Parole chiave

• Hilbert function
• linear code
• cryptography

Abstract The code equivalence problem asks whether two given linear codes are isometric. It plays a fundamental role in code-based cryptography and identifying an invariant that discriminates structured codes from random ones results in mining any security proof of a scheme and could potentially lead to an attack. In this talk, we will see how the Hilbert function can be used to distinguish non-equivalent families of linear codes even when it is not possible to do so with classical invariants. In the first part, we will focus on the case of linear codes with the Hamming metric, while in the second part we will see how these ideas can be extended to $\mathbb{F}_q^m$-linear rank-metric codes. In particular, we will show that it is possible to distinguish a Gabidulin code from a random one. This talk is based on a joint work with V. Astore, M. Borello, and M. Calderini.