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Adaptive Precision Floating Point LLL

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Information Security and Privacy (ACISP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7959))

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Abstract

The LLL algorithm is one of the most studied lattice basis reduction algorithms in the literature. Among all of its variants, the floating point version, also known as L2, is the most popular one, due to its efficiency and its practicality. In its classic setting, the floating point precision is a fixed value, determined by the dimension of the input basis at the initiation of the algorithm. We observe that a fixed precision overkills the problem, since one does not require a huge precision to handle the process at the beginning of the reduction. In this paper, we propose an adaptive way to handle the precision, where the precision is adaptive during the procedure. Although this optimization does not change the worst-case complexity, it reduces the average-case complexity by a constant factor. In practice, we observe an average 20% acceleration in our implementation.

This work is supported by ARC Future Fellowship FT0991397.

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References

  1. mpfr library, http://www.mpfr.org/

  2. SVP CHALLENGE, http://www.latticechallenge.org/svp-challenge/index.php

  3. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. the user language. J. Symbolic Comput. 24(3-4), 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bremner, M.: Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications. Pure and Applied Mathematics. CRC Press Inc. (2012)

    Google Scholar 

  5. Coppersmith, D.: Finding a small root of a bivariate integer equation; factoring with high bits known. In: Maurer (ed.) [13], pp. 178–189

    Google Scholar 

  6. Coppersmith, D.: Finding a small root of a univariate modular equation. In: Maurer (ed.) [13], pp. 155–165

    Google Scholar 

  7. Coppersmith, D., Shamir, A.: Lattice attacks on NTRU. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 52–61. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  8. Gama, N., Nguyen, P.Q.: Predicting lattice reduction. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  9. Goldstein, D., Mayer, A.: On the equidistribution of hecke points. Forum Mathematicum 15, 165–189 (2006)

    Article  MathSciNet  Google Scholar 

  10. Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: A ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  11. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 513–534 (1982)

    Article  Google Scholar 

  12. Lovász, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 50. SIAM Publications (1986)

    Google Scholar 

  13. Maurer, U.M. (ed.): EUROCRYPT 1996. LNCS, vol. 1070. Springer, Heidelberg (1996)

    MATH  Google Scholar 

  14. Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems, A Cryptographic Perspective. Kluwer Academic Publishers (2002)

    Google Scholar 

  15. Minkowski, H.: Geometrie der Zahlen. B. G. Teubner, Leipzig (1896)

    Google Scholar 

  16. Nguen, P.Q., Stehlé, D.: Floating-point LLL revisited. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 215–233. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  17. Nguyen, P.Q., Stehlé, D.: LLL on the average. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 238–256. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  18. Nguyen, P.Q., Stehlé, D.: An lll algorithm with quadratic complexity. SIAM J. Comput. 39(3), 874–903 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nguyen, P.Q., Valle, B.: The LLL Algorithm: Survey and Applications, 1st edn. Springer Publishing Company, Incorporated (2009)

    Google Scholar 

  20. Pujol, X., Stehlé, D., Cade, D.: fplll library, http://perso.ens-lyon.fr/xavier.pujol/fplll/

  21. Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schnorr, C.-P.: A more efficient algorithm for lattice basis reduction. J. Algorithms 9(1), 47–62 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schnorr, C.-P., Euchner, M.: Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Math. Program. 66, 181–199 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shoup, V.: NTL - A Library for Doing Number Theory, http://www.shoup.net/ntl/index.html

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Plantard, T., Susilo, W., Zhang, Z. (2013). Adaptive Precision Floating Point LLL. In: Boyd, C., Simpson, L. (eds) Information Security and Privacy. ACISP 2013. Lecture Notes in Computer Science, vol 7959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39059-3_8

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  • DOI: https://doi.org/10.1007/978-3-642-39059-3_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39058-6

  • Online ISBN: 978-3-642-39059-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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