Finding Short Integer Solutions When the Modulus Is Small

We present cryptanalysis of the inhomogenous short integer solution (ISIS) problem for anomalously small moduli q 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 q versions of the lattice signature scheme Falcon and its variant Mitaka. For one small q parametrisation of Falcon we reduce the estimated security against signature forgery by approximately 26 bits. For one small q parametrisation of Mitaka we successfully forge a signature in 15 seconds.