In this article, we explore the use of universal Gr\"obner bases in public-key cryptography by proposing a key establishment protocol that is resistant to quantum attacks. By utilizing a universal Gr\"obner basis $\mathcal{U}_I$ of a polynomial ideal $I$ as a private key, this protocol leverages the computational disparity between generating the universal Gr\"obner basis needed for decryption compared with the single Gr\"obner basis used for encryption. The security of the system lies in the difficulty of directly computing the Gr\"obner fan of $I$ required to construct $\mathcal{U}_I$. We provide an analysis of the security of the protocol and the complexity of its various parameters. Additionally, we provide efficient ways to recursively generate $\mathcal{U}_I$ for toric ideals of graphs with techniques which are also of independent interest to the study of these ideals.

翻译：本文探讨了在公钥密码学中应用通用Gröbner基的方法，提出了一种能够抵抗量子攻击的密钥建立协议。该协议将多项式理想$I$的通用Gröbner基$\mathcal{U}_I$作为私钥，利用解密所需的通用Gröbner基生成与加密所用的单一Gröbner基之间的计算差异。系统的安全性依赖于直接计算构造$\mathcal{U}_I$所需的$I$的Gröbner扇的困难性。我们对协议的安全性及其各项参数的复杂度进行了分析。此外，针对图的环面理想，我们提供了递归生成$\mathcal{U}_I$的高效方法，这些技术本身对这类理想的研究也具有独立价值。