MORSE: An Efficient Homomorphic Secret Sharing Scheme Enabling Non-Linear Operation

TL;DR

MORSE addresses the limitations of Paillier-based HSS by enabling non-linear operations with a compact key size and zero correctness error. It builds on a FastPai variant, introduces a perfectly correct distributed discrete log, and implements S2C and C2S conversion protocols to support continuous computation. The core innovations are SMULHSS for secure multiplication and SCMPHSS for secure comparison, complemented by modular reductions and conversion routines, with KDM security analyzed under a single-key, two-server model. Empirical results show MORSE outperforms prior schemes by up to 9.3x in secure multiplication and reduces communication in secure comparison by up to 16.6%, illustrating its practical impact for efficient private computation on integers.

Abstract

Homomorphic secret sharing (HSS) enables two servers to locally perform functions on encrypted data directly and obtain the results in the form of shares. A Paillier-based HSS solution seamlessly achieves multiplicative homomorphism and consumes less communication costs. Unfortunately, existing Paillier-based HSS schemes suffer from a large private key size, potential calculation error, expensive computation and storage overhead, and only valid on linear operations (e.g., addition and multiplication). To this end, inspired by the Paillier cryptosystem with fast encryption and decryption, we propose MORSE, an efficient homomorphic secret sharing scheme enabling non-linear operation, which enjoys a small key size, no calculation error and low overhead. In terms of functions, MORSE supports addition, subtraction, multiplication, scalar-multiplication, and comparison. Particularly, we carefully design two conversion protocols achieving the mutual conversion between one Paillier ciphertext and two secret shares, which allows MORSE to continuously perform the above operations. Rigorous analyses demonstrate that MORSE securely outputs correct results. Experimental results show that MORSE makes a runtime improvement of up to 9.3 times in terms of secure multiplication, and a communication costs reduction of up to 16.6% in secure comparison, compared to the state-of-the-art.

Figures (3)

• Figure 1: The system model of MORSE
• Figure 2: The performance of MORSE under a varying $N$ ($l=32$)
• Figure 3: The performance of MORSE under a varying $l$ ($|N|=3072$)

Theorems & Definitions (11)

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• Definition 1
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