Quantum-Resistant Cryptography via Universal Gr\"obner Bases

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.