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Polly cracker, revisited. (English) Zbl 1405.94039

Summary: 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 \(\mathsf {IND}\)-\(\mathsf{CPA}\)-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.
A preliminary version appeared in ASIACRYPT 2011 [Lect. Notes Comput. Sci. 7073, 179–196 (2011; Zbl 1227.94025)]. see also G. Herold [PKC 2012, Lect. Notes Comput. Sci. 7293, 17–33 (2012; Zbl 1290.94086)].

MSC:

94A60 Cryptography
93C35 Multivariable systems, multidimensional control systems
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

Software:

SageMath; QUAD; fhe
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Full Text: DOI

References:

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