Abstract
We present a GPU implementation of the Simple Sampling Reduction (SSR) algorithm that searches for short vectors in lattices. SSR makes use of the famous BKZ algorithm. It complements an exhaustive search in a suitable search region to insert random, short vectors to the lattice basis. The sampling of short vectors can be executed in parallel.
Our GPU implementation increases the number of sampled vectors per second from 5200 to more than 120,000. With this we are the first to present a parallel implementation of SSR and we make use of the computing capability of modern graphics cards to enhance the search for short vectors even more.
Chapter PDF
Similar content being viewed by others
References
Advanced Micro Devices. ATI CTM Guide. Technical report, ATI (2006)
Backes, W., Wetzel, S.: Parallel lattice basis reduction using a multi-threaded schnorr-euchner LLL algorithm. In: Sips, H., Epema, D., Lin, H.-X. (eds.) Euro-Par 2009. LNCS, vol. 5704, pp. 960–973. Springer, Heidelberg (2009)
Bernstein, D.J., Chen, T.-R., Cheng, C.-M., Lange, T., Yang, B.-Y.: ECM on graphics cards. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 483–501. Springer, Heidelberg (2009)
Bos, J.W., Stefan, D.: Performance analysis of the SHA-3 candidates on exotic multi-core architectures. In: Mangard, S., Standaert, F.-X. (eds.) CHES 2010. LNCS, vol. 6225, pp. 279–293. Springer, Heidelberg (2010)
Buchmann, J., Lindner, R.: Secure parameters for SWIFFT. In: Roy, B., Sendrier, N. (eds.) INDOCRYPT 2009. LNCS, vol. 5922, pp. 1–17. Springer, Heidelberg (2009)
Buchmann, J., Lindner, R., Rückert, M.: Explicit hard instances of the shortest vector problem. In: Buchmann, J., Ding, J. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 79–94. Springer, Heidelberg (2008)
Buchmann, J., Ludwig, C.: Practical lattice basis sampling reduction. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 222–237. Springer, Heidelberg (2006)
Cook, D.L., Ioannidis, J., Keromytis, A.D., Luck, J.: CryptoGraphics: Secret key cryptography using graphics cards. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 334–350. Springer, Heidelberg (2005)
Dagdelen, Ö., Schneider, M.: Parallel enumeration of shortest lattice vectors. In: D’Ambra, P., Guarracino, M., Talia, D. (eds.) Euro-Par 2010. LNCS, vol. 6272, pp. 211–222. Springer, Heidelberg (2010)
Detrey, J., Hanrot, G., Pujol, X., Stehlé, D.: Accelerating lattice reduction with FPGAs. In: Abdalla, M., Barreto, P.S.L.M. (eds.) LATINCRYPT 2010. LNCS, vol. 6212, pp. 124–143. Springer, Heidelberg (2010)
Fleissner, S.: GPU-accelerated montgomery exponentiation. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2007. LNCS, vol. 4487, pp. 213–220. Springer, Heidelberg (2007)
Gama, N., Nguyen, P.Q.: Predicting lattice reduction. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008)
Gama, N., Schneider, M.: SVP Challenge (2010), http://www.latticechallenge.org/svp-challenge
Harrison, O., Waldron, J.: AES encryption implementation and analysis on commodity graphics processing units. In: Paillier, P., Verbauwhede, I. (eds.) CHES 2007. LNCS, vol. 4727, pp. 209–226. Springer, Heidelberg (2007)
Hermans, J., Schneider, M., Buchmann, J., Vercauteren, F., Preneel, B.: Parallel shortest lattice vector enumeration on graphics cards. In: Bernstein, D.J., Lange, T. (eds.) AFRICACRYPT 2010. LNCS, vol. 6055, pp. 52–68. Springer, Heidelberg (2010)
Lenstra, A., Lenstra, H., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 4, 515–534 (1982)
Ludwig, C.: Practical Lattice Basis Sampling Reduction. PhD thesis, Technische Universität Darmstadt (2005), http://elib.tu-darmstadt.de/diss/000640/
Manavski, S.A.: CUDA compatible GPU as an efficient hardware accelerator for AES cryptography. In: ICSPC, pp. 65–68. IEEE Computer Society Press, Los Alamitos (2007)
Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems: a cryptographic perspective. Kluwer Academic Publishers, Dordrecht (2002)
Micciancio, D., Regev, O.: Lattice-based cryptography. In: Bernstein, D.J., Buchmann, J.A., Dahmen, E. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 147–191. Springer, Heidelberg (2008)
Moss, A., Page, D., Smart, N.P.: Toward acceleration of RSA using 3D graphics hardware. In: Galbraith, S.D. (ed.) Cryptography and Coding 2007. LNCS, vol. 4887, pp. 364–383. Springer, Heidelberg (2007)
Nguyen, P.Q., Vallée, B.: The LLL Algorithm - Survey and Applications. Springer, Heidelberg (2010)
NVIDIA. Compute Unified Device Architecture Programming Guide. Technical report, NVIDIA (2007)
NVIDIA. CUBLAS Library (2007)
Schnorr, C.-P.: Block reduced lattice bases and successive minima. Combinatorics, Probability & Computing 3, 507–522 (1994)
Schnorr, C.-P.: Lattice reduction by random sampling and birthday methods. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 146–156. Springer, Heidelberg (2003)
Schnorr, C.-P., Euchner, M.: Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Mathematical Programming 66, 181–199 (1994)
Shoup, V.: Number theory library (NTL) for C++, http://www.shoup.net/ntl/
Szerwinski, R., Güneysu, T.: Exploiting the power of gPUs for asymmetric cryptography. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 79–99. Springer, Heidelberg (2008)
Villard, G.: Parallel lattice basis reduction. In: ISSAC, pp. 269–277. ACM, New York (1992)
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
Schneider, M., Göttert, N. (2011). Random Sampling for Short Lattice Vectors on Graphics Cards. In: Preneel, B., Takagi, T. (eds) Cryptographic Hardware and Embedded Systems – CHES 2011. CHES 2011. Lecture Notes in Computer Science, vol 6917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23951-9_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-23951-9_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23950-2
Online ISBN: 978-3-642-23951-9
eBook Packages: Computer ScienceComputer Science (R0)