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Non-interactive Zero Knowledge Proofs in the Random Oracle Model

  • Vincenzo IovinoEmail author
  • Ivan Visconti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11445)

Abstract

The Fiat-Shamir (FS) transform is a well known and widely used technique to convert any constant-round public-coin honest-verifier zero-knowledge (HVZK) proof or argument system \(\mathsf {HVZK}=(\mathcal {P},\mathcal {V})\) in a non-interactive zero-knowledge (NIZK) argument system

\(\mathsf {NIZK}=(\mathsf {NIZK}.\mathsf {Prove}, \mathsf {NIZK}.\mathsf{Verify})\). The FS transform is secure in the random oracle (RO) model and is extremely efficient: it adds an evaluation of the RO for every message played by \(\mathcal {V}\).

While a major effort has been done to attack the soundness of the transform when the RO is instantiated with a “secure” hash function, here we focus on a different limitation of the FS transform that exists even when there is a secure instantiation of the random oracle: the soundness of \(\mathsf {NIZK}\) holds against polynomial-time adversarial provers only. Therefore even when \(\mathsf {HVZK}\) is a proof system, \(\mathsf {NIZK}\) is only an argument system.

In this paper we propose a new transform from 3-round public-coin HVZK proof systems for several practical relations to NIZK proof systems in the RO model. Our transform outperforms the FS transform protecting the honest verifier from unbounded adversarial provers with no restriction on the number of RO queries. The protocols our transform can be applied to are the ones for proving membership to the range of a one-way group homomorphism as defined by [Maurer - Design, Codes and Cryptography 2015] except that we additionally require the function to be endowed with a trapdoor and other natural properties. For instance, we obtain new efficient instantiations of NIZK proofs for relations related to quadratic residuosity and the RSA function.

As a byproduct, with our transform we obtain essentially for free the first efficient non-interactive zap (i.e., 1-round non-interactive witness indistinguishable proof system) for several practical languages in the non-programmable RO model and in an ideal-PUF model.

Our approach to NIZK proofs can be seen as an abstraction of the celebrated work of [Feige, Lapidot and Shamir - FOCS 1990].

Keywords

FS transform NIZK Random oracle model 

Supplementary material

482447_1_En_9_MOESM1_ESM.pdf (237 kb)
Supplementary material 1 (pdf 237 KB)

