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Gadget-Based iNTRU Lattice Trapdoors

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Progress in Cryptology – INDOCRYPT 2020 (INDOCRYPT 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12578))

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Abstract

We present two new related families of lattice trapdoors based on the inhomogeneous NTRU problem (iNTRU) defined in Genise et al. [16] (ASIACRYPT 2019). Our constructions are “gadget-based” and offer compact secret keys and preimages and compatibility with existing, efficient preimage sampling algorithms. Our trapdoors can be used as a fundamental building block in lattice-based schemes relying lattice trapdoors. In addition, we implemented our trapdoors using the PALISADE library.

This work was done at Rutgers supported by National Science Foundation grant SaTC-1815562.

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Notes

  1. 1.

    https://csrc.nist.gov/Projects/post-quantum-cryptography/round-2-submissions.

  2. 2.

    https://palisade-crypto.org/.

  3. 3.

    https://eprint.iacr.org/.

  4. 4.

    https://falcon-sign.info/.

  5. 5.

    The smoothing parameter of \(\eta _\epsilon (\varLambda _q^\perp (\mathbf {g}^t))\) can be lower-bounded by \(\approx b \cdot \eta _{\epsilon }(\mathbb {Z}^{nk})\) [28, Sect. 4]. And, \(\eta _{\epsilon }(\mathbb {Z}^{nk})\) is no more than \(\sqrt{ \log (2nk(1+1/\epsilon ))/\pi }\) by Lemma 2.

  6. 6.

    Notice how these are the best Gaussian shapes, in terms of spectral width, that one can hope for. The Gaussian shape of \(r\mathbf {x}\) is \(\sigma _g^2r\overline{r}\mathbf {I}_m\) and the Gaussian shape of \(\mathbf {e}^t\mathbf {x}\) is \(\sigma _g^2\mathbf {e}^t\mathbf {e}\).

  7. 7.

    We remark that the analogous proof in [20] amounts to s being above the smoothing parameter, \(s \ge \eta _{\epsilon }(\varLambda _q^\perp (\mathbf {A}))\), and Lemma 1. In our situation, we must carefully show the trapdoor is statistically hidden with the kernel lemma, Lemma 5.

  8. 8.

    Source code can be downloaded at https://www.dropbox.com/s/uz3g3atpqu7u87n/intrusign.zip?dl=0.

References

  1. Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: STOC, pp. 99–108. ACM (1996)

    Google Scholar 

  2. Albrecht, M., Bai, S., Ducas, L.: A subfield lattice attack on overstretched NTRU assumptions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 153–178. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_6

    Chapter  Google Scholar 

  3. 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 

  4. Becker, A., Ducas, L., Gama, N., Laarhoven, T.: New directions in nearest neighbor searching with applications to lattice sieving. In: SODA, pp. 10–24. SIAM (2016)

    Google Scholar 

  5. Bernstein, D.J., Chuengsatiansup, C., Lange, T., van Vredendaal, C.: NTRU prime: reducing attack surface at low cost. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 235–260. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72565-9_12

    Chapter  Google Scholar 

  6. Boneh, D., et al.: Fully key-homomorphic encryption, arithmetic circuit ABE and compact garbled circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 533–556. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_30

    Chapter  Google Scholar 

  7. 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 

  8. Chen, Y.: Réduction de réseau et sécurité concréte du chiffrement complétement homomorphe. PhD thesis, Paris 7 (2013)

    Google Scholar 

  9. Cheon, J.H., Kim, D., Kim, T., Son, Y.: A new trapdoor over module-NTRU lattice and its application to id-based encryption. IACR Cryptology ePrint Archive, 1468 (2019)

    Google Scholar 

  10. 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 

  11. Cousins, D.B.: Implementing conjunction obfuscation under entropic ring LWE. In: Proceedings of 2018 IEEE Symposium on Security and Privacy, SP 2018, 21–23 May 2018, San Francisco, California, USA, pp. 354–371 (2018)

    Google Scholar 

  12. Ducas, L., Galbraith, S., Prest, T., Yu, Y.: Integral matrix gram root and lattice gaussian sampling without floats. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12106, pp. 608–637. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45724-2_21

    Chapter  Google Scholar 

  13. 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 

  14. Ducas, L., Micciancio, D.: Improved short lattice signatures in the standard model. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 335–352. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_19

    Chapter  MATH  Google Scholar 

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

    Google Scholar 

  16. Genise, N., Gentry, C., Halevi, S., Li, B., Micciancio, D.: Homomorphic encryption for finite automata. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11922, pp. 473–502. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34621-8_17

    Chapter  Google Scholar 

  17. 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 

  18. Genise, N., Micciancio, D., Peikert, C., Walter, M.: Improved discrete Gaussian and sub Gaussian analysis for lattice cryptography. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020. LNCS, vol. 12110, pp. 623–651. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45374-9_21

    Chapter  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  21. Gorbunov, S., Vaikuntanathan, V., Wichs, D.: Leveled fully homomorphic signatures from standard lattices. In: STOC, pp. 469–477. ACM (2015)

    Google Scholar 

  22. 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 

  23. Kirchner, P., Fouque, P.-A.: Revisiting lattice attacks on overstretched NTRU parameters. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 3–26. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_1

    Chapter  Google Scholar 

  24. Laarhoven, T.: Search Problems in Cryptography. PhD thesis, Eindhoven University of Technology (2015)

    Google Scholar 

  25. 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). https://doi.org/10.1007/11787006_13

    Chapter  Google Scholar 

  26. 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 

  27. Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. In: FOCS, pp. 356–365. IEEE Computer Society (2002)

    Google Scholar 

  28. 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 

  29. 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 

  30. 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 

  31. Peikert, C.: An efficient and parallel Gaussian sampler for lattices. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 80–97. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_5

    Chapter  Google Scholar 

  32. Peikert, C.: A decade of lattice cryptography. Found. Trends Theor. Comput. Sci. 10(4), 283–424 (2016)

    Article  MathSciNet  Google Scholar 

  33. 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 

  34. Prest, T.: Sharper bounds in lattice-based cryptography using the Rényi divergence. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 347–374. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_13

    Chapter  Google Scholar 

  35. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC, pp. 84–93. ACM (2005)

    Google Scholar 

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

    Article  Google Scholar 

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Authors and Affiliations

  1. SRI International, San Diego, USA

    Nicholas Genise

  2. University of California, San Diego, USA

    Baiyu Li

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  1. Nicholas Genise
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Correspondence to Nicholas Genise .

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Editors and Affiliations

  1. Inria Paris, Paris, France

    Karthikeyan Bhargavan

  2. University of Klagenfurt, Klagenfurt, Austria

    Elisabeth Oswald

  3. Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, Maharashtra, India

    Manoj Prabhakaran

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Genise, N., Li, B. (2020). Gadget-Based iNTRU Lattice Trapdoors. In: Bhargavan, K., Oswald, E., Prabhakaran, M. (eds) Progress in Cryptology – INDOCRYPT 2020. INDOCRYPT 2020. Lecture Notes in Computer Science(), vol 12578. Springer, Cham. https://doi.org/10.1007/978-3-030-65277-7_27

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