Paper 2025/928

HAWK: Having Automorphisms Weakens Key

Abstract

The search rank-2 module Lattice Isomorphism Problem (smLIP), over a cyclotomic ring of degree a power of two, can be reduced to an instance of the Lattice Isomorphism Problem (LIP) of at most half the rank if an adversary knows a nontrivial automorphism of the underlying integer lattice. Knowledge of such a nontrivial automorphism speeds up the key recovery attack on HAWK at least quadratically, which would halve the number of security bits. Luo et al. (ASIACRYPT 2024) recently found an automorphism that breaks omSVP, the initial underlying hardness assumption of HAWK. The team of HAWK amended the definition of omSVP to include this so-called symplectic automorphism in their submission to the second round of NIST's standardization of additional signatures. This work provides confidence in the soundness of this updated definition, assuming smLIP is hard, since there are plausibly no more trivial automorphisms that allow winning the omSVP game easily. Although this work does not affect the security of HAWK, it opens up a new attack avenue involving the automorphism group that may be theoretically interesting on its own.