Abstract
We initiate the formal treatment of cryptographic constructions (“Polly Cracker”) based on the hardness of computing remainders modulo an ideal over multivariate polynomial rings. We start by formalising the relation between the ideal remainder 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. 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 membership, ideal remainder, and Gröbner basis problems. These problems can be seen as natural generalisations of the LWE problem and the approximate GCD problem over polynomial rings. We then show that noisy encoding of messages results in a fully IND-CPA-secure somewhat homomorphic encryption scheme. Our results provide a new family of somewhat homomorphic encryption schemes based on new, but natural, hard problems. Our results also imply that Regev’s LWE-based public-key encryption scheme is (somewhat) multiplicatively homomorphic for appropriate choices of parameters.
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). M. Albrecht, J-C. Faugère, and L. Perret were 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). P. Farshim was funded in part by the US Army Research laboratory, the UK Ministry of Defense and was accomplished under Agreement Number W911NF-06-3-0001. Due to space limitations this work is only an extended abstract of the full work available in [1].
Chapter PDF
Similar content being viewed by others
Keywords
References
Albrecht, M.R., Farshim, P., Faugère, J.-C., Perret, L.: Polly Cracker, revisited. Cryptology ePrint Archive, Report 2011/289 (2011)
Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast Cryptographic Primitives and Circular-Secure Encryption Based on Hard Learning Problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009)
Arora, S., Ge, R.: New Algorithms for Learning in Presence of Errors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 403–415. Springer, Heidelberg (2011)
Bard, G.V., Courtois, N.T., Jefferson, C.: Efficient methods for conversion and solution of sparse systems of low-degree multivariate polynomials over GF(2) via SAT-solvers. Cryptology ePrint Archive, Report 2007/024 (2007)
Barkee, B., Can, D.C., Ecks, J., Moriarty, T., Ree, R.F.: Why you cannot even hope to use Gröbner bases in public key cryptography: An open letter to a scientist who failed and a challenge to those who have not yet failed. J. of Symbolic Computations 18(6), 497–501 (1994)
Berbain, C., Gilbert, H., Patarin, J.: QUAD: A multivariate stream cipher with provable security. J. Symb. Comput. 44(12), 1703–1723 (2009)
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. To appear in FOCS 2011 (2011)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, Universität Innsbruck (1965)
Caboara, M., Caruso, F., Traverso, C.: Lattice Polly Cracker cryptosystems. Journal of Symbolic Computation 46, 534–549 (2011)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Springer, Heidelberg (2005)
Cramer, R., Shoup, V.: Design and analysis of practical public-key encryption schemes secure against adaptive chosen ciphertext attack. SIAM J. of Computing, 167–226 (2003)
Dickenstein, A., Fitchas, N., Giusti, M., Sessa, C.: The membership problem for unmixed polynomial ideals is solvable in single exponential time. Discrete Appl. Math. 33(1-3), 73–94 (1991)
van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully Homomorphic Encryption Over the Integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24–43. Springer, Heidelberg (2010)
Ding, J., Yang, B.-Y.: Multivariate public key cryptography. In: Post-Quantum Cryptography, pp. 193–234. Springer, Heidelberg (2009)
dit Vehel, F.L., Marinari, M.G., Perret, L., Traverso, C.: A survey on Polly Cracker systems. In: Gröbner Bases. Coding and Cryptography, pp. 285–305. Springer, Heidelberg (2009)
Faugère, J.-C., Gianni, P.M., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. Journal of Symbolic Computation 16, 329–344 (1993)
Fellows, M., Koblitz, N.: Combinatorial cryptosystems galore! In: Finite Fields: Theory, Applications, and Algorithms. Contemporary Mathematics, vol. 168, pp. 51–61. AMS (1994)
Gentry, C.: A fully homomorphic encryption scheme. PhD thesis, Stanford University (2009)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: ACM Symposium on Theory of Computing, pp. 169–178 (2009)
Gentry, C., Halevi, S.: Implementing Gentry’s Fully-Homomorphic Encryption Scheme. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 129–148. Springer, Heidelberg (2011)
Gouget, A., Patarin, J.: Probabilistic Multivariate Cryptography. In: Nguyên, P.Q. (ed.) VIETCRYPT 2006. LNCS, vol. 4341, pp. 1–18. Springer, Heidelberg (2006)
Lindner, R., Peikert, C.: Better Key Sizes (and Attacks) for LWE-Based Encryption. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 319–339. Springer, Heidelberg (2011)
Melchor, C.A., Gaborit, P., Herranz, J.: Additively Homomorphic Encryption with d-Operand Multiplications. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 138–154. Springer, Heidelberg (2010)
Micciancio, D., Regev, O.: Lattice-based cryptography. In: Post-Quantum Cryptography, pp. 147–191. Springer, Heidelberg (2009)
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. Journal of the ACM 56, 34:1–34:40 (2009)
Regev, O.: The learning with errors problem. In: IEEE Conference on Computational Complexity 2010, pp. 191–204 (2010)
Rothblum, R.: Homomorphic Encryption: From Private-Key to Public-Key. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 219–234. Springer, Heidelberg (2011)
Rückert, M., Schneider, M.: Estimating the security of lattice-based cryptosystems. Cryptology ePrint Archive, Report 2010/137 (2010)
Smart, N.P., Vercauteren, F.: Fully Homomorphic Encryption with Relatively Small Key and Ciphertext Sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010)
Stein, W.A., et al.: Sage Mathematics Software. The Sage Development Team, Version 4.7.0 (2011), http://www.sagemath.org
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 International Association for Cryptologic Research
About this paper
Cite this paper
Albrecht, M.R., Farshim, P., Faugère, JC., Perret, L. (2011). Polly Cracker, Revisited. In: Lee, D.H., Wang, X. (eds) Advances in Cryptology – ASIACRYPT 2011. ASIACRYPT 2011. Lecture Notes in Computer Science, vol 7073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25385-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-25385-0_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25384-3
Online ISBN: 978-3-642-25385-0
eBook Packages: Computer ScienceComputer Science (R0)