Finding short integer solutions when the modulus is small

We present cryptanalysis of the inhomogenous short integer solution (ISIS) problem for anomalously small moduli  by exploiting the geometry of BKZ reduced bases of q-ary lattices.

We apply this cryptanalysis to examples from the literature where taking such small moduli has been suggested. A recent work [Espitau–Tibouchi–Wallet–Yu, CRYPTO 2022] suggests small  versions of the lattice signature scheme FALCON and its variant MITAKA.

For one small  parametrisation of FALCON we reduce the estimated security against signature forgery by approximately 26 bits. For one small  parametrisation of MITAKA we successfully forge a signature in  seconds.