Designs, Codes and Cryptography

, Volume 79, Issue 2, pp 261–302 | Cite as

Polly Cracker, revisited

  • Martin R. Albrecht
  • Jean-Charles Faugère
  • Pooya Farshim
  • Gottfried Herold
  • Ludovic Perret
Article

Abstract

We formally treat cryptographic constructions based on the hardness of deciding ideal membership in multivariate polynomial rings. Of particular interest to us is a class of schemes known as “Polly Cracker.” We start by formalising and studying the relation between the ideal membership problem and the problem of computing a Gröbner basis. We show both positive and negative results. On the negative side, we define a symmetric Polly Cracker encryption scheme and prove that this scheme only achieves bounded \(\mathsf {CPA}\) security under the hardness of the ideal membership problem. Furthermore, we show that a large class of algebraic transformations cannot convert this scheme to a fully secure Polly Cracker-style scheme. On the positive side, we formalise noisy variants of the ideal-theoretic problems. These problems can be seen as natural generalisations of the learning with errors (\(\mathsf {LWE}\)) and the approximate GCD problems over polynomial rings. After formalising and justifying the hardness of the noisy assumptions, we show that noisy encoding of messages results in a fully \(\mathsf {IND}{\text {-}}\mathsf {CPA}\)-secure and somewhat homomorphic encryption scheme. Together with a standard symmetric-to-asymmetric transformation for additively homomorphic schemes, we provide a positive answer to the long-standing open problem of constructing a secure Polly Cracker-style cryptosystem reducible to the hardness of solving a random system of equations. Indeed, our results go beyond this and also provide a new family of somewhat homomorphic encryption schemes based on generalised hard problems. Our results also imply that Regev’s \(\mathsf {LWE}\)-based public-key encryption scheme is (somewhat) multiplicatively homomorphic for appropriate choices of parameters.

Keywords

Cryptography Multivariable systems Computational difficulty of problems 

Mathematics Subject Classification

94A60 93C35 68Q17 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Martin R. Albrecht
    • 1
  • Jean-Charles Faugère
    • 2
    • 3
    • 6
  • Pooya Farshim
    • 4
  • Gottfried Herold
    • 5
  • Ludovic Perret
    • 2
    • 3
    • 6
  1. 1.Information Security Group, Royal HollowayUniversity of LondonEghamUK
  2. 2.Sorbonne UniversitésUPMC Univ Paris 06, POLSYS, UMR 7606, LIP6ParisFrance
  3. 3.CNRS, UMR 7606, LIP6ParisFrance
  4. 4.Queen’s University BelfastBelfast, Northern IrelandUK
  5. 5.Horst Görtz Institut für IT-SicherheitRuhr-Universität BochumBochumGermany
  6. 6.INRIA, Paris-Rocquencourt CenterParisFrance

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