Abstract
At EUROCRYPT ’10, van Dijk et al. presented simple fully- homomorphic encryption (FHE) schemes based on the hardness of approximate integer common divisors problems, which were introduced in 2001 by Howgrave-Graham. There are two versions for these problems: the partial version (PACD) and the general version (GACD). The seemingly easier problem PACD was recently used by Coron et al. at CRYPTO ’11 to build a more efficient variant of the FHE scheme by van Dijk et al.. We present a new PACD algorithm whose running time is essentially the “square root” of that of exhaustive search, which was the best attack in practice. This allows us to experimentally break the FHE challenges proposed by Coron et al. Our PACD algorithm directly gives rise to a new GACD algorithm, which is exponentially faster than exhaustive search. Interestingly, our main technique can also be applied to other settings, such as noisy factoring and attacking low-exponent RSA.
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References
Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than \(N^{\mbox{0.292}}\). IEEE Transactions on Information Theory 46(4), 1339 (2000)
Bostan, A., Gaudry, P., Schost, E.: Linear recurrences with polynomial coefficients and application to integer factorization and Cartier-Manin operator. SIAM Journal on Computing 36(6), 1777–1806 (2007)
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. Cryptology ePrint Archive, Report 2011/344 (2011), http://eprint.iacr.org/
Chen, Y., Nguyen, P.Q.: BKZ 2.0: Better Lattice Security Estimates. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 1–20. Springer, Heidelberg (2011)
Chen, Y., Nguyen, P.Q.: Faster Algorithms for Approximate Common Divisors: Breaking Fully-Homomorphic-Encryption Challenges over the Integers. Cryptology ePrint Archive, Report 2011/436 (2011), http://eprint.iacr.org/
Cohn, H., Heninger, N.: Approximate common divisors via lattices. Cryptology ePrint Archive, Report 2011/437 (2011)
Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Cryptology 10(4), 233–260 (1997)
Coron, J.-S., Joux, A., Mandal, A., Naccache, D., Tibouchi, M.: Cryptanalysis of the RSA Subgroup Assumption from TCC 2005. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 147–155. Springer, Heidelberg (2011)
Coron, J.-S., Mandal, A., Naccache, D., Tibouchi, M.: Fully Homomorphic Encryption over the Integers with Shorter Public Keys. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 487–504. Springer, Heidelberg (2011)
Coron, J.-S., Naccache, D., Tibouchi, M.: Public Key Compression and Modulus Switching for Fully Homomorphic Encryption over the Integers. Cryptology ePrint Archive, Report 2011/440 (2011), http://eprint.iacr.org/
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proc. STOC 2009, pp. 169–178. ACM (2009)
Gentry, C., Halevi, S.: Public challenges for fully-homomorphic encryption. The implementation is described in [12] (2010), https://researcher.ibm.com/researcher/view_project.php?id=1548
Gentry, C., Halevi, S.: Fully homomorphic encryption without squashing using depth-3 arithmetic circuits. Cryptology ePrint Archive, Report 2011/279 (2011), http://eprint.iacr.org/
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)
Groth, J.: Cryptography in Subgroups of \({\mathbb{Z}_n^*}\). In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 50–65. Springer, Heidelberg (2005)
Harvey, D., Roche, D.S.: An in-place truncated fourier transform and applications to polynomial multiplication. In: Proc. ISSAC 2010, pp. 325–329. ACM (2010)
Howgrave-Graham, N.: Approximate Integer Common Divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001)
Mateer, T.: Fast Fourier Transform Algorithms with Applications. PhD thesis, Clemson University (2008)
May, A.: Using LLL-reduction for solving RSA and factorization problems: A survey. In: [21] (2010)
Montgomery, P.L.: An FFT Extension of the Elliptic Curve Method of Factorization. PhD thesis, University of California Los Angeles (1992)
Nguyen, P.Q.: Public-key cryptanalysis. In: Luengo, I. (ed.) Recent Trends in Cryptography. Contemporary Mathematics, vol. 477. AMS–RSME (2009)
Nguyen, P.Q., Vallée, B. (eds.): The LLL Algorithm: Survey and Applications. Information Security and Cryptography. Springer, Heidelberg (2010)
Pollard, J.M.: Theorems on factorization and primality testing. Proc. Cambridge Philos. Soc. 76, 521–528 (1974)
Qiao, G., Lam, K.-Y.: RSA Signature Algorithm for Microcontroller Implementation. In: Schneier, B., Quisquater, J.-J. (eds.) CARDIS 1998. LNCS, vol. 1820, pp. 353–356. Springer, Heidelberg (2000)
Roche, D.S.: Space- and time-efficient polynomial multiplication. In: Proc. ISSAC 2009, pp. 295–302. ACM (2009)
Shoup, V.: Number Theory C++ Library (NTL) version 5.4.1, http://www.shoup.net/ntl/
Stinson, D.R.: Some baby-step giant-step algorithms for the low hamming weight discrete logarithm problem. Math. Comput. 71(237), 379–391 (2002)
Strassen, V.: Einige Resultate über Berechnungskomplexität. Jber. Deutsch. Math.-Verein. 78(1), 1–8 (1976/1977)
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)
von Zur Gathen, J., Gerhard, J.: Modern computer algebra, 2nd edn. Cambridge University Press (2003)
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Chen, Y., Nguyen, P.Q. (2012). Faster Algorithms for Approximate Common Divisors: Breaking Fully-Homomorphic-Encryption Challenges over the Integers. In: Pointcheval, D., Johansson, T. (eds) Advances in Cryptology – EUROCRYPT 2012. EUROCRYPT 2012. Lecture Notes in Computer Science, vol 7237. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29011-4_30
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