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Sieving for Shortest Vectors in Ideal Lattices

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Progress in Cryptology – AFRICACRYPT 2013 (AFRICACRYPT 2013)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7918))

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Abstract

Lattice based cryptography is gaining more and more importance in the cryptographic community. It is a common approach to use a special class of lattices, so-called ideal lattices, as the basis of lattice based crypto systems. This speeds up computations and saves storage space for cryptographic keys. The most important underlying hard problem is the shortest vector problem. So far there is no algorithm known that solves the shortest vector problem in ideal lattices faster than in regular lattices. Therefore, crypto systems using ideal lattices are considered to be as secure as their regular counterparts.

In this paper we present IdealListSieve, a variant of the ListSieve algorithm, that is a randomized, exponential time sieving algorithm solving the shortest vector problem in lattices. Our variant makes use of the special structure of ideal lattices. We show that it is indeed possible to find a shortest vector in ideal lattices faster than in regular lattices without special structure. The practical speedup of our algorithm is linear in the degree of the field polynomial. We also propose an ideal lattice variant of the heuristic GaussSieve algorithm that allows for the same speedup.

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References

  1. Arbitman, Y., Dogon, G., Lyubashevsky, V., Micciancio, D., Peikert, C., Rosen, A.: SWIFFTX: A proposal for the SHA-3 standard. In: The First SHA-3 Candidate Conference (2008)

    Google Scholar 

  2. Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: STOC, pp. 601–610. ACM (2001)

    Google Scholar 

  3. Blömer, J., Naewe, S.: Sampling methods for shortest vectors, closest vectors and successive minima. Theor. Comput. Sci. 410(18), 1648–1665 (2009)

    Article  MATH  Google Scholar 

  4. Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178. ACM (2009)

    Google Scholar 

  5. Gama, N., Howgrave-Graham, N., Nguyên, P.Q.: Symplectic lattice reduction and NTRU. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 233–253. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

  7. Gama, N., Schneider, M.: SVP Challenge (2010), http://www.latticechallenge.org/svp-challenge

  8. 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 

  9. 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)

    Chapter  Google Scholar 

  10. Klein, P.N.: Finding the closest lattice vector when it’s unusually close. In: SODA 2000, pp. 937–941. ACM (2000)

    Google Scholar 

  11. Lenstra, A., Lenstra, H., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 4, 515–534 (1982)

    Article  Google Scholar 

  12. Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  13. Lyubashevsky, V., Micciancio, D.: Asymptotically efficient lattice-based digital signatures. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 37–54. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  14. Lyubashevsky, V., Micciancio, D., Peikert, C., Rosen, A.: Swifft: A modest proposal for FFT hashing. In: Nyberg, K. (ed.) FSE 2008. LNCS, vol. 5086, pp. 54–72. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  15. Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  16. Lyubashevsky, V.: Towards practical lattice-based cryptography. Phd thesis, University of California, San Diego (2008)

    Google Scholar 

  17. Lyubashevsky, V.: Fiat-Shamir with aborts: Applications to lattice and factoring-based signatures. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 598–616. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  18. Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Computational Complexity 16(4), 365–411 (2007); Preliminary version in FOCS (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Micciancio, D., Regev, O.: Lattice-based cryptography. In: Bernstein, D.J., Buchmann, J.A., Dahmen, E. (eds.) Post-Quantum Cryptography, pp. 147–191. Springer (2008)

    Google Scholar 

  20. May, A., Silverman, J.H.: Dimension reduction methods for convolution modular lattices. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 110–125. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  21. Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. In: STOC, pp. 351–358. ACM (2010)

    Google Scholar 

  22. Micciancio, D., Voulgaris, P.: Faster exponential time algorithms for the shortest vector problem. In: SODA, pp. 1468–1480. ACM/SIAM (2010)

    Google Scholar 

  23. Nguyen, P.Q., Vidick, T.: Sieve algorithms for the shortest vector problem are practical. J. of Mathematical Cryptology 2(2) (2008)

    Google Scholar 

  24. Peikert, C., Rosen, A.: Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145–166. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  25. Pujol, X., Stehlé, D.: Rigorous and efficient short lattice vectors enumeration. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 390–405. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  26. Pujol, X., Stehlé, D.: Solving the shortest lattice vector problem in time 22.465n. Cryptology ePrint Archive, Report 2009/605 (2009)

    Google Scholar 

  27. Stein, W.A., et al.: Sage Mathematics Software (Version 4.5.2). The Sage Development Team (2010), http://www.sagemath.org

  28. Victor Shoup. Number theory library (NTL) for C++ (2011), http://www.shoup.net/ntl/

  29. Stehlé, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Efficient Public Key Encryption Based on Ideal Lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 617–635. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  30. Voulgaris, P.: Gauss Sieve alpha V. 0.1 (2010), http://cseweb.ucsd.edu/~pvoulgar/impl.html

  31. Wang, X., Liu, M., Tian, C., Bi, J.: Improved Nguyen-Vidick heuristic sieve algorithm for shortest vector problem. Cryptology ePrint Archive, Report 2010/647 (2010), http://eprint.iacr.org/

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Authors and Affiliations

  1. Technische Universität Darmstadt, Germany

    Michael Schneider

Authors
  1. Michael Schneider
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Editor information

Editors and Affiliations

  1. Concordia Institute for Information Systems Engineering, Concordia University, 1515 St. Catherine Street West, H3G 2W1, Montreal, QC, Canada

    Amr Youssef

  2. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Basse-Normandie, BP 5186, 14032, Caen, France

    Abderrahmane Nitaj

  3. Department of Information Technology, Cairo University, 5 Dr. Ahmed Zewil Street, 12613, Cairo, Giza, Egypt

    Aboul Ella Hassanien

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Schneider, M. (2013). Sieving for Shortest Vectors in Ideal Lattices. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds) Progress in Cryptology – AFRICACRYPT 2013. AFRICACRYPT 2013. Lecture Notes in Computer Science, vol 7918. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38553-7_22

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  • DOI: https://doi.org/10.1007/978-3-642-38553-7_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38552-0

  • Online ISBN: 978-3-642-38553-7

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