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Building an Efficient Lattice Gadget Toolkit: Subgaussian Sampling and More

  • Nicholas GeniseEmail author
  • Daniele Micciancio
  • Yuriy Polyakov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11477)

Abstract

Many advanced lattice cryptography applications require efficient algorithms for inverting the so-called “gadget” matrices, which are used to formally describe a digit decomposition problem that produces an output with specific (statistical) properties. The common gadget inversion problems are the classical (often binary) digit decomposition, subgaussian decomposition, Learning with Errors (LWE) decoding, and discrete Gaussian sampling. In this work, we build and implement an efficient lattice gadget toolkit that provides a general treatment of gadget matrices and algorithms for their inversion/sampling. The main contribution of our work is a set of new gadget matrices and algorithms for efficient subgaussian sampling that have a number of major theoretical and practical advantages over previously known algorithms. Another contribution deals with efficient algorithms for LWE decoding and discrete Gaussian sampling in the Residue Number System (RNS) representation.

We implement the gadget toolkit in PALISADE and evaluate the performance of our algorithms both in terms of runtime and noise growth. We illustrate the improvements due to our algorithms by implementing a concrete complex application, key-policy attribute-based encryption (KP-ABE), which was previously considered impractical for CPU systems (except for a very small number of attributes). Our runtime improvements for the main bottleneck operation based on subgaussian sampling range from 18x (for 2 attributes) to 289x (for 16 attributes; the maximum number supported by a previous implementation). Our results are applicable to a wide range of other advanced applications in lattice cryptography, such as GSW-based homomorphic encryption schemes, leveled fully homomorphic signatures, other forms of ABE, some program obfuscation constructions, and more.

References

  1. 1.
    Ajtai, M.: Generating hard instances of the short basis problem. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 1–9. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48523-6_1CrossRefGoogle Scholar
  2. 2.
    Albrecht, M., et al.: Homomorphic encryption security standard. Technical report, HomomorphicEncryption.org, Cambridge, MA, March 2018Google Scholar
  3. 3.
    Albrecht, M., Scott, S., Player, R.: On the concrete hardness of learning with errors. J. Math. Cryptol. 9(3), 169–203 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alperin-Sheriff, J., Apon, D.: Weak is better: tightly secure short signatures from weak PRFs. IACR Cryptology ePrint Archive, 2017:563 (2017)Google Scholar
  5. 5.
    Alperin-Sheriff, J., Peikert, C.: Faster bootstrapping with polynomial error. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 297–314. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44371-2_17CrossRefGoogle Scholar
  6. 6.
    Alwen, J., Peikert, C.: Generating shorter bases for hard random lattices. Theory Comput. Syst. 48(3), 535–553 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Babai, L.: On lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1–13 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bajard, J.-C., Eynard, J., Hasan, M.A., Zucca, V.: A full RNS variant of FV like somewhat homomorphic encryption schemes. In: Avanzi, R., Heys, H. (eds.) SAC 2016. LNCS, vol. 10532, pp. 423–442. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-69453-5_23CrossRefGoogle Scholar
  9. 9.
    Bert, P., Fouque, P.-A., Roux-Langlois, A., Sabt, M.: Practical implementation of ring-SIS/LWE based signature and IBE. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 271–291. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-79063-3_13CrossRefGoogle Scholar
  10. 10.
    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_30CrossRefGoogle Scholar
  11. 11.
    Bonnoron, G., Ducas, L., Fillinger, M.: Large FHE gates from tensored homomorphic accumulator. In: Joux, A., Nitaj, A., Rachidi, T. (eds.) AFRICACRYPT 2018. LNCS, vol. 10831, pp. 217–251. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-89339-6_13CrossRefGoogle Scholar
  12. 12.
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Innovations in Theoretical Computer Science - ITCS 2012, pp. 309–325. ACM (2012)Google Scholar
  13. 13.
    Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Symposium on Theory of Computing - STOC 2013, pp. 575–584 (2013)Google Scholar
  14. 14.
    Brakerski, Z., Vaikuntanathan, V.: Lattice-based FHE as secure as PKE. In: Innovations in Theoretical Computer Science - ITCS 2014, pp. 1–12 (2014)Google Scholar
  15. 15.
    Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 3–33. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53887-6_1CrossRefzbMATHGoogle Scholar
  16. 16.
    Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster packed homomorphic operations and efficient circuit bootstrapping for TFHE. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 377–408. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70694-8_14CrossRefGoogle Scholar
  17. 17.
    Cousins, D.B., et al.: Implementing conjunction obfuscation under entropic ring LWE. In: Symposium on Security and Privacy - SSP 2018, pp. 354–371 (2018)Google Scholar
  18. 18.
    Crockett, E., Peikert, C.: \(\Lambda \)\(o\)\(\lambda \): functional lattice cryptography. In: Weippl, E.R., Katzenbeisser, S., Kruegel, C., Myers, A.C., Halevi, S. (eds.) Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, Vienna, Austria, 24–28 October 2016, pp. 993–1005. ACM (2016)Google Scholar
  19. 19.
    Dai, W., et al.: Implementation and evaluation of a lattice-based key-policy ABE scheme. IEEE Trans. Inf. Forensics Secur. 13(5), 1169–1184 (2018)CrossRefGoogle Scholar
  20. 20.
    del Pino, R., Lyubashevsky, V., Seiler, G.: Lattice-based group signatures and zero-knowledge proofs of automorphism stability. In: Lie, D., Mannan, M., Backes, M., Wang, X. (eds.) Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, CCS 2018, pp. 574–591. ACM (2018)Google Scholar
  21. 21.
    Ducas, L., Micciancio, D.: FHEW: bootstrapping homomorphic encryption in less than a second. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 617–640. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_24CrossRefzbMATHGoogle Scholar
  22. 22.
    Ducas, L., Prest, T.: Fast fourier orthogonalization. In: Abramov, S.A., Zima, E.V., Gao, X. (eds.) Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, pp. 191–198. ACM (2016)Google Scholar
  23. 23.
    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_7CrossRefGoogle Scholar
  24. 24.
    Genise, N., Micciancio, D., Polyakov, Y.: Building an efficient lattice gadget toolkit: Subgaussian sampling and more. IACR Cryptology ePrint Archive, 2018:946 (2018)Google Scholar
  25. 25.
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_49CrossRefGoogle Scholar
  26. 26.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Symposium on Theory of Computing - STOC 2008, pp. 197–206 (2008)Google Scholar
  27. 27.
    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
  28. 28.
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Predicate encryption for circuits from LWE. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 503–523. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48000-7_25CrossRefGoogle Scholar
  29. 29.
    Gorbunov, S., Vaikuntanathan, V., Wichs, D.: Leveled fully homomorphic signatures from standard lattices. In: Symposium on Theory of Computing - STOC 2015, pp. 469–477 (2015)Google Scholar
  30. 30.
    Gür, K.D., Polyakov, Y., Rohloff, K., Ryan, G.W., Sajjadpour, H., Savas, E.: Practical applications of improved Gaussian sampling for trapdoor lattices. IACR Cryptology ePrint Archive, 2017:1254 (2017)Google Scholar
  31. 31.
    Gür, K.D., Polyakov, Y., Rohloff, K., Ryan, G.W., Savas, E.: Implementation and evaluation of improved Gaussian sampling for lattice trapdoors. In: Proceedings of the 6th Workshop on Encrypted Computing & Applied Homomorphic Cryptography, WAHC 2018, pp. 61–71 (2018)Google Scholar
  32. 32.
    Halevi, S., Halevi, T., Shoup, V., Stephens-Davidowitz, N.: Implementing BP-obfuscation using graph-induced encoding. In: Computer and Communications Security - CCS 2017, pp. 783–798 (2017)Google Scholar
  33. 33.
    Halevi, S., Polyakov, Y., Shoup, V.: An improved RNS variant of the BFV homomorphic encryption scheme. IACR Cryptology ePrint Archive, 2018:117 (2018)Google Scholar
  34. 34.
    Klein, P.N.: Finding the closest lattice vector when it’s unusually close. In: Symposium on Discrete Algorithms - SODA 2000, pp. 937–941 (2000)Google Scholar
  35. 35.
    Lyubashevsky, V., Micciancio, D., Peikert, C., Rosen, A.: SWIFFT: a modest proposal for FFT hashing. In: Nyberg, K. (ed.) FSE 2008. LNCS, vol. 5086, pp. 54–72. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-71039-4_4CrossRefGoogle Scholar
  36. 36.
    Lyubashevsky, V., Peikert, C., Regev, O.: A toolkit for ring-LWE cryptography. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 35–54. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_3CrossRefGoogle Scholar
  37. 37.
    Aguilar-Melchor, C., Barrier, J., Guelton, S., Guinet, A., Killijian, M.-O., Lepoint, T.: NFLlib: NTT-based fast lattice library. In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 341–356. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29485-8_20CrossRefGoogle Scholar
  38. 38.
    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
  39. 39.
    Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. SIAM J. Comput. 37(1), 267–302 (2007)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Micciancio, D., Sorrell, J.: Ring packing and amortized FHEW bootstrapping. In: Automata, Languages, and Programming - ICALP 2018. LIPIcs, vol. 107, pp. 100:1–100:14 (2018)Google Scholar
  41. 41.
    Peikert, C.: Personal Communication (2018)Google Scholar
  42. 42.
    Polyakov, Y., Rohloff, K., Ryan, G.W.: PALISADE lattice cryptography library. https://git.njit.edu/palisade/PALISADE. Accessed Oct 2018
  43. 43.
    Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. CoRR, abs/1011.3027 (2010)Google Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  1. 1.University of California, San DiegoLa JollaUSA
  2. 2.New Jersey Institute of TechnologyNewarkUSA

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