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Implementation of Lattice Trapdoors on Modules and Applications

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Post-Quantum Cryptography (PQCrypto 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12841))

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Abstract

We develop and implement efficient Gaussian preimage sampling techniques on module lattices, which rely on the works of Micciancio and Peikert in 2012, and Micciancio and Genise in 2018. The main advantage of our implementation is its modularity, which makes it practical to use for signature schemes, but also for more advanced constructions using trapdoors such as identity-based encryption. In particular, it is easy to use in the ring or module setting, and to modify the arithmetic on \(\mathcal R_q\) (as different schemes have different conditions on q).

Relying on these tools, we also present two instantiations and implementations of proven trapdoor-based signature schemes in the module setting: GPV in the random oracle model and a variant of it in the standard model presented in Bert et al. in 2018. For that last scheme, we address a security issue and correct obsolescence problems in their implementation by building ours from scratch. To the best of our knowledge, this is the first efficient implementation of a lattice-based signature scheme in the standard model. Relying on that last signature, we also present the implementation of a standard model IBE in the module setting. We show that while the resulting schemes may not be competitive with the most efficient NIST candidates, they are practical and run on a standard laptop in acceptable time, which paves the way for practical advanced trapdoor-based constructions.

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References

  1. Alkim, E., Barreto, P.S.L.M., Bindel, N., Longa, P., Ricardini, J.E.: The lattice-based digital signature scheme qtesla. IACR Cryptology ePrint Archive 2019:85 (2019)

    Google Scholar 

  2. Agrawal, S., Boneh, D., Boyen, X.: Efficient lattice (H)IBE in the standard model. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 553–572. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_28

    Chapter  MATH  Google Scholar 

  3. Agrawal, S., Boneh, D., Boyen, X.: Lattice basis delegation in fixed dimension and shorter-ciphertext hierarchical IBE. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 98–115. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_6

    Chapter  MATH  Google Scholar 

  4. Alkim, E., Ducas, L., Pöppelmann, T., Schwabe, P.: Post-quantum key exchange - a new hope. In: USENIX Security Symposium, pp. 327–343. USENIX Association (2016)

    Google Scholar 

  5. Albrecht, M.R., Hanser, C., Höller, A., Pöppelmann, T., Virdia, F., Wallner, A.: Implementing RLWE-based schemes using an RSA co-processor. IACR TCHES 2019(1), 169–208 (2019)

    Google Scholar 

  6. 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). https://doi.org/10.1007/978-3-662-43414-7_3

    Chapter  Google Scholar 

  7. Bert, P., Fouque, P.-A., Roux-Langlois, A., Sabt, M.: Practical implementation of ring-SIS/LWE based signature and IBE. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 271–291. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_13

    Chapter  Google Scholar 

  8. Boyen, X.: Lattice mixing and vanishing trapdoors: a framework for fully secure short signatures and more. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 499–517. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_29

    Chapter  Google Scholar 

  9. Chen, Y., Genise, N., Mukherjee, P.: Approximate trapdoors for lattices and smaller hash-and-sign signatures. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11923, pp. 3–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34618-8_1

    Chapter  Google Scholar 

  10. Cash, D., Hofheinz, D., Kiltz, E., Peikert, C.: Bonsai trees, or how to delegate a lattice basis. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 523–552. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_27

    Chapter  Google Scholar 

  11. Chuengsatiansup, C., Prest, T., Stehlé, D., Wallet, A., Xagawa, K.: Modfalcon: compact signatures based on module NTRU lattices. IACR Cryptol. ePrint Arch. 2019:1456 (2019)

    Google Scholar 

  12. Ducas, L., et al.: Crystals-dilithium: a lattice-based digital signature scheme. TCHES 2018(1), 238–268 (2018)

    Article  MathSciNet  Google Scholar 

  13. D’Anvers, J.-P., Karmakar, A., Sinha Roy, S., Vercauteren, F.: Saber: module-LWR based key exchange, CPA-secure encryption and CCA-secure KEM. In: Joux, A., Nitaj, A., Rachidi, T. (eds.) AFRICACRYPT 2018. LNCS, vol. 10831, pp. 282–305. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89339-6_16

