Quantum-Resistant Cryptography via Universal Gröbner Bases
Sergio Da Silva, Aniya Stewart
TL;DR
This work proposes quantum-resistant public-key cryptography based on universal Gröbner bases, where a private $U_I$ serves as the decryption trapdoor and security relies on the hardness of computing the Gröbner fan $GFan(I)$ and related lattice problems. The protocol allows a public setup and private key derivation via a private monomial order, enabling encryption and decryption through initial ideals derivable from $U_I$, with optional leakage controlled by a hash $\tau$. It provides a rigorous complexity discussion and shows how to initialize large, secure instances using toric graph ideals, via graph-operations that systematically expand $U(I_G)$ while keeping computations tractable. The approach connects algebraic and combinatorial structures to post-quantum security, offering a new direction for constructing scalable, algebraically rooted cryptographic primitives with potential practical impact in environments requiring quantum resistance.
Abstract
In this article, we explore the use of universal Gröbner bases in public-key cryptography by proposing a key establishment protocol that is resistant to quantum attacks. By utilizing a universal Gröbner basis $\mathcal{U}_I$ of a polynomial ideal $I$ as a private key, this protocol leverages the computational disparity between generating the universal Gröbner basis needed for decryption compared with the single Gröbner basis used for encryption. The security of the system lies in the difficulty of directly computing the Gröbner 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.