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Practical Cryptanalysis of a Public-Key Encryption Scheme Based on New Multivariate Quadratic Assumptions

  • Martin R. Albrecht
  • Jean-Charles Faugére
  • Robert Fitzpatrick
  • Ludovic Perret
  • Yosuke Todo
  • Keita Xagawa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8383)

Abstract

In this paper, we investigate the security of a public-key encryption scheme introduced by Huang, Liu and Yang (HLY) at PKC’12. This new scheme can be provably reduced to the hardness of solving a set of quadratic equations whose coefficients of highest degree are chosen according to a discrete Gaussian distributions. The other terms being chosen uniformly at random. Such a problem is a variant of the classical problem of solving a system of non-linear equations (PoSSo), which is known to be hard for random systems. The main hypothesis of Huang, Liu and Yang is that their variant is not easier than solving PoSSo for random instances. In this paper, we disprove this hypothesis. To this end, we exploit the fact that the new problem proposed by Huang, Liu and Yang reduces to an easy instance of the Learning With Errors (LWE) problem. The main contribution of this paper is to show that security and efficiency are essentially incompatible for the HLY proposal. That is, one cannot find parameters which yield a secure and a practical scheme. For instance, we estimate that a public-key of at least \(1.03~\textnormal{GB}\) is required to achieve 80-bit security against the simplest of our attacks. As a proof of concept, we present 3 practical attacks against all the parameters proposed by Huang, Liu and Yang. With the most efficient attack, we have been able to recover the private-key in roughly 5 minutes for the first challenge (i.e. Case 1) proposed by HLY and less than 30 minutes for the second challenge (i.e. Case2).

Keywords

Lattice Reduction Homomorphic Encryption Dual Lattice Modulus Reduction Short Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  • Martin R. Albrecht
    • 1
  • Jean-Charles Faugére
    • 3
    • 2
    • 4
  • Robert Fitzpatrick
    • 5
  • Ludovic Perret
    • 2
    • 3
    • 4
  • Yosuke Todo
    • 6
  • Keita Xagawa
    • 6
  1. 1.Technical University of DenmarkDenmark
  2. 2.POLSYS, UMR 7606, LIP6Sorbonne Universités, UPMC Univ Paris 06ParisFrance
  3. 3.Paris-Rocquencourt Center, POLSYS ProjectINRIAFrance
  4. 4.UMR 7606, LIP6CNRSParisFrance
  5. 5.Information Security GroupRoyal Holloway, University of LondonEghamUnited Kingdom
  6. 6.NTT Secure Platform LaboratoriesJapan

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