    Chapter  Google Scholar 

  14. Ducas, L., Lyubashevsky, V., Prest, T.: Efficient identity-based encryption over NTRU lattices. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8874, pp. 22–41. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45608-8_2

    Chapter  Google Scholar 

  15. Ducas, L., Prest, T.: Fast fourier orthogonalization. In: ISSAC, pp. 191–198. ACM (2016)

    Google Scholar 

  16. Fouque, P.-A., et al.: Falcon: fast-fourier lattice-based compact signatures over NTRU (2017). https://falcon-sign.info/falcon.pdf

  17. Fouotsa, E.: Calcul des couplages et arithmetique des courbes elliptiques pour la cryptographie. Ph.D. thesis, Rennes 1 (2013)

    Google Scholar 

  18. Genise, N., Micciancio, D.: Faster Gaussian sampling for trapdoor lattices with arbitrary modulus. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 174–203. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_7

    Chapter  Google Scholar 

  19. Gür, K.D., Polyakov, Y., Rohloff, K., Ryan, G.W., Savas, E.: Implementation and evaluation of improved gaussian sampling for lattice trapdoors. In: WAHC@CCS, pp. 61–71. ACM (2018)

    Google Scholar 

  20. Gür, K.D., Polyakov, Y., Rohloff, K., Ryan, G.W., Sajjadpour, H., Savas, E.: Practical applications of improved gaussian sampling for trapdoor lattices. IEEE Trans. Comput. 68(4), 570–584 (2019)

    Article  MathSciNet  Google Scholar 

  21. Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197–206. ACM (2008)

    Google Scholar 

  22. Gorbunov, S., Vaikuntanathan, V., Wee, H.: Attribute-based encryption for circuits. In: STOC, pp. 545–554. ACM (2013)

    Google Scholar 

  23. 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). https://doi.org/10.1007/BFb0054868

    Chapter  Google Scholar 

  24. Karney, C.F.F.: Sampling exactly from the normal distribution. ACM Trans. Math. Softw. 42(1), 3:1-3:14 (2016)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  26. Lai, R.W.F., Cheung, H.K.F., Chow, S.S.M.: Trapdoors for ideal lattices with applications. In: Lin, D., Yung, M., Zhou, J. (eds.) Inscrypt 2014. LNCS, vol. 8957, pp. 239–256. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16745-9_14

    Chapter  Google Scholar 

  27. Laguillaumie, F., Langlois, A., Libert, B., Stehlé, D.: Lattice-based group signatures with logarithmic signature size. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8270, pp. 41–61. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42045-0_3

    Chapter  Google Scholar 

  28. 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). https://doi.org/10.1007/978-3-540-78524-8_3

    Chapter  Google Scholar 

  29. 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). https://doi.org/10.1007/978-3-642-13190-5_1

    Chapter  Google Scholar 

  30. Lyubashevsky, V., Peikert, C., Regev, O.: A toolkit for ring-LWE cryptography. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 35–54. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_3

    Chapter  Google Scholar 

  31. Langlois, A., Stehlé, D.: Worst-case to average-case reductions for module lattices. Des. Codes Cryptogr. 75(3), 565–599 (2014). https://doi.org/10.1007/s10623-014-9938-4

    Article  MathSciNet  MATH  Google Scholar 

  32. Lyubashevsky, V., Seiler, G.: Short, invertible elements in partially splitting cyclotomic rings and applications to lattice-based zero-knowledge proofs. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 204–224. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_8

