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Hunting and Gathering – Verifiable Random Functions from Standard Assumptions with Short Proofs

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Public-Key Cryptography – PKC 2019 (PKC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11443))

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Abstract

A verifiable random function (VRF) is a pseudorandom function, where outputs can be publicly verified. That is, given an output value together with a proof, one can check that the function was indeed correctly evaluated on the corresponding input. At the same time, the output of the function is computationally indistinguishable from random for all non-queried inputs.

We present the first construction of a VRF which meets the following properties at once: It supports an exponential-sized input space, it achieves full adaptive security based on a non-interactive constant-size assumption and its proofs consist of only a logarithmic number of group elements for inputs of arbitrary polynomial length.

Our construction can be instantiated in symmetric bilinear groups with security based on the decision linear assumption. We build on the work of Hofheinz and Jager (TCC 2016), who were the first to construct a verifiable random function with security based on a non-interactive constant-size assumption. Basically, their VRF is a matrix product in the exponent, where each matrix is chosen according to one bit of the input. In order to allow verification given a symmetric bilinear map, a proof consists of all intermediary results. This entails a proof size of \(\varOmega (L)\) group elements, where L is the bit-length of the input.

Our key technique, which we call hunting and gathering, allows us to break this barrier by rearranging the function, which – combined with the partitioning techniques of Bitansky (TCC 2017) – results in a proof size of \(\ell \) group elements for arbitrary \(\ell \in \omega (1)\).

Supported by ERC Project PREP-CRYPTO (724307), by DFG grant HO 4534/2-2 and by a DAAD scholarship. This work was done in part while visiting the FACT Center at IDC Herzliya, Israel.

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Notes

  1. 1.

    For matrix \(\mathbf {M}\in \mathbb {Z}_p^3\) and subspaces \(\mathcal {U},\mathcal {V}\subseteq \mathbb {Z}_p^3\), by \(\mathbf {M}^\top \cdot \mathcal {U}=\mathcal {V}\) we denote the property that for all \(\mathbf {u}\in \mathcal {U}\) we have \(\mathbf {M}^\top \mathbf {u}\in \mathcal {V}\) and for each \(\mathbf {v}\in \mathcal {V}\) there exists a \(\mathbf {u}\) with \(\mathbf {M}^\top \mathbf {u}=\mathbf {v}\).

  2. 2.

    More precisely, with probability at least \(1-(d-1)/p-1/p=1-d/p\).

References

  1. Abdalla, M., Catalano, D., Fiore, D.: Verifiable random functions from identity-based key encapsulation. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 554–571. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01001-9_32

    Chapter  Google Scholar 

  2. Abdalla, M., Catalano, D., Fiore, D.: Verifiable random functions: relations to identity-based key encapsulation and new constructions. J. Cryptol. 27(3), 544–593 (2014). https://doi.org/10.1007/s00145-013-9153-x

    Article  MathSciNet  MATH  Google Scholar 

  3. Abdalla, M., Fiore, D., Lyubashevsky, V.: From selective to full security: semi-generic transformations in the standard model. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 316–333. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30057-8_19

    Chapter  Google Scholar 

  4. Au, M.H., Susilo, W., Mu, Y.: Practical compact E-Cash. In: Pieprzyk, J., Ghodosi, H., Dawson, E. (eds.) ACISP 2007. LNCS, vol. 4586, pp. 431–445. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73458-1_31

    Chapter  Google Scholar 

  5. Badrinarayanan, S., Goyal, V., Jain, A., Sahai, A.: A note on VRFs from verifiable functional encryption. Cryptology ePrint Archive, Report 2017/051 (2017). http://eprint.iacr.org/2017/051

  6. Belenkiy, M., Chase, M., Kohlweiss, M., Lysyanskaya, A.: Compact E-Cash and simulatable VRFs revisited. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 114–131. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03298-1_9

    Chapter  Google Scholar 

  7. Bitansky, N.: Verifiable random functions from non-interactive witness-indistinguishable proofs. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part II. LNCS, vol. 10678, pp. 567–594. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70503-3_19

    Chapter  Google Scholar 

  8. Boneh, D., Boyen, X.: Secure identity based encryption without random oracles. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 443–459. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-28628-8_27

    Chapter  Google Scholar 

  9. Boneh, D., Montgomery, H.W., Raghunathan, A.: Algebraic pseudorandom functions with improved efficiency from the augmented cascade. In: Al-Shaer, E., Keromytis, A.D., Shmatikov, V. (eds.) ACM CCS 2010. ACM Press, October 2010, pp. 131–140 (2010). https://doi.org/10.1145/1866307.1866323

  10. Cash, D., Hofheinz, D., Kiltz, E., Peikert, C.: Bonsai trees, or how to delegate a lattice basis. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 523–552. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_27

    Chapter  Google Scholar 

  11. Dodis, Y.: Efficient construction of (distributed) verifiable random functions. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 1–17. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36288-6_1

    Chapter  Google Scholar 

  12. Dodis, Y., Yampolskiy, A.: A verifiable random function with short proofs and keys. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 416–431. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30580-4_28

