**A Generalization of Paillier's Public-Key System With Fast Decryption**

*Ying Guo and Zhenfu Cao and Xiaolei Dong*

**Abstract: **In Paillier's scheme, $c=y^{m}x^{n}\,\mathrm{mod}\,n^{2},\,m \in Z_{n},\,x \in Z_{n^{2}}^{*},\,n=PQ$ is a product of two large primes. Damgård and Jurik generalized Paillier's scheme to reduce the ciphertext expansion, $c=y^{m}x^{n^{s}}\,\mathrm{mod}\,n^{s+1},\,m \in Z_{n^{s}},\,x \in Z_{n^{s+1}}^{*}$. In this paper, we propose a new generalization of Paillier's scheme and prove that our scheme is IND-CPA secure under $k$-subgroup assumption for $\Pi_{k}$. Compared to Damgård and Jurik's generalization, our scheme has three advantages. (a)We use the modulus $P^{a}Q^{b}$ instead of $P^{a}Q^{a}$, so it is more general. (b)We use a general $y$ satisfying $P^{a-1} | order_{P^{a}}(y), \,Q^{b-1} | order_{Q^{b}}(y)$ instead of $y=(1+PQ)^{j}x \,\mathrm{mod}\,N$ which is used in Damgård and Jurik's generalization. (c)Our decryption scheme is more efficient than Damgård and Jurik's generalization system.

**Category / Keywords: **public-key cryptography / public-key cryptography, discrete logarithm problem,

**Date: **received 25 Jun 2020

**Contact author: **sjtuguoying at 126 com

**Available format(s): **PDF | BibTeX Citation

**Version: **20200627:185557 (All versions of this report)

**Short URL: **ia.cr/2020/796

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