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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
- 2.
- 3.
- 4.
- 5.
- 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.
- 8.
Source code can be downloaded at https://www.dropbox.com/s/uz3g3atpqu7u87n/intrusign.zip?dl=0.
References
Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: STOC, pp. 99–108. ACM (1996)
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
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)
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)
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
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
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
Chen, Y.: Réduction de réseau et sécurité concréte du chiffrement complétement homomorphe. PhD thesis, Paris 7 (2013)
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)
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)
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)
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
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
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
Ducas, L., Prest, T.: Fast fourier orthogonalization. In: ISSAC, pp. 191–198. ACM (2016)
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
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
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
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178. ACM (2009)
Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197–206. ACM (2008)
Gorbunov, S., Vaikuntanathan, V., Wichs, D.: Leveled fully homomorphic signatures from standard lattices. In: STOC, pp. 469–477. ACM (2015)
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
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
Laarhoven, T.: Search Problems in Cryptography. PhD thesis, Eindhoven University of Technology (2015)
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
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
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)
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
Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. SIAM J. Comput. 37(1), 267–302 (2007)
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
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
Peikert, C.: A decade of lattice cryptography. Found. Trends Theor. Comput. Sci. 10(4), 283–424 (2016)
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
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
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC, pp. 84–93. ACM (2005)
Schnorr, C.P., Euchner, M.: Improved practical algorithms and solving subset sum problems: lattice basis reduction. Math. Program. 66, 181–199 (1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-65277-7_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-65276-0
Online ISBN: 978-3-030-65277-7
eBook Packages: Computer ScienceComputer Science (R0)