Paper 2024/1355 Original: eprint.iacr.org/2024/1355 ↗

Direct Range Proofs for Paillier Cryptosystem and Their Applications

Abstract

The Paillier cryptosystem is renowned for its applications in electronic voting, threshold ECDSA, multi-party computation, and more, largely due to its additive homomorphism. In these applications, range proofs for the Paillier cryptosystem are crucial for maintaining security, because of the mismatch between the message space in the Paillier system and the operation space in application scenarios. In this paper, we present novel range proofs for the Paillier cryptosystem, specifically aimed at optimizing those for both Paillier plaintext and affine operation. We interpret encryptions and affine operations as commitments over integers, as opposed to solely over $\mathbb{Z}_{N}$. Consequently, we propose direct range proof for the updated cryptosystem, thereby eliminating the need for auxiliary integer commitments as required by the current state-of-the-art. Our work yields significant improvements: In the range proof for Paillier plaintext, our approach reduces communication overheads by approximately $60\%$, and computational overheads by $30\%$ and $10\%$ for the prover and verifier, respectively. In the range proof for Paillier affine operation, our method reduces the bandwidth by $70\%$, and computational overheads by $50\%$ and $30\%$ for the prover and verifier, respectively. Furthermore, we demonstrate that our techniques can be utilized to improve the performance of threshold ECDSA and the DCR-based instantiation of the Naor-Yung CCA2 paradigm.