On the Complexity of “Superdetermined” Minrank Instances

The Minrank (MR) problem is a computational problem closely related to attacks on code- and multivariate-based schemes. In this paper we revisit the so-called Kipnis-Shamir (KS) approach to this problem. We extend previous complexity analysis by exposing non-trivial syzygies through the analysis of the Jacobian of the resulting system, with respect to a group of variables. We focus on a particular set of instances that yield a very overdetermined system which we refer to as “superdetermined”. We provide a tighter complexity estimate for such instances and discuss its implications for the key recovery attack on some multivariate schemes. For example, in HFE the speedup is roughly a square root.