Paper 2014/811

A Polynomial-Time Key-Recovery Attack on MQQ Cryptosystems

Jean-Charles Faugere, Danilo Gligoroski, Ludovic Perret, Simona Samardjiska, and Enrico Thomae


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 to address in theory for MinRank attacks. Also, we emphasize that our attack works without any restriction on the number of polynomials removed from the public-key, that is, using the minus modifier. This was not the case for previous MinRank 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.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
MQ cryptographyMQQ cryptosystemsEquivalent keysGood keysMinRankGroebner bases
Contact author(s)
jean-charles faugere @ inria fr
danilog @ item ntno no
ludovic perret @ lip6 fr
simonas @ item ntno no
Enrico Thomae @ rub de
2014-10-11: received
Short URL
Creative Commons Attribution


      author = {Jean-Charles Faugere and Danilo Gligoroski and Ludovic Perret and Simona Samardjiska and Enrico Thomae},
      title = {A Polynomial-Time Key-Recovery Attack on MQQ Cryptosystems},
      howpublished = {Cryptology ePrint Archive, Paper 2014/811},
      year = {2014},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.