Abstract
The LWE problem with its ring variants is today the most prominent candidate for building efficient public key cryptosystems resistant to quantum computers. NTRU-type cryptosystems use an LWE-type variant with small max-norm secrets, usually with ternary coefficients from the set \(\{-1,0,1\}\). The presumably best attack on these schemes is a hybrid attack that combines lattice reduction techniques with Odlyzko’s Meet-in-the-Middle approach. Odlyzko’s algorithm is a classical combinatorial attack that for key space size \(\mathcal{S}\) runs in time \(\mathcal{S}^{0.5}\). We substantially improve on this Meet-in-the-Middle approach, using the representation technique developed for subset sum algorithms. Asymptotically, our heuristic Meet-in-the-Middle attack runs in time roughly \(\mathcal{S}^{0.25}\), which also beats the \(\mathcal{S}^{\frac{1}{3}}\) complexity of the best known quantum algorithm.
For the round-3 NIST post-quantum encryptions NTRU and NTRU Prime we obtain non-asymptotic instantiations of our attack with complexity roughly \(\mathcal{S}^{0.3}\). As opposed to other combinatorial attacks, our attack benefits from larger LWE field sizes q, as they are often used in modern lattice-based signatures. For example, for BLISS and GLP signatures we obtain non-asymptotic combinatorial attacks around \(\mathcal{S}^{0.28}\).
Our attacks do not invalidate the security claims of the aforementioned schemes. However, they establish improved combinatorial upper bounds for their security. We leave it is an open question whether our new Meet-in-the-Middle attack in combination with lattice reduction can be used to speed up the hybrid attack.
A. May—Funded by DFG under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albrecht, M., Bai, S., Ducas, L.: A subfield lattice attack on overstretched NTRU assumptions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part I. LNCS, vol. 9814, pp. 153–178. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_6
Albrecht, M., Cid, C., Faugere, J.C., Fitzpatrick, R., Perret, L.: Algebraic algorithms for LWE problems (2014)
Albrecht, M.R., Player, R., Scott, S.: On the concrete hardness of learning with errors. J. Math. Cryptol. 9(3), 169–203 (2015)
Babai, L.: On lovász’lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1–13 (1986)
Bonnetain, X., Bricout, R., Schrottenloher, A., Shen, Y.: Improved classical and quantum algorithms for subset-sum. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020, Part II. LNCS, vol. 12492, pp. 633–666. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_22
Becker, A., Coron, J.-S., Joux, A.: Improved generic algorithms for hard knapsacks. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 364–385. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_21
Bernstein, D.J., Chuengsatiansup, C., Lange, T., van Vredendaal, C.: NTRU prime: round 2 specification (2019)
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
Becker, A., Ducas,, L., Gama, N., Laarhoven, T.: New directions in nearest neighbor searching with applications to lattice sieving. In: Krauthgamer, R. (ed.) 27th SODA, ACM-SIAM, pp. 10–24, January 2016
Bos, J.W., et al.: CRYSTALS-Kyber: a CCA-secure module-lattice-based KEM. Cryptology ePrint Archive (20180716: 135545) (2017)
Bai, S., Galbraith, S.D.: Lattice decoding attacks on binary LWE. In: Susilo, W., Mu, Y. (eds.) ACISP 2014. LNCS, vol. 8544, pp. 322–337. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08344-5_21
Buchmann, J., Göpfert, F., Player, R., Wunderer, T.: On the hardness of LWE with binary error: revisiting the hybrid lattice-reduction and meet-in-the-middle attack. In: Pointcheval, D., Nitaj, A., Rachidi, T. (eds.) AFRICACRYPT 2016. LNCS, vol. 9646, pp. 24–43. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31517-1_2
Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in \(2^n/20\): how \(1+1=0\) improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_31
Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, pp. 575–584. ACM Press, June 2013
Chen, C., et al.: NTRU - algorithm specifications and supporting documentation (2019)
de Boer, K., Ducas, L., Jeffery, S., de Wolf, R.: Attacks on the AJPS Mersenne-based cryptosystem. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 101–120. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_5
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). https://doi.org/10.1007/978-3-642-40041-4_3
Ducas, L., Nguyen, P.Q.: Faster Gaussian lattice sampling using lazy floating-point arithmetic. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 415–432. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34961-4_26
D’Anvers, J.-P., Rossi, M., Virdia, F.: (One) Failure Is Not an Option: bootstrapping the search for failures in lattice-based encryption schemes. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 3–33. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_1
