Paper 2002/138

On the Security of HFE, HFEv- and Quartz

Nicolas T. Courtois, Magnus Daum, and Patrick Felke


Quartz is a signature scheme based on an HFEv- trapdoor function published at Eurocrypt 1996. In this paper we study "inversion" attacks for Quartz, i.e. attacks that solve the system of multivariate equations used in Quartz. We do not cover some special attacks that forge signatures without inversion. We are interested in methods to invert the HFEv- trapdoor function or at least to distinguish it from a random system of the same size. There are 4 types of attacks known on HFE: Shamir-Kipnis, Shamir-Kipnis-Courtois, Courtois, and attacks related to Gröbner bases such as the F5/2 attack by Jean Charles Faugère. No attack has been published so far on HFEv- and it was believed to be more secure than HFE. In this paper we show that even modified HFE systems can be successfully attacked. It seems that the complexity of the attack increases by at least a factor of $q^{tot}$ with $tot$ being the total number of perturbations in HFE. From this and all the other known attacks we will estimate what is the complexity of the best "inversion" attack for Quartz.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. -
asymmetric cryptographyfinite fieldsmultivariate cryptanalysisGröbner basesHidden Field EquationHFE problemQuartzNessie project
Contact author(s)
Magnus Daum @ ruhr-uni-bochum de
2002-09-17: revised
2002-09-12: received
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Creative Commons Attribution


      author = {Nicolas T.  Courtois and Magnus Daum and Patrick Felke},
      title = {On the Security of HFE, HFEv- and Quartz},
      howpublished = {Cryptology ePrint Archive, Paper 2002/138},
      year = {2002},
      note = {\url{}},
      url = {}
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