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Polly Cracker, revisited

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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.

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  1. Here we are slightly abusing notation and using \(P\) both for the polynomial ring and its description.

  2. Since the noise distribution \(\chi \) only enters our construction via \(\chi ^t\), this has the side effect of removing all dependencies between the coefficients. In particular, we may as well assume that \(\chi \) samples the coefficients of all monomials \(t\) independently.

  3. Here \(\mathcal {P}\) is a distribution which returns \(P = \mathbb {F} _q[x_0,\ldots ,x_{n-1}]\) with \(q\) as in the \(\mathsf {LWE}\) game. Algorithm \(\mathsf {GBGen}_{\mathsf {1={}}}(\cdot ) \) returns \([x_0 - s_0,\ldots ,x_{n-1} - s_{n-1}]\) for some \(s_i \in \mathbb {F} _q\), which is the only choice for \(d=1\).


  5. See


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We would like to thank Carlos Cid for valuable feedback and discussions on this work. We would also like to thank Frederik Armknecht for helpful discussions on an earlier draft of this work. The work described in this paper has been supported by the Royal Society Grant JP090728 and by the Commission of the European Communities through the ICT program under contract ICT-2007-216676 (ECRYPT-II). Martin R. Albrecht, Jean-Charles Faugère, and Ludovic Perret are also supported by the French ANR under the Computer Algebra and Cryptography (CAC) Project (ANR-09-JCJCJ-0064-01) and the EXACTA Project (ANR-09-BLAN-0371-01).

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Correspondence to Ludovic Perret.

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Albrecht, M.R., Faugère, JC., Farshim, P. et al. Polly Cracker, revisited. Des. Codes Cryptogr. 79, 261–302 (2016).

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