounds for \(k\) parties), or Function Secret Sharing (FSS) with lookup tables (introducing quantization errors and large storage, mainly limited to 2-4 parties). These approaches suffer from fundamental trade-offs among accuracy, communication rounds, and scalability. In this work, we propose a unified \emph{mask-broadcast-unmask} design pattern that enables constant-round, scalable and \emph{approximation-free} evaluation of a wide range of non-linear functions. Our contributions include: \begin{itemize} \item A \textbf{general multiplication} protocol for \(k\) variables in \emph{one round} with optimal \(O(kn)\) communication. When \(k=2\), it reduces to the classic Beaver triple multiplication; when each secret has only one non-zero share and \(k=2\), it becomes the well-known M2A protocol. \item \textbf{Power functions} (\(x^k\)) in one round. \item \textbf{Trigonometric functions} (\(\sin x, \cos x\)) and \textbf{exponential functions} (\(a^x\)) in 4 rounds. \item \textbf{Sigmoid, softmax} in 6 rounds. \end{itemize} All these protocols are provably secure in the semi-honest model, support arbitrary number of parties, introduce \textbf{no approximation error} beyond plaintext floating-point rounding, and require only constant communication rounds (1–6) independent of function complexity. Furthermore, by restricting the random mask to a suitable range (e.g., $0 \le r <2^l-2^{l_x}$), we can reduce the rounds from 1-6 to 1-3. This work fills the long-standing gap for general non-linear functions in standard MPC settings, making privacy-preserving machine learning more practical for modern DNNs.

@misc{cryptoeprint:2026/765,
      author = {Min Yang and Dongcan Guo and Zihang Zhou and Jinxuan Du and Qingshu Meng},
      title = {{MBU}:  Scalable and Constant-Round  Evaluation of Non-linear Functions in Standard {MPC} Setting},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/765},
      year = {2026},
      url = {https://eprint.iacr.org/2026/765}
}