Abstract
The construction of a verifiable random function (VRF) with large input space and full adaptive security from a static, non-interactive complexity assumption, like decisional Diffie-Hellman, has proven to be a challenging task. To date it is not even clear that such a VRF exists. Most known constructions either allow only a small input space of polynomially-bounded size, or do not achieve full adaptive security under a static, non-interactive complexity assumption.
The only known constructions without these restrictions are based on non-static, so-called “q-type” assumptions, which are parametrized by an integer q. Since q-type assumptions get stronger with larger q, it is desirable to have q as small as possible. In current constructions, q is either a polynomial (e.g., Hohenberger and Waters, Eurocrypt 2010) or at least linear (e.g., Boneh et al., CCS 2010) in the security parameter.
We show that it is possible to construct relatively simple and efficient verifiable random functions with full adaptive security and large input space from non-interactive q-type assumptions, where q is only logarithmic in the security parameter. Interestingly, our VRF is essentially identical to the verifiable unpredictable function (VUF) by Lysyanskaya (Crypto 2002), but very different from Lysyanskaya’s VRF from the same paper. Thus, our result can also be viewed as a new, direct VRF-security proof for Lysyanskaya’s VUF. As a technical tool, we introduce and construct balanced admissible hash functions.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
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)
Abdalla, M., Catalano, D., Fiore, D.: Verifiable random functions: Relations to identity-based key encapsulation and new constructions. Journal of Cryptology 27(3), 544–593 (2014)
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)
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)
Bellare, M., Ristenpart, T.: Simulation without the artificial abort: Simplified proof and improved concrete security for Waters’ IBE scheme. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 407–424. Springer, Heidelberg (2009)
Bellare, M., Ristenpart, T.: Simulation without the artificial abort: Simplified proof and improved concrete security for Waters’ IBE scheme. Cryptology ePrint Archive, Report 2009/084 (2009), http://eprint.iacr.org/
Bellare, M., Rogaway, P.: The security of triple encryption and a framework for code-based game-playing proofs. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 409–426. Springer, Heidelberg (2006)
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)
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, Chicago, Illinois, USA, October 4–8, pp. 131–140. ACM Press (2010)
Boneh, D., Shen, E., Waters, B.: Strongly unforgeable signatures based on computational Diffie-Hellman. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 229–240. Springer, Heidelberg (2006)
Brakerski, Z., Goldwasser, S., Rothblum, G.N., Vaikuntanathan, V.: Weak verifiable random functions. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 558–576. Springer, Heidelberg (2009)
Chase, M., Lysyanskaya, A.: Simulatable VRFs with applications to multi-theorem NIZK. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 303–322. Springer, Heidelberg (2007)
Chase, M., Meiklejohn, S.: Déjà Q: Using dual systems to revisit q-type assumptions. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 622–639. Springer, Heidelberg (2014)
Chatterjee, S., Sarkar, P.: HIBE with short public parameters without random oracle. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 145–160. Springer, Heidelberg (2006)
Cheon, J.H.: Security analysis of the strong Diffie-Hellman problem. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 1–11. Springer, Heidelberg (2006)
Dodis, Y.: Efficient construction of (distributed) verifiable random functions. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 1–17. Springer, Heidelberg (2002)
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)
Fiore, D., Schröder, D.: Uniqueness Is a Different Story: Impossibility of Verifiable Random Functions from Trapdoor Permutations. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 636–653. Springer, Heidelberg (2012)
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)
Freire, E.S.V., Hofheinz, D., Paterson, K.G., Striecks, C.: Programmable hash functions in the multilinear setting. Cryptology ePrint Archive, Report 2013/354 (2013), http://eprint.iacr.org/
Fuchsbauer, G.: Constrained Verifiable Random Functions. In: Abdalla, M., De Prisco, R. (eds.) SCN 2014. LNCS, vol. 8642, pp. 95–114. Springer, Heidelberg (2014)
Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: 21st ACM STOC, Seattle, Washington, USA, May 15–17, pp. 25–32. ACM Press (1989)
Goldwasser, S., Ostrovsky, R.: Invariant signatures and non-interactive zero-knowledge proofs are equivalent (extended abstract). In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 228–245. Springer, Heidelberg (1993)
Hohenberger, S., Waters, B.: Realizing Hash-and-Sign Signatures under Standard Assumptions. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 333–350. Springer, Heidelberg (2009)
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)
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)
Liskov, M.: Updatable zero-knowledge databases. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 174–198. Springer, Heidelberg (2005)
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)
Micali, S., Rabin, M.O., Vadhan, S.P.: Verifiable random functions. In: 40th FOCS, October 17–19, pp. 120–130. IEEE Computer Society Press, New York (1999)
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)
Micali, S., Rivest, R.L.: Micropayments revisited. In: Preneel, B. (ed.) CT-RSA 2002. LNCS, vol. 2271, pp. 149–163. Springer, Heidelberg (2002)
Naor, M.: On cryptographic assumptions and challenges (invited talk). In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003)
Naor, M., Reingold, O.: Number-theoretic constructions of efficient pseudo-random functions. In: 38th FOCS, Miami Beach, Florida, October 19–22, pp. 458–467. IEEE Computer Society Press (1997)
Naor, M., Reingold, O.: From unpredictability to indistinguishability: A simple construction of pseudo-random functions from MACs (extended abstract). In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 267–282. Springer, Heidelberg (1998)
Steinfeld, R., Pieprzyk, J., Wang, H.: How to Strengthen Any Weakly Unforgeable Signature into a Strongly Unforgeable Signature. In: Abe, M. (ed.) CT-RSA 2007. LNCS, vol. 4377, pp. 357–371. Springer, Heidelberg (2006)
Waters, B.: Dual system encryption: Realizing fully secure IBE and HIBE under simple assumptions. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 619–636. Springer, Heidelberg (2009)
Waters, B.: Efficient identity-based encryption without random oracles. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 International Association for Cryptologic Research
About this paper
Cite this paper
Jager, T. (2015). Verifiable Random Functions from Weaker Assumptions. In: Dodis, Y., Nielsen, J.B. (eds) Theory of Cryptography. TCC 2015. Lecture Notes in Computer Science, vol 9015. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46497-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-46497-7_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-46496-0
Online ISBN: 978-3-662-46497-7
eBook Packages: Computer ScienceComputer Science (R0)