(2026-05) Cryptanalysis of Definite and Indefinite Lattice Isomorphism Problems With Applications to HAWK and DEFI
2026-05-06
We study the Lattice Isomorphism Problem (LIP) for both indefinite and definite quadratic forms, with applications to the signature schemes DEFI and HAWK. By combining arithmetic and algorithmic techniques, we obtain efficient attacks on DEFIv2, an efficient digital signature scheme based on isotropic quadratic forms. Our approach to the Decision/Distinguishing-LIP draws on the arithmetic theory of quadratic forms, with particular emphasis on indefinite forms of dimension at least~. We show that such forms arise naturally in the analysis of DEFI and prove that, under suitable assumptions, the genus, spinor genus, and equivalence class coincide. This structural collapse leads to a classical polynomial-time algorithm for the Decision/Distinguishing-LIP instances obtained from DEFI. In addition, we present a quantum polynomial-time algorithm for recovering the secret key of DEFIv2 and demonstrate practical signature forgeries within minutes using the authors' public challenge instances. Finally, we evaluate t