Abstract: Homomorphic encryption(HE) is considered to be one of the most important cryptographic techniques in privacy protection applications. The SM9 algorithm is a commercial cryptographic algorithm standard published in China, but it does not has the homomorphic property. In this paper, based on the algorithm of SM9, we designs a cryptographic scheme with homomorphic properties based on PKI system. The newly designed algorithm simultaneously possesses the homomorphic properties of addition and multiplication. It satisfy the addition homomorphic property of arbitrary ciphertext and a ciphertext multiplication operation,  Additionally,  the ciphertext still satisfies the addition homomorphic property and scalar multiplication after multiplication operations. This scheme is suitable for privacy calculation: such asquadratic polynomials and provides selective plaintext attack (INDCPA) security.

Key words: homomorphic encryption, elliptic curves, bilinear pairing, SM9, privacy computing

摘要： 同态加密(homomorphic encryption, HE)被认为是隐私保护应用中最重要的密码技术之一.SM9算法是我国发布的商用密码算法标准，但SM9算法并不满足同态的属性.以SM9的算法为基础，设计一种基于PKI体制且具有同态性质的加密方案，使得新设计的算法同时具备加法和乘法同态性质，即满足任意次密文加法运算和1次密文乘法运算，且密文经过乘法运算后仍然满足加法同态性质以及标量乘，适合2次多项式的隐私计算.该方案具备选择明文攻击(INDCPA)安全性.

关键词: 同态加密, 椭圆曲线, 双线性对, SM9, 隐私计算