    Chapter  MATH  Google Scholar 

  33. Mennink, B., Neves, S.: Optimal PRFs from blockcipher designs. IACR ToSC 2017(3), 228–252 (2017)

    Article  Google Scholar 

  34. Micciancio, D., Peikert, C.: Trapdoors for lattices: simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_41

    Chapter  Google Scholar 

  35. Micciancio, D., Regev, O.: Worst-case to average-case reductions based on gaussian measures. SIAM J. Comput. 37(1), 267–302 (2007)

    Article  MathSciNet  Google Scholar 

  36. McCarthy, S., Smyth, N., O’Sullivan, E.: A practical implementation of identity-based encryption over NTRU lattices. In: O’Neill, M. (ed.) IMACC 2017. LNCS, vol. 10655, pp. 227–246. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71045-7_12

    Chapter  Google Scholar 

  37. Micciancio, D., Walter, M.: Gaussian sampling over the integers: efficient, generic, constant-time. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10402, pp. 455–485. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63715-0_16

    Chapter  Google Scholar 

  38. Pellet-Mary, A., Hanrot, G., Stehlé, D.: Approx-svp in ideal lattices with pre-processing. IACR Cryptology ePrint Archive 2019:215 (2019)

    Google Scholar 

  39. 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). https://doi.org/10.1007/11681878_8

    Chapter  Google Scholar 

  40. Seiler, G.: Faster AVX2 optimized NTT multiplication for ringlwe lattice cryptography. IACR Cryptology ePrint Archive 2018:39 (2018)

    Google Scholar 

  41. Shamir, A.: Identity-based cryptosystems and signature schemes. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 47–53. Springer, Heidelberg (1985). https://doi.org/10.1007/3-540-39568-7_5

    Chapter  Google Scholar 

  42. Shor, P.W.: Polynomial time algorithms for discrete logarithms and factoring on a quantum computer. In: Adleman, L.M., Huang, M.-D. (eds.) ANTS 1994. LNCS, vol. 877, pp. 289–289. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-58691-1_68

    Chapter  Google Scholar 

  43. Stehlé, D., Steinfeld, R.: Making ntruencrypt and ntrusign as secure as standard worst-case problems over ideal lattices. IACR Cryptology ePrint Archive 2013:4 (2013)

    Google Scholar 

  44. 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). https://doi.org/10.1007/978-3-642-10366-7_36

    Chapter  Google Scholar 

  45. Yamada, S.: Adaptively secure identity-based encryption from lattices with asymptotically shorter public parameters. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 32–62. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_2

    Chapter  Google Scholar 

  46. Zhao, R.K., McCarthy, S., Steinfeld, R., Sakzad, A., O’Neill, M.: Quantum-safe hibe: does it cost a latte? Cryptology ePrint Archive, Report 2021/222 (2021)

    Google Scholar 

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Acknowledgements

This work was supported by the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701). Lucas Prabel is funded by the Direction Générale de l’Armement (Pôle de Recherche CYBER).

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Authors and Affiliations

  1. Univ Rennes, CNRS, IRISA, Rennes, France

    Pauline Bert, Gautier Eberhart, Lucas Prabel, Adeline Roux-Langlois & Mohamed Sabt

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  1. Pauline Bert
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  2. Gautier Eberhart
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  4. Adeline Roux-Langlois
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Correspondence to Lucas Prabel .

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  1. Seoul National University, Seoul, Korea (Republic of)

    Jung Hee Cheon

  2. Inria, Paris, France

    Jean-Pierre Tillich

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Bert, P., Eberhart, G., Prabel, L., Roux-Langlois, A., Sabt, M. (2021). Implementation of Lattice Trapdoors on Modules and Applications. In: Cheon, J.H., Tillich, JP. (eds) Post-Quantum Cryptography. PQCrypto 2021. Lecture Notes in Computer Science(), vol 12841. Springer, Cham. https://doi.org/10.1007/978-3-030-81293-5_11

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  • DOI: https://doi.org/10.1007/978-3-030-81293-5_11

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