SCIENCE CHINA Information Sciences, Volume 61, Issue 3: 032115(2018) https://doi.org/10.1007/s11432-017-9307-0

## Orthogonalized lattice enumeration for solving SVP

More info
• ReceivedSep 6, 2017
• AcceptedNov 30, 2017
• PublishedJan 17, 2018
Share
Rating

### Abstract

The orthogonalized integer representation wasindependently proposed by Ding et al. using genetic algorithm and Fukase et al. using sampling technique to solve shortest vector problem (SVP).Their results are promising.In this paper, we consider sparse orthogonalized integer representations for shortest vectors and propose a new enumeration method,called orthognalized enumeration, by integratingsuch a representation. Furthermore, we present a mixed BKZ method, called MBKZ, by alternately applying orthognalized enumeration andother existing enumeration methods. Compared to the existing ones,our methods have greater efficiency and achieve exponential speedups both in theory and in practice for solving SVP.Implementations of our algorithms have been tested to be effective in solving challenging lattice problems.

### References

[1] Ajtai M, Kumar R, Sivakumar D. A sieve algorithm for the shortest lattice vector problem. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Hersonissos, 2001. 601--610. Google Scholar

[2] Kannan R. Improved algorithms for integer programming and related lattice problems. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing. New York: ACM, 1983. 193--206. Google Scholar

[3] Nguyen P Q, Vidick T. Sieve algorithms for the shortest vector problem are practical. J Math Cryptol, 2008, 2: 181--207. Google Scholar

[4] Pujol X, Stehl${\rm~\acute{e}}$ D. Solving the Shortest Lattice Vector Problem in Time $2^{2.465~n}$. IACR Cryptology ePrint Archive, Report 2009/605. http://perso.ens-lyon.fr/damien.stehle/downloads/2465.pdf. Google Scholar

[5] Wang X, Liu M, Tian C, et al. Improved Nguyen-Vidick heuristic sieve algorithm for shortest vector problem. In: Proceedings of the 6th ACM Symposium on Information, Computer and Communications Security, Hong Kong, 2011. 1--9. Google Scholar

[6] Micciancio D, Voulgaris P. A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations. SIAM J Comput, 2013, 42: 1364-1391 CrossRef Google Scholar

[7] Aggarwal D, Dadush D, Regev O, et al. Solving the shortest vector problem in ${2^n}$ time using discrete Gaussian sampling. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing, Portland, 2015. 733--742. Google Scholar

[8] Lenstra A K, Lenstra Jr. H W, Lov?sz L. Factoring polynomials with rational coefficients. Math Ann, 1982, 261: 515-534 CrossRef Google Scholar

[9] Schnorr C P, Euchner M. Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Math Programming, 1994, 66: 181-199 CrossRef Google Scholar

[10] Helfrich B. Algorithms to construct minkowski reduced and hermite reduced lattice bases. Theor Comput Sci, 1985, 41: 125-139 CrossRef Google Scholar

[11] Hanrot G, Stehl${\rm~\acute{e}}$ D. Improved analysis of Kannan's shortest lattice vector algorithm. In: Proceedings of the 27th Annual International Cryptology Conference, Santa Barbara, 2007. 170--186. Google Scholar

[12] Fincke U, Pohst M. A procedure for determining algebraic integers of given norm. In: Proceedings of the European Computer Algebra Conference on Computer Algebra. Berlin: Springer, 1983. 194--202. Google Scholar

[13] Gama N, Nguyen P Q, Regev O. Lattice enumeration using extreme pruning. In: Proceedings of Annual International Conference on the Theory and Applications of Cryptographic Techniques, Monaco and Nice, 2010. 257--278. Google Scholar

[14] Shoup V. Number theory c+ library (ntl). http://www.shoup.net/ntl/. Google Scholar

[15] Lindner R, Peikert C. Better key sizes (and attacks) for LWE-based encryption. In: Proceedings of Cryptographers' Track at the RSA Conference, San Francisco, 2011. 319--339. Google Scholar

[16] Micciancio D, Regev O. Lattice-based cryptography. In: Post-Quantum Cryptography. Berlin: Springer, 2009. łinebreak 147--191. Google Scholar

