Advertisement

High-Speed Signatures from Standard Lattices

  • Özgür Dagdelen
  • Rachid El Bansarkhani
  • Florian Göpfert
  • Tim Güneysu
  • Tobias Oder
  • Thomas Pöppelmann
  • Ana Helena Sánchez
  • Peter Schwabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8895)

Abstract

At CT-RSA 2014 Bai and Galbraith proposed a lattice-based signature scheme optimized for short signatures and with a security reduction to hard standard lattice problems. In this work we first refine the security analysis of the original work and propose a new 128-bit secure parameter set chosen for software efficiency. Moreover, we increase the acceptance probability of the signing algorithm through an improved rejection condition on the secret keys. Our software implementation targeting Intel CPUs with AVX/AVX2 and ARM CPUs with NEON vector instructions shows that even though we do not rely on ideal lattices, we are able to achieve high performance. For this we optimize the matrix-vector operations and several other aspects of the scheme and finally compare our work with the state of the art.

Keywords

Signature scheme Standard lattices Vectorization Ivy bridge 

Notes

Acknowledgment

We would like to thank Patrick Weiden, Rafael Misoczki, Shi Bai, and Steven Galbraith for useful discussions. We would further like to thank the anonymous reviewers for their suggestions and comments.

References

  1. 1.
    Melchor, C.A., Boyen, X., Deneuville, J.-C., Gaborit, P.: Sealing the leak on classical NTRU signatures. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 1–21. Springer, Heidelberg (2014). 99 Google Scholar
  2. 2.
    Albrecht, M.R., Fitzpatrick, R., Göpfert, F.: On the efficacy of solving LWE by reduction to unique-SVP. Cryptology ePrint Archive, Report 2013/602 (2013). http://eprint.iacr.org/2013/602/. 92
  3. 3.
    Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast cryptographic primitives and circular-secure encryption based on hard learning problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009). 87 CrossRefGoogle Scholar
  4. 4.
    Babai, L.: On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1–13 (1986). http://www.csie.nuk.edu.tw/~cychen/Lattices/Onlovaszlatticereductionandthenearestlatticepointproblem.pdf. 90, 102
  5. 5.
    Bai, S., Galbraith, S.: Personal communication and e-mail exchanges (2014). 86, 90 Google Scholar
  6. 6.
    Bai, S., Galbraith, S.D.: An improved compression technique for signatures based on learning with errors. In: Benaloh, J. (ed.) CT-RSA 2014. LNCS, vol. 8366, pp. 28–47. Springer, Heidelberg (2014). 85, 86, 87, 88, 89, 90, 92, 93, 102 CrossRefGoogle Scholar
  7. 7.
    El Bansarkhani, R., Buchmann, J.: Improvement and efficient implementation of a lattice-based signature scheme. In: Lange, T., Lauter, K., Lisoněk, P. (eds.) SAC 2013. LNCS, vol. 8282, pp. 48–67. Springer, Heidelberg (2014). 84, 85, 99 CrossRefGoogle Scholar
  8. 8.
    Bernstein, D.J.: A subfield-logarithm attack against ideal lattices, Feb 2014. http://blog.cr.yp.to/20140213-ideal.html. 85
  9. 9.
    Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.): Post-Quantum Cryptography. Mathematics and Statistics. Springer, Heidelberg (2009). 84, 85, 91, 93 Google Scholar
  10. 10.
    Bernstein, D.J., Lange, T.: eBACS: ECRYPT benchmarking of cryptographic systems. http://bench.cr.yp.to. Accessed 25 Jan 2013. 86, 98
  11. 11.
    Boorghany, A., Jalili, R.: Implementation and comparison of lattice-based identification protocols on smart cards and microcontrollers. IACR Cryptology ePrint Archive, 2014. http://eprint.iacr.org/2014/078/. 85
  12. 12.
    Bos, J.W., Costello, C., Naehrig, M., Stebila, D.: Post-quantum key exchange for the TLS protocol from the ring learning with errors problem. IACR Cryptology ePrint Archive (2014). http://eprint.iacr.org/2014/599. 86, 99
  13. 13.
    Brumley, D., Boneh, D.: Remote timing attacks are practical. In: SSYM 2003 Proceedings of the 12th Conference on USENIX Security Symposium. USENIX Association (2003). http://crypto.stanford.edu/dabo/pubs/papers/ssl-timing.pdf. 86
  14. 14.
    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). 92 CrossRefGoogle Scholar
  15. 15.
    Couvreur, A., Otmani, A., Tillich, J.P.: Polynomial time attack on wild McEliece over quadratic extensions. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 17–39. Springer, Heidelberg (2014). 85 CrossRefGoogle Scholar
  16. 16.
    Ducas, L., Durmus, A., Lepoint, T., Lyubashevsky, V.: Lattice signatures and bimodal Gaussians. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 40–56. Springer, Heidelberg (2013). 85, 86, 88, 89, 99 CrossRefGoogle Scholar
  17. 17.
    Dwarakanath, N.C., Galbraith, S.D.: Sampling from discrete Gaussians for lattice-based cryptography on a constrained device. Appl. Algebra Eng. Commun. Comput. 25(3), 159–180 (2014). 86 CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). 88 CrossRefGoogle Scholar
