Abstract. We investigate the security of the family of MQQ public key cryptosystems using multivariate quadratic quasigroups (MQQ). These cryptosystems show especially good performance properties. In particular, the MQQ-SIG signature scheme is the fastest scheme in the ECRYPT benchmarking of cryptographic systems (eBACS). We show that both the signature scheme MQQ-SIG and the encryption scheme MQQ-ENC, although using different types of MQQs, share a common algebraic structure that introduces a weakness in both schemes. We use this weakness to mount a successful polynomial time key-recovery attack. Our key-recovery attack finds an equivalent key using the idea of so-called good keys that reveals the structure gradually. In the process we need to solve a MinRank problem that, because of the structure, can be solved in polynomial-time assuming some mild algebraic assumptions. We highlight that our theoretical results work in characteristic 2 which is known to be the most difficult case ...