Skip to Main content Skip to Navigation
Journal articles

Polly Cracker, Revisited

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 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 (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 IND-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 LWE-based public-key encryption scheme is (somewhat) multiplicatively homomorphic for appropriate choices of parameters.
Document type :
Journal articles
Complete list of metadata

Cited literature [66 references]  Display  Hide  Download

https://hal.inria.fr/hal-01112976
Contributor : Ludovic Perret <>
Submitted on : Wednesday, February 4, 2015 - 9:27:34 AM
Last modification on : Thursday, March 25, 2021 - 11:44:02 AM
Long-term archiving on: : Wednesday, May 27, 2015 - 4:41:04 PM

File

mq-homomorphic (1).pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01112976, version 1

Citation

Martin Albrecht, Jean-Charles Faugère, Pooya Farshim, Gottfried Herold, Ludovic Perret. Polly Cracker, Revisited. Designs, Codes and Cryptography, Springer Verlag, 2011, pp.43. ⟨hal-01112976⟩

Share

Metrics

Record views

416

Files downloads

323