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International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2019: Advances in Cryptology – ASIACRYPT 2019 pp 325–355Cite as

On Kilian’s Randomization of Multilinear Map Encodings

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11922))

Abstract

Indistinguishability obfuscation constructions based on matrix branching programs generally proceed in two steps: first apply Kilian’s randomization of the matrix product computation, and then encode the matrices using a multilinear map scheme. In this paper we observe that by applying Kilian’s randomization after encoding, the complexity of the best attacks is significantly increased for CLT13 multilinear maps. This implies that much smaller parameters can be used, which improves the efficiency of the constructions by several orders of magnitude.

As an application, we describe the first concrete implementation of multiparty non-interactive Diffie-Hellman key exchange secure against existing attacks. Key exchange was originally the most straightforward application of multilinear maps; however it was quickly broken for the three known families of multilinear maps (GGH13, CLT13 and GGH15). Here we describe the first implementation of key exchange that is resistant against known attacks, based on CLT13 multilinear maps. For \(N=4\) users and a medium level of security, our implementation requires 18 GB of public parameters, and a few minutes for the derivation of a shared key.

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References

  1. Ananth, P.V., Gupta, D., Ishai, Y., Sahai, A.: Optimizing obfuscation: avoiding Barrington’s theorem. In: ACM CCS. ACM (2014)

    Google Scholar 

  2. Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC\({^1}\). In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing, Berkeley, California, USA, 28–30 May 1986 (1986)

    Google Scholar 

  3. Becker, A., Ducas, L., Gama, N., Laarhoven, T.: New directions in nearest neighbor searching with applications to lattice sieving. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, 10–12 January 2016 (2016)

    Google Scholar 

  4. Bernstein, D.J.: Fast multiplication and its applications. Algorithmic Number Theory 44, 325–384 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Barak, B., Garg, S., Kalai, Y.T., Paneth, O., Sahai, A.: Protecting obfuscation against algebraic attacks. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 221–238. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_13

    Chapter  Google Scholar 

  6. Boneh, D., Ishai, Y., Sahai, A., Wu, D.J.: Lattice-based SNARGs and their application to more efficient obfuscation. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 247–277. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56617-7_9

    Chapter  Google Scholar 

  7. Badrinarayanan, S., Miles, E., Sahai, A., Zhandry, M.: Post-zeroizing obfuscation: new mathematical tools, and the case of evasive circuits. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 764–791. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_27

    Chapter  Google Scholar 

  8. Boneh, D., Silverberg, A.: Applications of multilinear forms to cryptography. Contemp. Math. 324, 71–90 (2003)

    Article  MathSciNet  Google Scholar 

  9. Coron, J.-S., et al.: Zeroizing without low-level zeroes: new MMAP attacks and their limitations. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 247–266. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_12

    Chapter  Google Scholar 

  10. Chen, Y., Gentry, C., Halevi, S.: Cryptanalyses of candidate branching program obfuscators. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 278–307. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56617-7_10

    Chapter  Google Scholar 

  11. Cheon, J.H., Han, K., Lee, C., Ryu, H., Stehlé, D.: Cryptanalysis of the multilinear map over the integers. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 3–12. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_1

    Chapter  Google Scholar 

  12. Coron, J.-S., Lee, M.S., Lepoint, T., Tibouchi, M.: Cryptanalysis of GGH15 multilinear maps. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 607–628. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53008-5_21

    Chapter  Google Scholar 

  13. Coron, J.-S., Lee, M.S., Lepoint, T., Tibouchi, M.: Zeroizing attacks on indistinguishability obfuscation over CLT13. In: Fehr, S. (ed.) PKC 2017. LNCS, vol. 10174, pp. 41–58. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54365-8_3

    Chapter  Google Scholar 

  14. Coron, J.-S., Lepoint, T., Tibouchi, M.: Practical multilinear maps over the integers. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 476–493. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_26

    Chapter  Google Scholar 

  15. Chen, Y., Nguyen, P.Q.: BKZ 2.0: better lattice security estimates. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 1–20. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_1

    Chapter  Google Scholar 

  16. Chen, Y., Nguyen, P.Q.: Faster algorithms for approximate common divisors: breaking fully-homomorphic-encryption challenges over the integers. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 502–519. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_30

    Chapter  Google Scholar 

  17. Coron, J.-S., Pereira, H.V.L.: On kilian’s randomization of multilinear map encodings. Cryptology ePrint Archive, Report 2018/1129 (2018). https://eprint.iacr.org/2018/1129

  18. Coron, J.-S., Pereira, H.V.L.: Implementation of key-exchange based on CLT13 multilinear maps (2019). https://github.com/coron/cltexchangeimpl

  19. Chen, Y., Vaikuntanathan, V., Wee, H.: GGH15 beyond permutation branching programs: proofs, attacks, and candidates. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 577–607. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_20

    Chapter  Google Scholar 

  20. 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_2

    Chapter  Google Scholar 

  21. Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_1

    Chapter  Google Scholar 

  22. Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: FOCS. IEEE Computer Society (2013)

    Google Scholar 

  23. Gentry, C., Gorbunov, S., Halevi, S.: Graph-induced multilinear maps from lattices. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 498–527. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_20

    Chapter  Google Scholar 

  24. Galbraith, S.D., Gebregiyorgis, S.W., Murphy, S.: Algorithms for the approximate common divisor problem. LMS J. Comput. Math. 19(A), 58–72 (2016)

    Article  MathSciNet  Google Scholar 

  25. Hu, Y., Jia, H.: Cryptanalysis of GGH map. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9665, pp. 537–565. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49890-3_21

    Chapter  Google Scholar 

  26. Hanrot, G., Pujol, X., Stehlé, D.: Analyzing blockwise lattice algorithms using dynamical systems. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 447–464. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_25

    Chapter  Google Scholar 

  27. Kilian, J.: Founding cryptography on oblivious transfer. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988 (1988)

    Google Scholar 

  28. Lewi, K., et al.: 5Gen: a framework for prototyping applications using multilinear maps and matrix branching programs. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, CCS 2016 (2016)

    Google Scholar 

  29. Lee, H.T., Seo, J.H.: Security analysis of multilinear maps over the integers. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 224–240. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_13

    Chapter  Google Scholar 

  30. Miles, E., Sahai, A., Weiss, M.: Protecting obfuscation against arithmetic attacks. Cryptology ePrint Archive, Report 2014/878 (2014). https://eprint.iacr.org/2014/878

  31. Miles, E., Sahai, A., Zhandry, M.: Annihilation attacks for multilinear maps: cryptanalysis of indistinguishability obfuscation over GGH13. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 629–658. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53008-5_22

    Chapter  Google Scholar 

  32. Ma, F., Zhandry, M.: The MMap strikes back: obfuscation and new multilinear maps immune to CLT13 zeroizing attacks. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018. LNCS, vol. 11240, pp. 513–543. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03810-6_19

    Chapter  Google Scholar 

  33. Nguyen, P., Stern, J.: Merkle-Hellman revisited: a cryptanalysis of the Qu-Vanstone cryptosystem based on group factorizations. In: Kaliski, B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 198–212. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0052236

    Chapter  Google Scholar 

  34. Nguyen, P.Q., Stern, J.: The two faces of lattices in cryptology. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 146–180. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44670-2_12

    Chapter  MATH  Google Scholar 

  35. Pass, R., Seth, K., Telang, S.: Indistinguishability obfuscation from semantically-secure multilinear encodings. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 500–517. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_28

    Chapter  Google Scholar 

  36. Stein, W.A., et al.: Sage Mathematics Software (Version 8.0). The Sage Development Team (2017). http://www.sagemath.org

  37. Sahai, A., Waters, B.: How to use indistinguishability obfuscation: deniable encryption, and more. In: Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC 2014, New York, NY, USA. ACM (2014)

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Luxembourg, Luxembourg City, Luxembourg

    Jean-Sébastien Coron & Hilder V. L. Pereira

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  1. Jean-Sébastien Coron
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  2. Hilder V. L. Pereira
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Correspondence to Jean-Sébastien Coron .

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Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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© 2019 International Association for Cryptologic Research

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Coron, JS., Pereira, H.V.L. (2019). On Kilian’s Randomization of Multilinear Map Encodings. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11922. Springer, Cham. https://doi.org/10.1007/978-3-030-34621-8_12

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  • DOI: https://doi.org/10.1007/978-3-030-34621-8_12

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  • Online ISBN: 978-3-030-34621-8

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