The development of secure cryptographic protocols and the subsequent attack mechanisms have been placed in the literature with the utmost curiosity. While sophisticated quantum attacks bring a concern to the classical cryptographic protocols present in the applications used in everyday life, the necessity of developing post-quantum protocols is felt primarily. In post-quantum cryptography, elliptic curve-base protocols are exciting to the researchers. While the comprehensive study of elliptic curves over finite fields is well known, the extended study over finite rings is still missing. In this work, we generalize the study of Weierstrass elliptic curves over finite ring $\mathbb{Z}_n$ through classification. Several expressions to compute critical factors in studying elliptic curves are conferred. An all-around computational classification on the Weierstrass elliptic curves over $\mathbb{Z}_n$ for rigorous understanding is also attached to this work.

翻译：密码协议的安全性构建及其对应的攻击机制始终是文献中备受关注的核心议题。随着量子攻击手段日趋复杂，日常生活应用中的经典密码协议面临严峻挑战，因此开发后量子密码协议的需求尤为迫切。在后量子密码学领域，基于椭圆曲线的协议深受研究者青睐。尽管有限域上椭圆曲线的系统性研究已臻成熟，但有限环上的扩展研究仍属空白。本研究通过分类方法，系统性地推广了有限环 $\mathbb{Z}_n$ 上魏尔斯特拉斯椭圆曲线的研究。文中给出了若干用于计算椭圆曲线关键因子的表达式，并附带了一套针对 $\mathbb{Z}_n$ 上魏尔斯特拉斯椭圆曲线的全方位计算分类，以期深入理解该课题。