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Arithmetic Garbling from Bilinear Maps

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Computer Security – ESORICS 2019 (ESORICS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11736))

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Abstract

We consider the problem of garbling arithmetic circuits and present a garbling scheme for inner-product predicates over exponentially large fields. Our construction stems from a generic transformation from predicate encryption which makes only blackbox calls to the underlying primitive. The resulting garbling scheme has practical efficiency and can be used as a garbling gadget to securely compute common arithmetic subroutines. We also show that inner-product predicates are complete by generically bootstrapping our construction to arithmetic garbling for polynomial-size circuits, albeit with a loss of concrete efficiency.

In the process of instantiating our construction we propose two new predicate encryption schemes, which might be of independent interest. More specifically, we construct (i) the first pairing-free (weakly) attribute-hiding non-zero inner-product predicate encryption scheme, and (ii) a key-homomorphic encryption scheme for linear functions from bilinear maps. Both schemes feature constant-size keys and practical efficiency.

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Notes

  1. 1.

    A garbling scheme is said to be projective if the encoding information is in the form of a list of labels \((X^0_1, X^1_1, \ldots , X^0_n, X^1_n)\), which are selected according to the binary representation of the input.

References

  1. Abdalla, M., Bourse, F., De Caro, A., Pointcheval, D.: Simple functional encryption schemes for inner products. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 733–751. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46447-2_33

    Chapter  Google Scholar 

  2. Agrawal, S., Bhattacherjee, S., Phan, D.H., Stehlé, D., Yamada, S.: Efficient public trace and revoke from standard assumptions: extended abstract. In: ACM CCS 2017, pp. 2277–2293 (2017)

    Google Scholar 

  3. Agrawal, S., Libert, B., Stehlé, D.: Fully secure functional encryption for inner products, from standard assumptions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 333–362. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_12

    Chapter  Google Scholar 

  4. Applebaum, B., Avron, J., Brzuska, C.: Arithmetic cryptography: extended abstract. In: ITCS 2015, pp. 143–151 (2015)

    Google Scholar 

  5. Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC\(^0\). In: 45th FOCS, pp. 166–175 (2004)

    Google Scholar 

  6. Applebaum, B., Ishai, Y., Kushilevitz, E.: How to garble arithmetic circuits. In: 52nd FOCS, pp. 120–129 (2011)

    Google Scholar 

  7. Applebaum, B., Ishai, Y., Kushilevitz, E., Waters, B.: Encoding functions with constant online rate or how to compress garbled circuits keys. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 166–184. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_10

    Chapter  Google Scholar 

  8. Attrapadung, N., Libert, B.: Functional encryption for inner product: achieving constant-size ciphertexts with adaptive security or support for negation. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 384–402. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_23

    Chapter  Google Scholar 

  9. Ball, M., Malkin, T., Rosulek, M.: Garbling gadgets for Boolean and arithmetic circuits. In: ACM CCS 2016, pp. 565–577 (2016)

    Google Scholar 

  10. Baltico, C.E.Z., Catalano, D., Fiore, D., Gay, R.: Practical functional encryption for quadratic functions with applications to predicate encryption. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 67–98. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_3

    Chapter  Google Scholar 

  11. Bellare, M., Hoang, V.T., Rogaway, P.: Foundations of garbled circuits. In: ACM CCS 2012, pp. 784–796 (2012)

    Google Scholar 

  12. Bishop, A., Jain, A., Kowalczyk, L.: Function-hiding inner product encryption. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 470–491. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_20

    Chapter  Google Scholar 

  13. 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_30

    Chapter  Google Scholar 

  14. Cachin, C., Camenisch, J., Kilian, J., Müller, J.: One-round secure computation and secure autonomous mobile agents. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 512–523. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45022-X_43

    Chapter  MATH  Google Scholar 

  15. Chen, J., Gay, R., Wee, H.: Improved dual system ABE in prime-order groups via predicate encodings. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 595–624. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_20

    Chapter  Google Scholar 

  16. Chen, J., Libert, B., Ramanna, S.C.: Non-zero inner product encryption with short ciphertexts and private keys. In: Zikas, V., De Prisco, R. (eds.) SCN 2016. LNCS, vol. 9841, pp. 23–41. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44618-9_2

    Chapter  MATH  Google Scholar 

  17. ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985). https://doi.org/10.1007/3-540-39568-7_2

    Chapter  Google Scholar 

  18. Escala, A., Herranz, J., Libert, B., Ràfols, C.: Identity-based lossy trapdoor functions: new definitions, hierarchical extensions, and implications. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 239–256. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54631-0_14

    Chapter  Google Scholar 

  19. Fleischhacker, N., Malavolta, G., Schröder, D.: Arithmetic garbling from bilinear maps. Cryptology ePrint Archive, Report 2019/082 (2019). https://eprint.iacr.org/2019/082

  20. Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th FOCS, pp. 40–49 (2013)

    Google Scholar 

  21. Gennaro, R., Gentry, C., Parno, B.: Non-interactive verifiable computing: outsourcing computation to untrusted workers. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 465–482. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_25

    Chapter  Google Scholar 

  22. 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_5

    Chapter  Google Scholar 

  23. Gorbunov, S., Vaikuntanathan, V., Wee, H.: Functional encryption with bounded collusions via multi-party computation. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 162–179. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_11

    Chapter  Google Scholar 

  24. 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_25

    Chapter  Google Scholar 

  25. Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: ACM CCS 2006, pp. 89–98 (2006). Available as Cryptology ePrint Archive Report 2006/309

    Google Scholar 

  26. Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: 41st FOCS, pp. 294–304 (2000)

    Google Scholar 

  27. Jawurek, M., Kerschbaum, F., Orlandi, C.: Zero-knowledge using garbled circuits: how to prove non-algebraic statements efficiently. In: ACM CCS 2013, pp. 955–966 (2013)

    Google Scholar 

  28. Katsumata, S., Yamada, S.: Non-zero inner product encryption schemes from various assumptions: LWE, DDH and DCR. In: Lin, D., Sako, K. (eds.) PKC 2019. LNCS, vol. 11443, pp. 158–188. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17259-6_6

    Chapter  Google Scholar 

  29. Katz, J., Sahai, A., Waters, B.: Predicate encryption supporting disjunctions, polynomial equations, and inner products. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 146–162. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_9

    Chapter  Google Scholar 

  30. Kolesnikov, V., Schneider, T.: Improved garbled circuit: free XOR gates and applications. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 486–498. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_40

    Chapter  MATH  Google Scholar 

  31. Naor, M., Pinkas, B.: Oblivious transfer and polynomial evaluation. In: 31st ACM STOC, pp. 245–254 (1999)

    Google Scholar 

  32. Okamoto, T., Takashima, K.: Achieving short ciphertexts or short secret-keys for adaptively secure general inner-product encryption. In: Lin, D., Tsudik, G., Wang, X. (eds.) CANS 2011. LNCS, vol. 7092, pp. 138–159. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25513-7_11

    Chapter  Google Scholar 

  33. Ostrovsky, R., Sahai, A., Waters, B.: Attribute-based encryption with non-monotonic access structures. In: ACM CCS 2007, pp. 195–203 (2007)

    Google Scholar 

  34. Parno, B., Raykova, M., Vaikuntanathan, V.: How to delegate and verify in public: verifiable computation from attribute-based encryption. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 422–439. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28914-9_24

    Chapter  Google Scholar 

  35. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: 37th ACM STOC, pp. 84–93 (2005)

    Google Scholar 

  36. Sahai, A., Seyalioglu, H.: Worry-free encryption: functional encryption with public keys. In: ACM CCS 2010, pp. 463–472 (2010)

    Google Scholar 

  37. Sahai, A., Waters, B.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_27

    Chapter  Google Scholar 

  38. Valiant, L.G.: Universal circuits (preliminary report). In: ACM STOC, pp. 196–203. ACM (1976)

    Google Scholar 

  39. Wee, H.: Attribute-hiding predicate encryption in bilinear groups, revisited. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 206–233. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_8

    Chapter  Google Scholar 

  40. Yao, A.C.-C.: How to generate and exchange secrets (extended abstract). In: 27th FOCS, pp. 162–167 (1986)

    Google Scholar 

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Acknowledgements

Research supported in part by a gift from Ripple, a gift from DoS Networks, a grant from Northrop Grumman, a Cylab seed funding award, and a JP Morgan Faculty Fellowship. Research based upon work supported by the German research foundation (DFG) through the collaborative research center 1223 and the training school 2475 and by the state of Bavaria at the Nuremberg Campus of Technology (NCT). NCT is a research cooperation between the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) and the Technische Hochschule Nürnberg Georg Simon Ohm (THN).

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Authors and Affiliations

  1. Ruhr University Bochum, Bochum, Germany

    Nils Fleischhacker

  2. Carnegie Mellon University, Pittsburgh, USA

    Giulio Malavolta

  3. Friedrich Alexander Universität Erlangen-Nürnberg, Erlangen, Germany

    Dominique Schröder

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  1. Nils Fleischhacker
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  2. Giulio Malavolta
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  3. Dominique Schröder
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Correspondence to Giulio Malavolta .

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Editors and Affiliations

  1. NEC Corporation, Kawasaki, Japan

    Kazue Sako

  2. University of Surrey, Guildford, UK

    Steve Schneider

  3. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Peter Y. A. Ryan

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Fleischhacker, N., Malavolta, G., Schröder, D. (2019). Arithmetic Garbling from Bilinear Maps. In: Sako, K., Schneider, S., Ryan, P. (eds) Computer Security – ESORICS 2019. ESORICS 2019. Lecture Notes in Computer Science(), vol 11736. Springer, Cham. https://doi.org/10.1007/978-3-030-29962-0_9

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  • DOI: https://doi.org/10.1007/978-3-030-29962-0_9

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