Secure and Efficient $L^p$-Norm Computation for Two-Party Learning Applications

2025-09-05

AI Summary

Existing works lack a general-purpose secure multi-party computation (MPC) framework supporting arbitrary $L^p$-norms—particularly $L^1$ and $L^infty$—often treating norm computation as a black box, resulting in poor efficiency and limited applicability. This paper introduces Crypto-$L^p$, the first unified and customized two-party secure computation framework enabling efficient, provably secure evaluation of $L^1$, $L^2$, and $L^infty$ norms. By synergistically integrating secret sharing, homomorphic encryption, and norm-specific arithmetic circuit design, Crypto-$L^p$ achieves deep protocol-level optimizations. Experiments demonstrate speedups of 82–271× and 4–36× reductions in communication overhead over state-of-the-art baselines. Moreover, Crypto-$L^p$ is the first to enable secure machine learning inference with $L^p$-norms, achieving a 3× reduction in communication cost while preserving model accuracy and inference latency.

Technology Category

• Machine Learning: Privacy
• Natural Language Processing: Safety and Robustness
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Secure norm computation is becoming increasingly important in many real-world learning applications. However, existing cryptographic systems often lack a general framework for securely computing the $L^p$-norm over private inputs held by different parties. These systems often treat secure norm computation as a black-box process, neglecting to design tailored cryptographic protocols that optimize performance. Moreover, they predominantly focus on the $L^2$-norm, paying little attention to other popular $L^p$-norms, such as $L^1$ and $L^infty$, which are commonly used in practice, such as machine learning tasks and location-based services. To our best knowledge, we propose the first comprehensive framework for secure two-party $L^p$-norm computations ($L^1$, $L^2$, and $L^infty$), denoted as mbox{Crypto-$L^p$}, designed to be versatile across various applications. We have designed, implemented, and thoroughly evaluated our framework across a wide range of benchmarking applications, state-of-the-art (SOTA) cryptographic protocols, and real-world datasets to validate its effectiveness and practical applicability. In summary, mbox{Crypto-$L^p$} outperforms prior works on secure $L^p$-norm computation, achieving $82 imes$, $271 imes$, and $42 imes$ improvements in runtime while reducing communication overhead by $36 imes$, $4 imes$, and $21 imes$ for $p=1$, $2$, and $infty$, respectively. Furthermore, we take the first step in adapting our Crypto-$L^p$ framework for secure machine learning inference, reducing communication costs by $3 imes$ compared to SOTA systems while maintaining comparable runtime and accuracy.

Problem

• Lack of secure computation framework for L^p-norms
• Existing systems focus only on L^2-norm, ignoring others
• Need optimized protocols for practical two-party learning applications

Innovation

• A comprehensive framework for secure two-party L^p-norm computations
• Designed versatile for L^1, L^2, and L^∞ norms across applications
• Optimized cryptographic protocols reducing runtime and communication overhead