Gröbner Bases Techniques in Post-Quantum Cryptography

After the publication of Shor's algorithm, it became evident the most popular public-key cryptographic systems that rely on the integer factorization problem or on the discrete logarithm problem would be easily solvable using large enough quantum computers (if such quantum computers are ever built). That triggered a vivid interest in the research of cryptographic algorithms (mostly public-key cryptographic systems) that are resistant to quantum computers. There are several approaches for designing post-quantum public key algorithms. The main candidates Hash-Based signatures, are Multivariate cryptosystems, Code-based cryptosystems and Lattice-based cryptosystems.<br/> Although Multivariate, Code-based and Lattice-based Cryptosystems rely on different algebraic problems, the goal of this talk is to show that Gr\"obner bases, via algebraic attacks, is a common tool which is underlying the security of these post-quantum public-key algorithms. As such, Gröbner bases is then a fundamental tool to design and analyze public-key schemes in the post-quantum era. In the talk, we will describe two applications. First, an efficient attack against a quantum money scheme proposed at STOC'12 whose security is based on a variant of the Isomorphism of Polynomial. This is [1] a joint work with Marta Conde, and Jean-Charles Faugère. Secondly, we will also show that Gröbner Bases can be used to analyze the security of the Learning With Errors (LWE) problem. Arora and Ge proved few years ago that LWE can be solved by linearizing a system of algebraic equations. We will describe a refined complexity analysis for solving Arora-Ge-style systems of non-linear equations with Gröbner bases for LWE and a variant of LWE BinaryError-LWE. This is [2] a joint work with Martin Albrecht, Carlos Cid, Jean-Charles Faugère, and Robert Fitzpatrick.<br/> [1] Marta Conde Pena, Jean-Charles Faugère, Ludovic Perret. Algebraic Cryptanalysis of a Quantum Money Scheme The Noise-Free Case. PKC'15. [2] Martin R. Albrecht, Carlos Cid, Jean-Charles Faugère, Robert Fitzpatrick, Ludovic Perret. Algebraic Algorithms for LWE Problems. IACR Cryptology ePrint Archive 2014: 1018 (2014).