Abstract
In this paper, we consider the Polly Cracker with Noise (PCN) cryptosystem by Albrecht, Farshim, Faugère, and Perret (Asiacrypt 2011), which is a public-key cryptosystem based on the hardness of computing Gröbner bases for noisy random systems of multivariate equations. We examine four settings, covering all possible parameter ranges of PCN with zero-degree noise. In the first setting, the PCN cryptosystem is known to be equivalent to Regev’s LWE-based scheme. In the second, it is known to be at most as secure as Regev’s scheme. We show that for one other settings it is equivalent to a variants of Regev’s with less efficiency and in the last setting it is completely insecure and we give an efficient key-recovery attack. Unrelated to the attack, we also fix some flaws in the security proofs of PCN.
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Albrecht, M.R., Farshim, P., Faugère, J.-C., Perret, L.: Polly Cracker, Revisited. In: Lee, D.H. (ed.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 179–196. Springer, Heidelberg (2011); Cryptology ePrint Archive, Report 2011/289, http://eprint.iacr.org/
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)
Becker, T., Weispfenning, V.: Gröbner bases: a computational approach to commutative algebra. Graduate Texts in Mathematics. Springer (1993)
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. To appear in FOCS 2011 (2011); Cryptology ePrint Archive, Report 2011/344 (2011), http://eprint.iacr.org/
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, Universität Innsbruck (1965)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Springer (2005)
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 (2009)
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)
Lyubashevsky, V., Peikert, C., Regev, O.: On Ideal Lattices and Learning with Errors over Rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010)
Peikert, C., Vaikuntanathan, V., Waters, B.: A Framework for Efficient and Composable Oblivious Transfer. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 554–571. Springer, Heidelberg (2008)
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2005), pp. 84–93. ACM (2005)
Regev, O.: The learning with errors problem (invited survey). In: IEEE Conference on Computational Complexity, pp. 191–204. IEEE Computer Society Press (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)
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Herold, G. (2012). Polly Cracker, Revisited, Revisited. In: Fischlin, M., Buchmann, J., Manulis, M. (eds) Public Key Cryptography – PKC 2012. PKC 2012. Lecture Notes in Computer Science, vol 7293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30057-8_2
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DOI: https://doi.org/10.1007/978-3-642-30057-8_2
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