Revisiting the minrank problem on multivariate cryptography

The minrank problem is often considered in the cryptanalysis of multivariate cryptography and code-based cryptography. There have been many multivariate cryptosystems proven insecure due to their weakness against the minrank attack, which is an attack that transforms breaking a cryptosystem into solving a minrank problem instance. In this paper, we review two existing methods, the Kipnis-Shamir method (KS), and minors modeling for solving a minrank instance, and then propose a mixed method that merges these two methods. Our method uses a bilinear subsystem from the KS method and a subsystem from minors modeling. It is at least as effective as the KS method, and does not require as many minors as minors modeling. Moreover, we consider applying the hybrid approach on multivariate polynomials solved in our mixed method to further improve our method. We then revisit the minrank attack on Rainbow and conclude the previous complexity analysis of the minrank attack on Rainbow is overestimated, and provide the correct complexity of the minrank attack on NIST PQC 2nd round Rainbow parameters.