Advertisement

Secure Parameters for SWIFFT

  • Johannes Buchmann
  • Richard Lindner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5922)

Abstract

The SWIFFT compression functions, proposed by Lyubashevsky et al. at FSE 2008, are very efficient instantiations of generalized compact knapsacks for a specific set of parameters. They have the property that, asymptotically, finding collisions for a randomly chosen compression function implies being able to solve computationally hard ideal lattice problems in the worst-case.

We present three results. First, we present new average-case problems, which may be used for all lattice schemes whose security is proven with the worst-case to average-case reduction in either general or ideal lattices. The new average-case problems require less description bits, resulting in improved keysize and speed for these schemes. Second, we propose a parameter generation algorithm for SWIFFT where the main parameter n can be any integer in the image of Euler’s totient function, and not necessarily a power of 2 as before. Third, we give experimental evidence that finding pseudo-collisions for SWIFFT is as hard as breaking a 68-bit symmetric cipher according to the well-known heuristic by Lenstra and Verheul. We also recommend conservative parameters corresponding to a 127-bit symmetric cipher.

Keywords

post-quantum cryptography hash functions lattices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: Proceedings of the Annual Symposium on the Theory of Computing (STOC) 1996, pp. 99–108. ACM Press, New York (1996)Google Scholar
  2. 2.
    Arbitman, Y., Dogon, G., Lyubashevsky, V., Micciancio, D., Peikert, C., Rosen, A.: SWIFFTX: A proposal for the SHA-3 standard (2008), http://www.eecs.harvard.edu/~alon/PAPERS/lattices/swifftx.pdf
  3. 3.
    Buchmann, J., Döring, M., Lindner, R.: Efficiency improvement for NTRU. In: Alkassar, A., Siekmann, J. (eds.) SICHERHEIT 2008. Lecture Notes in Informatics, vol. 128, pp. 79–94. Bonner Köllen Verlag (2008)Google Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Buchmann, J., Ludwig, C.: Practical lattice basis sampling reduction. In: Hess, F., Pauli, S., Pohst, M.E. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 222–237. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    De Cannière, C., Mendel, F., Rechberger, C.: Collisions for 70-step sha-1: On the full cost of collision search. In: Adams, C.M., Miri, A., Wiener, M.J. (eds.) SAC 2007. LNCS, vol. 4876, pp. 56–73. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Coppersmith, D., Shamir, A.: Lattice attacks on NTRU. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 52–61. Springer, Heidelberg (1997)Google Scholar
  8. 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)CrossRefGoogle Scholar
  9. 9.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the Annual Symposium on the Theory of Computing (STOC) 2008, pp. 197–206. ACM Press, New York (2008)Google Scholar
  10. 10.
    Goldreich, O., Goldwasser, S., Halevi, S.: Collision-free hashing from lattice problems. Electronic Colloquium on Computational Complexity (ECCC) 3(42) (1996)Google Scholar
  11. 11.
    Hirschhorn, P., Hoffstein, J., Howgrave-Graham, N., Whyte, W.: Choosing NTRU parameters in light of combined lattice reduction and MITM approaches (will be published at ACNS) (2009), http://www.ntru.com/cryptolab/pdf/params.pdf
  12. 12.
    Howgrave-Graham, N.: A hybrid lattice-reduction and meet-in-the-middle attack against ntru. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 150–169. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Howgrave-Graham, N., Silverman, J.H., Whyte, W.: A meet-in-the-middle attack on an NTRU private key, http://www.ntru.com/cryptolab/tech_notes.htm#004
  14. 14.
    Lenstra, A.K., Verheul, E.R.: Selecting cryptographic key sizes. J. Cryptology 14(4), 255–293 (2001)zbMATHMathSciNetGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Micciancio, D., Regev, O.: Lattice-based Cryptography. In: Post Quantum Cryptography. Springer, Heidelberg (2009)Google Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) STOC, pp. 84–93. ACM, New York (2005)Google Scholar
  21. 21.
    Schnorr, C.-P.: Lattice reduction by random sampling and birthday methods. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 145–156. Springer, Heidelberg (2003)Google Scholar
  22. 22.
    Shoup, V.: Number theory library (NTL) for C++, http://www.shoup.net/ntl/
  23. 23.
    Stehlé, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Efficient public key encryption based on ideal lattices. Technical Report 285, Cryptology ePrint Archive (2009)Google Scholar
  24. 24.
    Stevens, M., Lenstra, A.K., de Weger, B.: Chosen-prefix collisions for MD5 and colliding X.509 certificates for different identities. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 1–22. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Xagawa, K., Tanaka, K.: A compact signature scheme with ideal lattice. In: Asian Assiciation for Algorithms and Computation (AAAC) (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Johannes Buchmann
    • 1
  • Richard Lindner
    • 1
  1. 1.Department of Computer ScienceTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations