Advertisement

Compressible FHE with Applications to PIR

  • Craig GentryEmail author
  • Shai HaleviEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11892)

Abstract

Homomorphic encryption (HE) is often viewed as impractical, both in communication and computation. Here we provide an additively homomorphic encryption scheme based on (ring) LWE with nearly optimal rate (\(1-\epsilon \) for any \(\epsilon >0\)). Moreover, we describe how to compress many Gentry-Sahai-Waters (GSW) ciphertexts (e.g., ciphertexts that may have come from a homomorphic evaluation) into (fewer) high-rate ciphertexts.

Using our high-rate HE scheme, we are able for the first time to describe a single-server private information retrieval (PIR) scheme with sufficiently low computational overhead so as to be practical for large databases. Single-server PIR inherently requires the server to perform at least one bit operation per database bit, and we describe a rate-(4/9) scheme with computation which is not so much worse than this inherent lower bound. In fact it is probably less than whole-database AES encryption – specifically about 2.3 mod-q multiplication per database byte, where q is about 50 to 60 bits. Asymptotically, the computational overhead of our PIR scheme is \(\tilde{O}(\log \log \mathsf {\lambda }+ \log \log \log N)\), where \(\mathsf {\lambda }\) is the security parameter and N is the number of database files, which are assumed to be sufficiently large.

Notes

Acknowledgment

We thank Yuval Ishai for badgering us over the last four years to figure out the achievable rate in LWE-based constructions, until we could bare it no longer and did this work. We also thank Samir Menon and the anonymous reviewers for their useful comments.

References

  1. 1.
    Aguilar-Melchor, C., Barrier, J., Fousse, L., Killijian, M.-O.: XPIR: private information retrieval for everyone. Proc. Priv. Enhancing Technol. 2016(2), 155–174 (2016)CrossRefGoogle Scholar
  2. 2.
    Albrecht, M., et al.: Homomorphic encryption standard, November 2018. http://homomorphicencryption.org/. Accessed Feb 2019
  3. 3.
    Angel, S., Chen, H., Laine, K., Setty, S.: PIR with compressed queries and amortized query processing. In: 2018 IEEE Symposium on Security and Privacy (SP), pp. 962–979. IEEE (2018)Google Scholar
  4. 4.
    Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast cryptographic primitives and circular-secure encryption based on hard learning problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-03356-8_35CrossRefGoogle Scholar
  5. 5.
    Arora, S., Ge, R.: New algorithms for learning in presence of errors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 403–415. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22006-7_34CrossRefGoogle Scholar
  6. 6.
    Badrinarayanan, S., Garg, S., Ishai, Y., Sahai, A., Wadia, A.: Two-message witness indistinguishability and secure computation in the plain model from new assumptions. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10626, pp. 275–303. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70700-6_10CrossRefzbMATHGoogle Scholar
  7. 7.
    Brakerski, Z., Döttling, N., Garg, S., Malavolta, G.: Leveraging linear decryption: Rate-1 fully-homomorphic encryption and time-lock puzzles. Private communications (2019)Google Scholar
  8. 8.
    Brakerski, Z., Gentry, C., Halevi, S.: Packed ciphertexts in LWE-based homomorphic encryption. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 1–13. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36362-7_1CrossRefGoogle Scholar
  9. 9.
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: Fully homomorphic encryption without bootstrapping. In: Innovations in Theoretical Computer Science (ITCS 2012) (2012). http://eprint.iacr.org/2011/277
  10. 10.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. SIAM J. Comput. 43(2), 831–871 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brakerski, Z., Vaikuntanathan, V.: Lattice-based FHE as secure as PKE. In: Naor, M. (ed.) Innovations in Theoretical Computer Science, ITCS 2014, pp. 1–12. ACM (2014)Google Scholar
  12. 12.
    Chor, B., Goldreich, O., Kushilevitz, E., Sudan, M.: Private information retrieval. In: Proceedings, 36th Annual Symposium on Foundations of Computer Science 1995, pp. 41–50. IEEE (1995)Google Scholar
  13. 13.
    Crypto++ 5.6.0, pentium 4 benchmarks (2009). https://www.cryptopp.com/benchmarks-p4.html. Accessed Feb 2019
  14. 14.
    Damgård, I., Jurik, M.: A generalisation, a simpli.cation and some applications of paillier’s probabilistic public-key system. In: Kim, K. (ed.) PKC 2001. LNCS, vol. 1992, pp. 119–136. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44586-2_9CrossRefzbMATHGoogle Scholar
  15. 15.
    Döttling, N., Garg, S., Ishai, Y., Malavolta, G., Mour, T., Ostrovsky, R.: Trapdoor hash functions and their applications. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 3–32. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-26954-8_1CrossRefGoogle Scholar
  16. 16.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197–206. ACM (2008)Google Scholar
  17. 17.
    Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_5CrossRefGoogle Scholar
  18. 18.
    Green, M., Hohenberger, S., Waters, B.: Outsourcing the decryption of ABE ciphertexts. In: Proceedings 20th USENIX Security Symposium, San Francisco, CA, USA, 8–12 August 2011. USENIX Association (2011)Google Scholar
  19. 19.
    Halevi, S.: Homomorphic encryption. Tutorials on the Foundations of Cryptography. ISC, pp. 219–276. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-57048-8_5CrossRefGoogle Scholar
  20. 20.
    Hiromasa, R., Abe, M., Okamoto, T.: Packing messages and optimizing bootstrapping in gsw-fhe. IEICE TRANS. Fundam. Electron. Commun. Comput. Sci. 99(1), 73–82 (2016)CrossRefGoogle Scholar
  21. 21.
    Kiayias, A., Leonardos, N., Lipmaa, H., Pavlyk, K., Tang, Q.: Optimal rate private information retrieval from homomorphic encryption. Proc. Priv. Enhancing Technol. 2015(2), 222–243 (2015)CrossRefGoogle Scholar
  22. 22.
    Klinc, D., Hazay, C., Jagmohan, A., Krawczyk, H., Rabin, T.: On compression of data encrypted with block ciphers. IEEE Trans. Inf. Theory 58(11), 6989–7001 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kushilevitz, E., Ostrovsky, R.: Replication is not needed: Single database, computationally-private information retrieval. In: Proceedings, 38th Annual Symposium on Foundations of Computer Science 1997, pp. 364–373. IEEE (1997)Google Scholar
  24. 24.
    Laderman, J.D.: A noncommutative algorithm for multiplying \(3 \times 3\) matrices using 23 multiplications. Bull. Amer. Math. Soc. 82(1), 126–128 (1976)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lipmaa, H., Pavlyk, K.: A simpler rate-optimal CPIR protocol. In: Kiayias, A. (ed.) FC 2017. LNCS, vol. 10322, pp. 621–638. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70972-7_35CrossRefGoogle Scholar
  26. 26.
    Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. J. ACM 60(6), 43 (2013). Early version in EUROCRYPT 2010MathSciNetCrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    Olumofin, F., Goldberg, I.: Revisiting the computational practicality of private information retrieval. In: Danezis, G. (ed.) FC 2011. LNCS, vol. 7035, pp. 158–172. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-27576-0_13CrossRefGoogle Scholar
  29. 29.
    Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48910-X_16CrossRefGoogle Scholar
  30. 30.
    Peikert, C., Vaikuntanathan, V., Waters, B.: A framework for efficient and composable oblivious transfer. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 554–571. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85174-5_31CrossRefGoogle Scholar
  31. 31.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6), 34:1–34:40 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Schmid, P., Roos, A.: AES-NI performance analyzed; limited to 32nm core i5 CPUs (2010). https://www.tomshardware.com/reviews/clarkdale-aes-ni-encryption,2538.html. Accessed Feb 2019
  33. 33.
    Sion, R., Carbunar, B.: On the practicality of private information retrieval. In: Proceedings of the Network and Distributed System Security Symposium, NDSS 2007, San Diego, California, USA, 28 February–2nd March 2007 (2007)Google Scholar
  34. 34.
    Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Crypt. 71(1), 57–81 (2014). Early verion at http://eprint.iacr.org/2011/133CrossRefGoogle Scholar
  35. 35.
    Stern, J.P.: A new and efficient all-or-nothing disclosure of secrets protocol. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 357–371. Springer, Heidelberg (1998).  https://doi.org/10.1007/3-540-49649-1_28CrossRefGoogle Scholar
  36. 36.
    van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24–43. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_2CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  1. 1.Algorand FoundationNew-York CityUSA

Personalised recommendations