    Chapter  Google Scholar 

  13. Escala, A., Herold, G., Kiltz, E., Ràfols, C., Villar, J.: An algebraic framework for Diffie-Hellman assumptions. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 129–147. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_8

    Chapter  Google Scholar 

  14. Freire, E.S.V., Hofheinz, D., Paterson, K.G., Striecks, C.: Programmable hash functions in the multilinear setting. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 513–530. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_28

    Chapter  MATH  Google Scholar 

  15. Goldreich, O.: Computational complexity: a conceptual perspective. ACM Sigact News 39(3), 35–39 (2008)

    Article  Google Scholar 

  16. Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)

    Article  MathSciNet  Google Scholar 

  17. Goldreich, O., Oren, Y.: Definitions and properties of zero-knowledge proof systems. J. Cryptol. 7(1), 1–32 (1994)

    Google Scholar 

  18. Goyal, R., Hohenberger, S., Koppula, V., Waters, B.: A generic approach to constructing and proving verifiable random functions. Cryptology ePrint Archive, Report 2017/021 (2017). http://eprint.iacr.org/2017/021

  19. Groth, J., Ostrovsky, R., Sahai, A.: New techniques for noninteractive zero-knowledge. J. ACM 59(3), 11:1–11:35 (2012). https://doi.org/10.1145/2220357.2220358. ISSN 0004–5411

    Article  MathSciNet  MATH  Google Scholar 

  20. Hofheinz, D., Jager, T.: Verifiable random functions from standard assumptions. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016, Part I. LNCS, vol. 9562, pp. 336–362. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49096-9_14

    Chapter  Google Scholar 

  21. Hohenberger, S., Waters, B.: Constructing verifiable random functions with large input spaces. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 656–672. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_33

    Chapter  Google Scholar 

  22. Hofheinz, D., Jager, T.: Verifiable random functions from standard assumptions. Cryptology ePrint Archive, Report 2015/1048 (2015). http://eprint.iacr.org/2015/1048

  23. Jager, T.: Verifiable random functions from weaker assumptions. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part II. LNCS, vol. 9015, pp. 121–143. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_5

    Chapter  Google Scholar 

  24. Jarecki, S., Shmatikov, V.: Handcuffing big brother: an abuse-resilient transaction escrow scheme. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 590–608. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24676-3_35

    Chapter  Google Scholar 

  25. Katsumata, S.: On the untapped potential of encoding predicates by arithmetic circuits and their applications. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017, Part III. LNCS, vol. 10626, pp. 95–125. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70700-6_4

    Chapter  Google Scholar 

  26. Liskov, M.: Updatable zero-knowledge databases. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 174–198. Springer, Heidelberg (2005). https://doi.org/10.1007/11593447_10

    Chapter  Google Scholar 

  27. Lysyanskaya, A.: Unique signatures and verifiable random functions from the DH-DDH separation. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 597–612. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45708-9_38

    Chapter  Google Scholar 

  28. Micali, S., Rabin, M.O., Vadhan, S.P.: Verifiable random functions. In: 40th FOCS. IEEE Computer Society Press, pp. 120–130, October 1999. https://doi.org/10.1109/SFFCS.1999.814584

  29. Micali, S., Reyzin, L.: Soundness in the public-key model. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 542–565. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44647-8_32

    Chapter  MATH  Google Scholar 

  30. Micali, S., Rivest, R.L.: Micropayments revisited. In: Preneel, B. (ed.) CT-RSA 2002. LNCS, vol. 2271, pp. 149–163. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45760-7_11

    Chapter  Google Scholar 

  31. Naor, M., Reingold, O.: Number-theoretic constructions of efficient pseudo-random functions. J. ACM 51(2), 231–262 (2004)

    Google Scholar 

  32. Reed, I.S., Solomon, G.: Polynomial codes over certain finite fields. J. Soc. Ind. Appl. Math. 8(2), 300–304 (1960)

    Article  MathSciNet  Google Scholar 

  33. Roşie, R.: Adaptive-secure VRFs. Cryptology ePrint Archive, Report 2017/750 (2017). http://eprint.iacr.org/2017/750

  34. Waters, B.: Efficient identity-based encryption without random oracles. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_7

    Chapter  Google Scholar 

  35. Yamada, S.: Asymptotically compact adaptively secure lattice IBEs and verifiable random functions via generalized partitioning techniques. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part III. LNCS, vol. 10403, pp. 161–193. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63697-9_6

    Chapter  Google Scholar 

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Acknowledgments

I would like to thank the anonymous reviewers of TCC 2018 and PKC 2019 for their helpful comments. Further, I would like to thank my advisor Dennis Hofheinz for his support and helpful feedback.

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  1. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Lisa Kohl

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  1. Lisa Kohl
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Correspondence to Lisa Kohl .

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  1. Institute of Information Engineering, SKLOIS, Beijing, China

    Dongdai Lin

  2. NEC Corporation, Kawasaki, Japan

    Kazue Sako

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Kohl, L. (2019). Hunting and Gathering – Verifiable Random Functions from Standard Assumptions with Short Proofs. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11443. Springer, Cham. https://doi.org/10.1007/978-3-030-17259-6_14

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  • DOI: https://doi.org/10.1007/978-3-030-17259-6_14

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