A Simulated Annealing Algorithm for SVP Challenge Through y-Sparse Representations of Short Lattice Vectors

  • Dan DingEmail author
  • Guizhen Zhu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9473)


In this paper, we propose a novel simulated annealing algorithm for the shortest vector problem through y-sparse representations of short lattice vectors. A Markov analysis proves that the algorithm guarantees to converge to the shortest vector at a probability 1, under certain conditions to ensure strong ergodicity of its inhomogeneous Markov chain. After that, we propose a polynomial-time approximation version of our algorithm, and the experimental results under benchmarks in SVP challenge [27] show that the simulated annealing one outperforms the famous Kannan’s algorithm in two aspects: it runs exponentially faster and it succeeds in searching the shortest vectors in lattices of higher dimensions. Therefore, our newly-proposed algorithm is a fast and efficient SVP solver and paves a completely new road for SVP algorithms.


Lattice-based cryptography Simulated annealing Shortest vector problem Inhomogeneous markov chain Strong ergodicity 


  1. 1.
    Aarts, E.H., Laarhoven, V.P.: Statistical cooling: a general approach to combinatorial optimization problems. Philips J. Res. 40(4), 193–226 (1985)MathSciNetGoogle Scholar
  2. 2.
    Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: STOC, pp. 99–108 (1996)Google Scholar
  3. 3.
    Ajtai, M.: The shortest vector problem in \(\ell _2\) is np-hard for randomized reductions. In: Proceeding of the \(30^{th}\) Symposium on the Theory of Computing (STOC 1998), pp. 284–406 (1998)Google Scholar
  4. 4.
    Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In STOC, pp. 284–293 (1997)Google Scholar
  5. 5.
    Ajtai, M., Kumar, R., Sivaumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Proceedings of the \(33^{th}\) annual ACM symposium on Theory of computing (STOC 2001) 33, pp. 601–610 (2001)Google Scholar
  6. 6.
    Anily, S., Federgruen, A.: Simulated annealing methods with general acceptance probabilities. J. Appl. Probab. 24, 657–667 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Černỳ, V.: Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl. 45(1), 41–51 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    Ding, D., Zhu, G., Wang, X.: A genetic algorithm for searching shortest lattice vector of svp challenge. Cryptology ePrint Archive, Report 2014/489 (2014).
  10. 10.
    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) CrossRefGoogle Scholar
  11. 11.
    Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)zbMATHCrossRefGoogle Scholar
  12. 12.
    Goldreich, O., Goldwasser, S., Halevi, S.: Public-key cryptosystems from lattice reduction problems. In: Kaliski Jr, B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 112–131. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  13. 13.
    Hanrot, G., Stehlé, D.: Improved analysis of kannan’s shortest lattice vector algorithm. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 170–186. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  14. 14.
    Kannan, R.: Improved algorithms for integer programming and related lattice problems. In: Proceedings of the \(15^{th}\) Symposium on the Theory of Computing (STOC 1983) 15, pp. 99–108 (1983)Google Scholar
  15. 15.
    Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., et al.: Optimization by simmulated annealing. Science 220(4598), 671–680 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Lawler, G. F. Introduction to Stochastic Processes. CRC Press, Boca Raton (1995)Google Scholar
  18. 18.
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 513–534 (1982)CrossRefGoogle Scholar
  19. 19.
    Lundy, M., Mees, A.: Convergence of an annealing algorithm. Math. Prog. 34(1), 111–124 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems: A Cryptographic Perspective. The Springer International Series in Engineering and Computer Science, vol. 671. Kluwer Academic Publishers, Boston (2002) CrossRefGoogle Scholar
  21. 21.
    Micciancio, D., Regev, O.: Worst-case to average-case reductions based on gaussian measure. In: Proceedings of the 45rd annual symposium on foundations of computer science - FOCS 2004 (Rome, Italy), October 2004, pp. 371–381. IEEE. Journal verion in SIAM Journal on ComputingGoogle Scholar
  22. 22.
    Micciancio, D., Regev, O.: Worst-case to average-case reductions based on gaussian measure. SIAM J. Comput. 37(1), 267–302 (2007). Preliminary version in FOCS 2004zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations. In: Proceedings of the \(42^{th}\) annual ACM symposium on Theory of computing (STOC 2010) 42, pp. 351–358 (2010)Google Scholar
  24. 24.
    Mitra, D., Romeo, F., Sangiovanni-Vincentelli, A.: Convergence and finite-time behavior of simulated annealing. In: 24th IEEE Conference on Decision and Control, vol. 24, pp. 761–767. IEEE (1985)Google Scholar
  25. 25.
    Nguyen, P.Q., Vidick, T.: Sieve algorithms for the shortest vector problem are practical. J. Math. Crypt. 2(2), 181–207 (2008)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Regev, O.: New lattice-based cryptographic constructions. J. ACM 51(6), 899–942 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Schneider, M., Gamma, N.: Svp challenge (2010).
  28. 28.
    Schnorr, C.P.: A hierarchy of polynomial lattice basis reduction algorithms. Theor. Comput. Sci. 53, 201–224 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Seneta, E.: Non-negative Matrices and Markov Chains, 2nd edn. Springer Publishers, New York (2006) zbMATHGoogle Scholar
  30. 30.
    Shoup, V.: Number theory c++ library (ntl) vesion 6.0.0 (2010).
  31. 31.
    van Emde Boas, P.: Another np-complete partition problem and the complexity of computing short vectors in a lattice. Technical Report, Mathematisch Instituut, Universiteit van Amsterdam 81–04 (1981)Google Scholar
  32. 32.
    Wang, X., Liu, M., Tian, C., Bi, J.: Improved nguyen-vidick heuristic sieve algorithm for shortest vector problem. In: Proceedings of the 6th ACM Symposium on Information, Computer and Communications Security. ACM, pp. 1–9 (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer Science and TechnologyTsinghua UniversityBeijingChina
  2. 2.Data Communication Science and Technology Research InstituteBeijingChina

Personalised recommendations