  19. 19.
    Gama, N., Nguyen, P.Q.: Predicting lattice reduction. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008). 90, 91 CrossRefGoogle Scholar
  20. 20.
    Göttert, N., Feller, T., Schneider, M., Buchmann, J., Huss, S.: On the design of hardware building blocks for modern lattice-based encryption schemes. In: Prouff, E., Schaumont, P. (eds.) CHES 2012. LNCS, vol. 7428, pp. 512–529. Springer, Heidelberg (2012). 85 CrossRefGoogle Scholar
  21. 21.
    Güneysu, T., Lyubashevsky, V., Pöppelmann, T.: Practical lattice-based cryptography: a signature scheme for embedded systems. In: Prouff, E., Schaumont, P. (eds.) CHES 2012. LNCS, vol. 7428, pp. 530–547. Springer, Heidelberg (2012). 84, 85, 88 CrossRefGoogle Scholar
  22. 22.
    Güneysu, T., Oder, T., Pöppelmann, T., Schwabe, P.: Software speed records for lattice-based signatures. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 67–82. Springer, Heidelberg (2013). 85, 86, 95, 96, 97, 98, 99 CrossRefGoogle Scholar
  23. 23.
    Hoffstein, J., Pipher, J., Schanck, J.M., Silverman, J.H., Whyte, W.: Practical signatures from the Partial Fourier recovery problem. In: Boureanu, I., Owesarski, P., Vaudenay, S. (eds.) ACNS 2014. LNCS, vol. 8479, pp. 476–493. Springer, Heidelberg (2014). 99 CrossRefGoogle Scholar
  24. 24.
    Ishiguro, T., Kiyomoto, S., Miyake, Y., Takagi, T.: Parallel Gauss Sieve algorithm: solving the SVP challenge over a 128-Dimensional ideal lattice. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 411–428. Springer, Heidelberg (2014). 85 CrossRefGoogle Scholar
  25. 25.
    Kocher, P.C.: Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996). 86 Google Scholar
  26. 26.
    Lindner, R., Peikert, C.: Better key sizes (and attacks) for LWE-based encryption. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 319–339. Springer, Heidelberg (2011). 90, 92, 102, 103 CrossRefGoogle Scholar
  27. 27.
    Liu, M., Nguyen, P.Q.: Solving BDD by enumeration: an update. In: Dawson, E. (ed.) CT-RSA 2013. LNCS, vol. 7779, pp. 293–309. Springer, Heidelberg (2013). 90, 93, 102, 103 CrossRefGoogle Scholar
  28. 28.
    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). 88 CrossRefGoogle Scholar
  29. 29.
    Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 738–755. Springer, Heidelberg (2012). 88 CrossRefGoogle Scholar
  30. 30.
    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). 84 CrossRefGoogle Scholar
  31. 31.
    Misoczki, R., Barreto, P.S.L.M.: Compact McEliece keys from Goppa codes. In: Jacobson Jr, M.J., Rijmen, V., Safavi-Naini, R. (eds.) SAC 2009. LNCS, vol. 5867, pp. 376–392. Springer, Heidelberg (2009). 85 CrossRefGoogle Scholar
  32. 32.
    Oder, T., Pöppelmann, T., Güneysu, T.: Beyond ECDSA and RSA: Lattice-based digital signatures on constrained devices. In: DAC 2014 Proceedings of the The 51st Annual Design Automation Conference on Design Automation Conference, pp. 1–6. ACM (2014). https://www.sha.rub.de/media/attachments/files/2014/06/bliss_arm.pdf. 85
  33. 33.
    Pöppelmann, T., Ducas, L., Güneysu, T.: Enhanced lattice-based signatures on reconfigurable hardware. In: Batina, L., Robshaw, M. (eds.) CHES 2014. LNCS, vol. 8731, pp. 353–370. Springer, Heidelberg (2014). 85, 86 CrossRefGoogle Scholar
  34. 34.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) STOC 2005 Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of computing, pp. 84–93. ACM (2005). http://www.cims.nyu.edu/~regev/papers/qcrypto.pdf. 85
  35. 35.
    Roy, S.S., Vercauteren, F., Mentens, N., Chen, D.D., Verbauwhede, I.: Compact ring-LWE cryptoprocessor. In: Batina, L., Robshaw, M. (eds.) CHES 2014. LNCS, vol. 8731, pp. 371–391. Springer, Heidelberg (2014). 85 CrossRefGoogle Scholar
  36. 36.
    Schneider, M.: Sieving for shortest vectors in ideal lattices. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds.) AFRICACRYPT 2013. LNCS, vol. 7918, pp. 375–391. Springer, Heidelberg (2013). 85 CrossRefGoogle Scholar
  37. 37.
    Schnorr, C.-P., Euchner, M.: Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Math. Program. 66, 181–199 (1994). http://www.csie.nuk.edu.tw/~cychen/Lattices/LatticeBasisReductionImprovedPracticalAlgorithmsandSolvingSubsetSumProblems.pdf92

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Özgür Dagdelen
    • 1
  • Rachid El Bansarkhani
    • 1
  • Florian Göpfert
    • 1
  • Tim Güneysu
    • 2
  • Tobias Oder
    • 2
  • Thomas Pöppelmann
    • 2
  • Ana Helena Sánchez
    • 3
  • Peter Schwabe
    • 3
  1. 1.Technische Universität DarmstadtDarmstadtGermany
  2. 2.Horst Görtz Institute for IT-SecurityRuhr-University BochumBochumGermany
  3. 3.Digital Security GroupRadboud University NijmegenNijmegenThe Netherlands

Personalised recommendations