A Nonstandard Variant of Learning with Rounding with Polynomial Modulus and Unbounded Samples

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10786)

Abstract

The Learning with Rounding Problem (LWR) has become a popular cryptographic assumption to study recently due to its determinism and resistance to known quantum attacks. Unfortunately, LWR is only known to be provably hard for instances of the problem where the LWR modulus q is at least as large as some polynomial function of the number of samples given to an adversary, meaning LWR is provably hard only when (1) an adversary can only see a fixed, predetermined amount of samples or (2) the modulus q is superpolynomial in the security parameter, meaning that the hardness reduction is from superpolynomial approximation factors on worst-case lattices.

In this work, we show that there exists a (still fully deterministic) variant of the LWR problem that allows for both unbounded queries and a polynomial modulus q, breaking an important theoretical barrier. To our knowledge, our new assumption, which we call the Nearby Learning with Lattice Rounding Problem (NLWLR), is the first fully deterministic version of the learning with errors (LWE) problem that allows for both unbounded queries and a polynomial modulus. We note that our assumption is not practical for any kind of use and is mainly intended as a theoretical proof of concept to show that provably hard deterministic forms of LWE can exist with a modulus that does not grow polynomially with the number of samples.

Keywords

Lattices LWE Learning with rounding 

Notes

Acknowledgements

We would like to thank Arnab Roy, Avradip Mandal, Sam Kim, David Wu, and Mike Hamburg for discussing lattices with us. We also want to thank the anonymous reviewers of the PQCrypto committee for their useful comments and suggestions on the paper and, in particular, anonymous reviewer #2 for their extremely helpful and detailed review.

