The concrete security of multivariate post-quantum signature schemes is coming under increasing scrutiny as the NIST standardisation process for additional signatures approaches its final stages. Among the leading candidates, the security of MAYO and QR-UOV relies on the hardness of the underdetermined multivariate quadratic (MQ) problem. This work revisits Hashimoto's algorithm for solving underdetermined systems of MQ equations, reinterpreting it as a computation of a pseudo-oil subspace. In light of this geometric point of view, we design a new algorithm that, by computing richer pseudo-oil structures, distributes algebraic work across more than two Gröbner Basis steps, subdividing the initial MQ problem into multiple subproblems that can be solved separately, while linearising multiple equations. Optimising a set of discrete parameters, we select the best trade-off between algebraic solving and combinatorial search. Concretely, our approach lowers the cost of the direct attack against Security Level I parameter sets of MAYO and QR-UOV by 7 and 8 bits, respectively.
BibTeX
@misc{cryptoeprint:2026/1122,
author = {Massimo Ostuzzi},
title = {Pseudo-Oil Subspaces and the Geometry of Underdetermined {MQ} Problems},
howpublished = {Cryptology {ePrint} Archive, Paper 2026/1122},
year = {2026},
url = {https://eprint.iacr.org/2026/1122}
}
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