[17] R${\rm~\ddot{u}}$ckert M, Schneider M. Estimating the Security of Lattice-based Cryptosystems. IACR Cryptology ePrint Archive, Report 2010/137. https://pdfs.semanticscholar.org/72de/2153c2f5fbe5769a739dfe7a4eb7cc9271de.pdf. Google Scholar

[18] Schnorr C P, H${\rm~\ddot{o}}$rner H H. Attacking the Chor-Rivest cryptosystem by improved lattice reduction. In: Proceedings of the 14th Annual International Conference on Theory and Application of Cryptographic Techniques, Saint-Malo, 1995. 1--12. Google Scholar

[19] Haque M, Rahman M O, Pieprzyk J. Analysing progressive-BKZ lattice reduction algorithm. In: Proceedings of the 1st National Conference on Intelligent Computing and Information Technology, Chittagong, 2013. 73--80. Google Scholar

[20] Kuo P C, Schneider M, Dagdelen ${\rm~\ddot{O}}$, et al. Extreme enumeration on GPU and in clouds. In: Proceedings of the 13th International Workshop on Cryptographic Hardware and Embedded Systems, Nara, 2011. 176--191. Google Scholar

[21] Chen Y M, Nguyen P Q. BKZ 2.0: better lattice security estimates. In: Proceedings of International Conference on the Theory and Application of Cryptology and Information Security, Seoul, 2011. 1--20. Google Scholar

[22] Aono Y, Wang Y T, Hayashi T, et al. Improved progressive BKZ algorithms and their precise cost estimation by sharp simulator. In: Proceedings of Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, 2016. 789--819. Google Scholar

[23] Micciancio D, Walter M. Practical, predictable lattice basis reduction. In: Proceedings of Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, 2016. 820--849. Google Scholar

[24] Schnorr C P. Lattice reduction by random sampling and birthday methods. In: Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science. Berlin: Springer, 2003. 145--156. Google Scholar

[25] Ding D, Zhu G Z, Wang X Y. A genetic algorithm for searching the shortest lattice vector of SVP challenge. In: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, Madrid, 2015, 823--830. Google Scholar

[26] Fukase M, Kashiwabara K. An accelerated algorithm for solving SVP based on statistical analysis. J Inf Process, 2015, 23: 67--80. Google Scholar

[27] Holland J H. Adaptation in Natural and Artificial Systems. Ann Arbor: The University of Michigan Press, 1975. Google Scholar

[28] Booker L B, Goldberg D E, Holland J H. Classifier systems and genetic algorithms. Artificial Intelligence, 1989, 40: 235-282 CrossRef Google Scholar

[29] Eiben A E, Aarts E H, Van Hee K M. Global convergence of genetic algorithms: a Markov chain analysis. In: Proceedings of International Conference on Parallel Problem Solving from Nature, Dortmund, 1990. 4--12. Google Scholar

[30] Goldberg D E, Holland J H. Genetic algorithms and machine learning. Mach Learn, 1988, 3: 95--99. Google Scholar

[31] Schneider M, Gama N. SVP CHALLENGE. http://www.latticechallenge.org/svp-challenge/. Google Scholar

[32] Aono Y, Wang Y, Hayashi T, et al. Progressive BKZ library. http://www2.nict.go.jp/security/pbkzcode/index.html. Google Scholar

• Table 1   MBKZ's results in solving SVP challenge
 Dimension Previous norm Our results CPU used CPU frequency 2635 (seed 997) 3 CPUs respectively 99 2642 (seed 0) 2606 (seed 998) running in seed 2.5 GHz 2604 (seed 999) 997998 and 999 2655 (seed 997) 3 CPUs respectively 105 2659 (seed 0) running in seed 2.5 GHz 2643 (seed 997) 997998 and 999 3 CPUs respectively 113 2804 (seed 0) 2739 (seed 999) running in seed 2.5 GHz 997998 and 999 2921 (seed 72) 100 CPUs respectively 121 – running in seed 2.93 GHz 2910 (seed 62) 0,1,…,99
• #### 2

Citations

• Altmetric

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有