A polynomial-time key-recovery attack on MQQ cryptosystems

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 that finds an equivalent key using the idea of so-called good keys. 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 to address in theory for MinRank attacks and also without any restriction on the number of polynomials removed from the public-key. This was not the case for previous Min- Rank like-attacks against MQ schemes. From a practical point of view, we are able to break an MQQ-SIG instance of 80 bits security in less than 2 days, and one of the more conservative MQQ-ENC instances of 128 bits security in little bit over 9 days. Altogether, our attack shows that it is very hard to design a secure public key scheme based on an easily invertible MQQ structure.