Paper 2026/1778

HOVER: Higher-Order Vanishing Endomorphism Recovery

Abstract

Higher-Order Vanishing (HOV), introduced by Hemmert and Wiemers at CRYPTO 2026, distinguishes Goppa codes and was recently extended to McEliece key recovery. We introduce a new variant of HOV, HOVER (Higher-Order Vanishing Endomorphism Recovery), that replaces the costly low-rank direction-search phase of HOV with a much faster linear algebra step. Our attack is based on the observation that the first catalecticants (the coefficient matrices of first partial derivatives) of the public HOV kernel define a contraction tensor; HOVER computes the coefficient endomorphisms that preserve all relations of this tensor. In the clean case, in which the scalar-extended HOV kernel is spanned by exactly the $m$ hidden Frobenius directions, this public algebra is $\mathbb{F}_{2^m}$, and the eigenspaces of a field generator are exactly those directions, already ordered by Frobenius. We give a basis-invariant construction, a structural classification of its possible matrix-algebra outputs, and explicit acceptance conditions that return $\bot$ rather than an uncertified output. During our end-to-end public-only experiments, we recovered keys for five Technology Innovation Institute (TII) McEliece challenges, including TII-252, whose label originally represented a $2^{252}$ brute-force enumeration model. However, our analysis does not indicate that HOVER would threaten Classic McEliece parameters in its present form. Lastly, we note that HOVER's main cryptanalytic shortcut was discovered with a large language model, underscoring the strong potential of AI cryptanalysis.

Note: Solved secret keys for TII Challenges broken with HOVER: https://github.com/mjosaarinen/tii-solved