References

  1. [AA16]
    Alperin-Sheriff, J., Apon, D.: Dimension-preserving reductions from LWE to LWR. IACR Cryptology ePrint Archive, 2016:589 (2016)Google Scholar
  2. [AGHS13]
    Agrawal, S., Gentry, C., Halevi, S., Sahai, A.: Discrete Gaussian leftover hash lemma over infinite domains. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8269, pp. 97–116. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-42033-7_6CrossRefGoogle Scholar
  3. [AKPW13]
    Alwen, J., Krenn, S., Pietrzak, K., Wichs, D.: Learning with rounding, revisited - new reduction, properties and applications. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 57–74. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_4CrossRefGoogle Scholar
  4. [BCD+16]
    Bos, J.W., Costello, C., Ducas, L., Mironov, I., Naehrig, M., Nikolaenko, V., Raghunathan, A., Stebila, D.: Frodo: take off the ring! Practical, quantum-secure key exchange from LWE. In: ACM CCS 2016: 23rd Conference on Computer and Communications Security, pp. 1006–1018. ACM Press (2016)Google Scholar
  5. [BDK+11]
    Barak, B., Dodis, Y., Krawczyk, H., Pereira, O., Pietrzak, K., Standaert, F.-X., Yu, Y.: Leftover hash lemma, revisited. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 1–20. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_1CrossRefGoogle Scholar
  6. [BDK+17]
    Bos, J.W., Ducas, L., Kiltz, E., Lepoint, T., Lyubashevsky, V., Schanck, J.M., Schwabe, P., Stehlé, D.: CRYSTALS - Kyber: a CCA-secure module-lattice-based KEM. IACR Cryptology ePrint Archive, 2017:634 (2017)Google Scholar
  7. [BFP+15]
    Banerjee, A., Fuchsbauer, G., Peikert, C., Pietrzak, K., Stevens, S.: Key-homomorphic constrained pseudorandom functions. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 31–60. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46497-7_2CrossRefGoogle Scholar
  8. [BGG+14]
    Boneh, D., Gentry, C., Gorbunov, S., Halevi, S., Nikolaenko, V., Segev, G., Vaikuntanathan, V., Vinayagamurthy, D.: 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_30CrossRefGoogle Scholar
  9. [BGM+16]
    Bogdanov, A., Guo, S., Masny, D., Richelson, S., Rosen, A.: On the hardness of learning with rounding over small modulus. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9562, pp. 209–224. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49096-9_9CrossRefGoogle Scholar
  10. [BKM17]
    Boneh, D., Kim, S., Montgomery, H.: Private puncturable PRFs from standard lattice assumptions. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 415–445. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56620-7_15CrossRefGoogle Scholar
  11. [BLL+15]
    Bai, S., Langlois, A., Lepoint, T., Stehlé, D., Steinfeld, R.: Improved security proofs in lattice-based cryptography: using the rényi divergence rather than the statistical distance. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 3–24. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48797-6_1CrossRefGoogle Scholar
  12. [BLMR13]
    Boneh, D., Lewi, K., Montgomery, H., Raghunathan, A.: Key homomorphic PRFs and their applications. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 410–428. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_23CrossRefGoogle Scholar
  13. [BLP+13]
    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 Annual ACM Symposium on Theory of Computing, Palo Alto, CA, USA, 1–4 June 2013, pp. 575–584. ACM Press (2013)Google Scholar
  14. [BP14]
    Banerjee, A., Peikert, C.: New and improved key-homomorphic pseudorandom functions. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 353–370. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44371-2_20CrossRefGoogle Scholar
  15. [BPR12]
    Banerjee, A., Peikert, C., Rosen, A.: Pseudorandom functions and lattices. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 719–737. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_42CrossRefGoogle Scholar
  16. [BV11]
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Ostrovsky, R. (ed.) 52nd Annual Symposium on Foundations of Computer Science, Palm Springs, California, USA, 22–25 October 2011, pp. 97–106. IEEE Computer Society Press (2011)Google Scholar
  17. [CC17]
    Canetti, R., Chen, Y.: Constraint-hiding constrained PRFs for NC\(^1\) from LWE. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 446–476. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56620-7_16CrossRefGoogle Scholar
  18. [CKLS16]
    Cheon, J.H., Kim, D., Lee, J., Song, Y.: Lizard: cut off the tail! practical post-quantum public-key encryption from LWE and LWR (2016). http://eprint.iacr.org/2016/1126
  19. [DFH+16]
    Dziembowski, S., Faust, S., Herold, G., Journault, A., Masny, D., Standaert, F.-X.: Towards sound fresh re-keying with hard (physical) learning problems. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 272–301. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53008-5_10CrossRefGoogle Scholar
  20. [GGM84]
    Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions (extended abstract). In: 25th Annual Symposium on Foundations of Computer Science, Singer Island, Florida, 24–26 October 1984, pp. 464–479. IEEE Computer Society Press (1984)Google Scholar
  21. [GPV08]
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Ladner, R.E., Dwork, C. (eds.) 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, 17–20 May 2008, pp. 197–206. ACM Press (2008)Google Scholar
  22. [MP12]
    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_41CrossRefGoogle Scholar
  23. [MR04]
    Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. In: 45th Annual Symposium on Foundations of Computer Science, Rome, Italy, 17–19 October 2004, pp. 372–381. IEEE Computer Society Press (2004)Google Scholar
  24. [NR06]
    Nguyen, P.Q., Regev, O.: Learning a parallelepiped: cryptanalysis of GGH and NTRU signatures. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 271–288. Springer, Heidelberg (2006).  https://doi.org/10.1007/11761679_17CrossRefGoogle Scholar
  25. [Pei14]
    Peikert, C.: Lattice cryptography for the internet. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 197–219. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-11659-4_12Google Scholar
  26. [Reg05]
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) 37th Annual ACM Symposium on Theory of Computing, Baltimore, Maryland, USA, 22–24 May 2005, pp. 84–93. ACM Press (2005)Google Scholar
  27. [RRVV14]
    Roy, S.S., Reparaz, O., Vercauteren, F., Verbauwhede, I.: Compact and side channel secure discrete Gaussian sampling. Cryptology ePrint Archive, Report 2014/591 (2014). http://eprint.iacr.org/2014/591

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fujitsu Laboratories of AmericaSunnyvaleUSA

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