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) 41st ACM STOC, pp. 169–178. ACM Press, May/June 2009
Guo, Q., Johansson, T., Stankovski, P.: Coded-BKW: solving LWE using lattice codes. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015, Part I. LNCS, vol. 9215, pp. 23–42. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_2
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). https://doi.org/10.1007/978-3-642-33027-8_31
Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Ladner, R.E., Dwork, C. (eds.) 40th ACM STOC, pp. 197–206. ACM Press, May 2008
Howgrave-Graham, N., Silverman, J.H., Whyte, W.: A meet-in-the-middle attack on an NTRU private key. Technical report, NTRU Cryptosystems, June 2003
Howgrave-Graham, N., Joux, A.: New generic algorithms for hard knapsacks. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 235–256. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_12
Herold, G., Kirshanova, E., Laarhoven, T.: Speed-ups and time–memory trade-offs for tuple lattice sieving. In: Abdalla, M., Dahab, R. (eds.) PKC 2018, Part I. LNCS, vol. 10769, pp. 407–436. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76578-5_14
Howgrave-Graham, N., et al.: The impact of decryption failures on the security of NTRU encryption. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 226–246. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_14
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). https://doi.org/10.1007/978-3-540-74143-5_9
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.: An improved BKW algorithm for LWE with applications to cryptography and lattices. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015, Part I. LNCS, vol. 9215, pp. 43–62. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_3
Laarhoven, T.: Sieving for shortest vectors in lattices using angular locality-sensitive hashing. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015, Part I. LNCS, vol. 9215, pp. 3–22. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_1
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
Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 738–755. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_43
May, A.: How to meet ternary LWE keys. Cryptology ePrint Archive, Report 2021/216 (2021). https://eprint.iacr.org/2021/216
May, A., Ozerov, I.: On computing nearest neighbors with applications to decoding of binary linear codes. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part I. LNCS, vol. 9056, pp. 203–228. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_9
Micciancio, D., Peikert, C.: Hardness of SIS and LWE with small parameters. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 21–39. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_2
May, A., Silverman, J.H.: Dimension reduction methods for convolution modular lattices. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 110–125. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44670-2_10
Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis. Cambridge University Press, Cambridge (2017)
Nguyen, P.Q.: Boosting the hybrid attack on NTRU: torus LSH, permuted HNF and boxed sphere. In: NIST Third PQC Standardization Conference (2021)
1-2008 - IEEE standard specification for public key cryptographic techniques based on hard problems over lattices (2008)
Nguyen, P.Q., Vidick, T.: Sieve algorithms for the shortest vector problem are practical. J. Math. Cryptol. 2(2), 181–207 (2008)
Regev, O.: New lattice based cryptographic constructions. In: 35th ACM STOC, pp. 407–416. ACM Press, June 2003
Stehlé, D., Steinfeld, R.: Making NTRU as secure as worst-case problems over ideal lattices. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 27–47. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_4
van Vredendaal, C.: Reduced memory meet-in-the-middle attack against the NTRU private key. LMS J. Comput. Math. 19(A), 43–57 (2016)
van Oorschot, P.C., Wiener, M.J.: Parallel collision search with cryptanalytic applications. J. Cryptol. 12(1), 1–28 (1999)
Wang, H., Ma, Z., Ma, C.G.: An efficient quantum meet-in-the-middle attack against NTRU-2005. Chin. Sci. Bull. 58(28), 3514–3518 (2013)
Wunderer, T.: A detailed analysis of the hybrid lattice-reduction and meet-in-the-middle attack. J. Math. Cryptol. 13(1), 1–26 (2019)
Acknowledgements
The author wants to thank Elena Kirshanova, John Schank and Andre Esser for discussions and estimations concerning lattice reduction and the Hybrid attack, and the anonymous reviewers for their valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 International Association for Cryptologic Research
About this paper
Cite this paper
May, A. (2021). How to Meet Ternary LWE Keys. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12826. Springer, Cham. https://doi.org/10.1007/978-3-030-84245-1_24
Download citation
DOI: https://doi.org/10.1007/978-3-030-84245-1_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-84244-4
Online ISBN: 978-3-030-84245-1
eBook Packages: Computer ScienceComputer Science (R0)