Paper 2026/408

Smoothing the degree of regularity for polynomial systems

Abstract

The complexity of many algebraic algorithms for solving non-linear polynomial systems of equations over finite fields, such as the XL (eXtended Linearization) algorithm or variants of the F4/F5 algorithms, is directly determined by the so called degree of regularity. In essence, we can form the Macaulay matrix at this degree, which can be thought of as the linearization of monomial multiples of the polynomials from the problem instance, and then solve the obtained linear system. The degree of regularity guarantees we can solve the system, however, it is a rather coarse parameter. This means that, sometimes, in order to provably show that we achieve overdeterminess, we end up with a heavily overdetermined Macaulay matrix which unnecessarily increases the computational cost. To reduce this coarseness, and thus avoid the unnecessary computational cost, we propose a technique for ``smoothing'' the degree of regularity that can be seen as operating at a degree in-between two integer values. Instead of the full Macaulay matrix, we consider specific submatrices that we show are sufficient to solve the given system. Under a mild assumption that generalizes the notion of semi-regularity, which we experimentally verify for a range of parameters, we show that our approach smooths the complexity of XL for the MQ (Multivariate Quadratic) problem. Finally, we apply our results to a recent UOV attack and demonstrate improvement of a few bits.