Multi-Input Ciphertext Multiplication for Homomorphic Encryption

2026-01-21

AI Summary

This work addresses the inefficiency of existing homomorphic encryption schemes in supporting multi-input ciphertext multiplication, which hinders the performance of privacy-preserving computation. Building upon the CKKS scheme, we propose a scalable and efficient method for arbitrary numbers of input operands. Our approach restructures the three-input multiplication circuit, introduces auxiliary evaluation keys to linearize the output, and incorporates a multi-level rescaling strategy along with an input partitioning criterion to jointly control noise growth and computational complexity. When integrated with a customized hardware architecture, the proposed three-input multiplier reduces logic area by 15% and latency by 50%. For cases involving 4 to 12 inputs, the design achieves average savings of 32% in area and 45% in delay.

Technology Category

• Machine Learning: Hardware-aware ML
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Homomorphic encryption (HE) enables arithmetic operations to be performed directly on encrypted data. It is essential for privacy-preserving applications such as machine learning, medical diagnosis, and financial data analysis. In popular HE schemes, ciphertext multiplication is only defined for two inputs. However, the multiplication of multiple inputs is needed in many HE applications. In our previous work, a three-input ciphertext multiplication method for the CKKS HE scheme was developed. This paper first reformulates the three-input ciphertext multiplication to enable the combination of computations in order to further reduce the complexity. The second contribution is extending the multiplication to multiple inputs without compromising the noise overhead. Additional evaluation keys are introduced to achieve relinearization of polynomial multiplication results. To minimize the complexity of the large number of rescaling units in the multiplier, a theoretical analysis is developed to relocate the rescaling, and a multi-level rescaling approach is proposed to implement combined rescaling with complexity similar to that of a single rescaling unit. Guidelines and examples are provided on the input partition to enable the combination of more rescaling. Additionally, efficient hardware architectures are designed to implement our proposed multipliers. The improved three-input ciphertext multiplier reduces the logic area and latency by 15% and 50%, respectively, compared to the best prior design. For multipliers with more inputs, ranging from 4 to 12, the architectural analysis reveals 32% savings in area and 45% shorter latency, on average, compared to prior work.

Problem

• Homomorphic Encryption
• Ciphertext Multiplication
• Multi-Input
• Privacy-Preserving Computation
• CKKS

Innovation

• Homomorphic Encryption
• Multi-Input Ciphertext Multiplication
• CKKS Scheme
• Relinearization
• Rescaling Optimization