Abstract
Lattice-based cryptanalysis is an important field in cryptography since lattice problems are among the most robust assumptions, and have been used to construct most cryptographic primitives. In this research, we focus on the Gauss Sieve algorithm, a heuristic lattice sieving algorithm proposed by Micciancio and Voulgaris. We propose the technique of lifting computations in prime-cyclotomic ideals into that in cyclic ideals. Lifting makes rotations easier to compute and reduces the complexity of inner products from \(O(n^3)\) to \(O(n^2)\). We implemented our Gauss Sieve on GPUs by adapting the framework of Ishiguro et al. in a single GPU, and the one of Bos et al. among multiple GPUs. We found a short vector at dimension 130 in the Darmstadt Ideal SVP Challenge (currently in first place in the Hall of Fame) using 8 GPUs in 824 h using our implementation.
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References
Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: STOC 1997, pp. 284–293. ACM, New York (1997)
Ajtai, M.: The shortest vector problem in \({\rm l}_{\text{2}}\) is np-hard for randomized reductions. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 4, no. 47 (1997)
Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: STOC 2001, pp. 601–610. ACM, New York (2001)
Becker, A., Ducas, L., Gama, N., Laarhoven, T.: New directions in nearest neighbor searching with applications to lattice sieving. In: SODA 2016, pp. 10–24 (2016)
Becker, A., Laarhoven, T.: Efficient (ideal) lattice sieving using cross-polytope LSH. In: Pointcheval, D., Nitaj, A., Rachidi, T. (eds.) AFRICACRYPT 2016. LNCS, vol. 9646, pp. 3–23. Springer, Heidelberg (2016). doi:10.1007/978-3-319-31517-1_1
Bos, J.W., Naehrig, M., van de Pol, J.: Sieving for shortest vectors in ideal lattices: a practical perspective. IACR Cryptology ePrint Archive 2014, 880 (2014)
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). doi:10.1007/978-3-642-25385-0_1
Crandall, R.E.: Topics in Advanced Scientific Computation. Springer-Telos, New York (1996)
CUDA C programming guide 7.5 (2015). http://docs.nvidia.com/cuda/cuda-c-programming-guide/
Fukase, M., Kashiwabara, K.: An accelerated algorithm for solving SVP based on statistical analysis. JIP 23(1), 67–80 (2015)
Gauss Sieve implementation by panagiotis voulgaris. https://cseweb.ucsd.edu/~pvoulgar/impl.html
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Annual ACM Symposium on Theory of Computing – STOC, pp. 169–178 (2009)
Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38348-9_1
Gama, N., Nguyen, P.Q., Regev, O.: Lattice enumeration using extreme pruning. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 257–278. Springer, Heidelberg (2010). doi:10.1007/978-3-642-13190-5_13
Idea Lattice Challenge. http://www.latticechallenge.org/ideallattice-challenge/
Ishiguro, T., Kiyomoto, S., Miyake, Y., Takagi, T.: Parallel Gauss Sieve algorithm: solving the SVP challenge over a 128-dimensional ideal lattice. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 411–428. Springer, Heidelberg (2014). doi:10.1007/978-3-642-54631-0_24
Kuo, P.-C., Schneider, M., Dagdelen, Ö., Reichelt, J., Buchmann, J., Cheng, C.-M., Yang, B.-Y.: Extreme enumeration on GPU and in clouds. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 176–191. Springer, Heidelberg (2011). doi:10.1007/978-3-642-23951-9_12
Laarhoven, T.: Sieving for shortest vectors in lattices using angular locality-sensitive hashing. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 3–22. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47989-6_1
SVP Challenge. http://www.latticechallenge.org/svp-challenge/
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). doi:10.1007/978-3-642-19074-2_21
Mariano, A., Bischof, C.H., Laarhoven, T.: Parallel (probable) lock-free hash sieve: a practical sieving algorithm for the SVP. ICPP 2015, 590–599 (2015)
Mariano, A., Dagdelen, Ö., Bischof, C.: A comprehensive empirical comparison of parallel ListSieve and GaussSieve. In: Lopes, L., et al. (eds.) Euro-Par 2014. LNCS, vol. 8805, pp. 48–59. Springer, Heidelberg (2014). doi:10.1007/978-3-319-14325-5_5
Duane (Nvidia Coorporation) Merrill. The CUB Library
Milde, B., Schneider, M.: A parallel implementation of GaussSieve for the shortest vector problem in lattices. In: Malyshkin, V. (ed.) PaCT 2011. LNCS, vol. 6873, pp. 452–458. Springer, Heidelberg (2011). doi:10.1007/978-3-642-23178-0_40
Micciancio, D., Voulgaris, P.: Faster exponential time algorithms for the shortest vector problem. In: SODA 2010, pp. 1468–1480 (2010)
Nguyen, P.Q., Vidick, T.: Sieve algorithms for the shortest vector problem are practical. J. Math. Cryptol. 2(2), 181–207 (2008)
Schneider, M.: Analysis of Gauss-Sieve for solving the shortest vector problem in lattices. In: Katoh, N., Kumar, A. (eds.) WALCOM 2011. LNCS, vol. 6552, pp. 89–97. Springer, Heidelberg (2011). doi:10.1007/978-3-642-19094-0_11
Schneider, M.: Sieving for shortest vectors in ideal lattices. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds.) AFRICACRYPT 2013. LNCS, vol. 7918, pp. 375–391. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38553-7_22
Acknowledgement
Partially sponsored by MoST projects 105-2923-E-001-003-MY3 and 105-2221-E-001-020-MY3. We would also like to thank Dr. Shinsaku Kiyomoto of KDDI Research for the fruitful discussion.
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Yang, SY., Kuo, PC., Yang, BY., Cheng, CM. (2017). Gauss Sieve Algorithm on GPUs. In: Handschuh, H. (eds) Topics in Cryptology – CT-RSA 2017. CT-RSA 2017. Lecture Notes in Computer Science(), vol 10159. Springer, Cham. https://doi.org/10.1007/978-3-319-52153-4_3
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DOI: https://doi.org/10.1007/978-3-319-52153-4_3
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