References

  1. [AABN02]
    Abdalla, M., An, J.H., Bellare, M., Namprempre, C.: From identification to signatures via the Fiat-Shamir transform: minimizing assumptions for security and forward-security. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 418–433. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-46035-7_28CrossRefGoogle Scholar
  2. [AABN08]
    Abdalla, M., An, J.H., Bellare, M., Namprempre, C.: From identification to signatures via the Fiat-Shamir transform: necessary and sufficient conditions for security and forward-security. IEEE Trans. Inf. Theory 54(8), 3631–3646 (2008)MathSciNetCrossRefGoogle Scholar
  3. [ABB+10]
    Almeida, J.B., Bangerter, E., Barbosa, M., Krenn, S., Sadeghi, A.-R., Schneider, T.: A certifying compiler for zero-knowledge proofs of knowledge based on \(\Sigma \)-protocols. In: Gritzalis, D., Preneel, B., Theoharidou, M. (eds.) ESORICS 2010. LNCS, vol. 6345, pp. 151–167. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15497-3_10CrossRefGoogle Scholar
  4. [AF07]
    Abe, M., Fehr, S.: Perfect NIZK with adaptive soundness. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 118–136. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-70936-7_7CrossRefGoogle Scholar
  5. [AMS+09]
    Armknecht, F., Maes, R., Sadeghi, A.-R., Sunar, B., Tuyls, P.: Memory leakage-resilient encryption based on physically unclonable functions. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 685–702. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-10366-7_40CrossRefGoogle Scholar
  6. [Bar01]
    Barak, B.: How to go beyond the black-box simulation barrier. In: 42nd Annual Symposium on Foundations of Computer Science, pp. 106–115. IEEE Computer Society Press, October 2001Google Scholar
  7. [BCNP04]
    Barak, B., Canetti, R., Nielsen, J.B., Pass, R.: Universally composable protocols with relaxed set-up assumptions. In: 45th Annual Symposium on Foundations of Computer Science, pp. 186–195. IEEE Computer Society Press, October 2004Google Scholar
  8. [BDSG+13]
    Bitansky, N., et al.: Why “Fiat-Shamir for proofs” lacks a proof. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 182–201. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36594-2_11CrossRefGoogle Scholar
  9. [BFM88]
    Blum, M., Feldman, P., Micali, S.: Non-interactive zero-knowledge and its applications (extended abstract). In: 20th Annual ACM Symposium on Theory of Computing, pp. 103–112. ACM Press, May 1988Google Scholar
  10. [BFS16]
    Bellare, M., Fuchsbauer, G., Scafuro, A.: NIZKs with an untrusted CRS: security in the face of parameter subversion. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 777–804. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53890-6_26CrossRefzbMATHGoogle Scholar
  11. [BFSK11]
    Brzuska, C., Fischlin, M., Schröder, H., Katzenbeisser, S.: Physically uncloneable functions in the universal composition framework. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 51–70. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_4CrossRefzbMATHGoogle Scholar
  12. [BFW15]
    Bernhard, D., Fischlin, M., Warinschi, B.: Adaptive proofs of knowledge in the random oracle model. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 629–649. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46447-2_28CrossRefGoogle Scholar
  13. [BG93]
    Bellare, M., Goldreich, O.: On defining proofs of knowledge. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 390–420. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-48071-4_28CrossRefGoogle Scholar
  14. [BLV03]
    Barak, B., Lindell, Y., Vadhan, S.P.: Lower bounds for non-black-box zero knowledge. In: 44th Annual Symposium on Foundations of Computer Science, pp. 384–393. IEEE Computer Society Press, October 2003Google Scholar
  15. [BM88]
    Babai, L., Moran, S.: Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes. J. Comput. Syst. Sci. 36(2), 254–276 (1988)MathSciNetCrossRefGoogle Scholar
  16. [BN06]
    Bellare, M., Neven, G.: Multi-signatures in the plain public-key model and a general forking lemma. In: Juels, A., Wright, R.N., De Capitani di Vimercati, S. (eds.) 13th ACM Conference on Computer and Communications Security, CCS 2006 pp. 390–399. ACM Press, October/November 2006Google Scholar
  17. [BPW12]
    Bernhard, D., Pereira, O., Warinschi, B.: How not to prove yourself: pitfalls of the Fiat-Shamir heuristic and applications to helios. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 626–643. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-34961-4_38CrossRefGoogle Scholar
  18. [BR93]
    Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Ashby, V. (ed.) 1st ACM Conference on Computer and Communications Security, CCS 1993, pp. 62–73. ACM Press, November 1993Google Scholar
  19. [BR08]
    Bellare, M., Ristov, T.: Hash functions from sigma protocols and improvements to VSH. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 125–142. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-89255-7_9CrossRefGoogle Scholar
  20. [BY96]
    Bellare, M., Yung, M.: Certifying permutations: noninteractive zero-knowledge based on any trapdoor permutation. J. Cryptol. 9(3), 149–166 (1996)MathSciNetCrossRefGoogle Scholar
  21. [CDS94]
    Cramer, R., Damgård, I., Schoenmakers, B.: Proofs of partial knowledge and simplified design of witness hiding protocols. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 174–187. Springer, Heidelberg (1994).  https://doi.org/10.1007/3-540-48658-5_19CrossRefGoogle Scholar
  22. [CG15]
    Chaidos, P., Groth, J.: Making sigma-protocols non-interactive without random oracles. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 650–670. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46447-2_29CrossRefGoogle Scholar
  23. [CGH98]
    Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited (preliminary version). In: 30th Annual ACM Symposium on Theory of Computing, pp. 209–218. ACM Press, May 1998Google Scholar
  24. [CP93]
    Chaum, D., Pedersen, T.P.: Wallet databases with observers. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 89–105. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-48071-4_7CrossRefGoogle Scholar
  25. [CPS+16]
    Ciampi, M., Persiano, G., Scafuro, A., Siniscalchi, L., Visconti, I.: Online/offline OR composition of sigma protocols. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 63–92. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_3CrossRefGoogle Scholar
  26. [CPSV16]
    Ciampi, M., Persiano, G., Siniscalchi, L., Visconti, I.: A transform for NIZK almost as efficient and general as the Fiat-Shamir transform without programmable random oracles. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9563, pp. 83–111. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49099-0_4CrossRefzbMATHGoogle Scholar
  27. [CS98]
    Cramer, R., Shoup, V.: A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 13–25. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0055717CrossRefGoogle Scholar
  28. [CS03]
    Cramer, R., Shoup, V.: Design and analysis of practical public-key encryption schemes secure against adaptive chosen ciphertext attack. SIAM J. Comput. 33(1), 167–226 (2003)MathSciNetCrossRefGoogle Scholar
  29. [Dam10]
    Damgård, I.: On \(\varSigma \)-protocol (2010). http://www.cs.au.dk/~ivan/Sigma.pdf
  30. [DFN06]
    Damgård, I., Fazio, N., Nicolosi, A.: Non-interactive zero-knowledge from homomorphic encryption. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 41–59. Springer, Heidelberg (2006).  https://doi.org/10.1007/11681878_3CrossRefGoogle Scholar
  31. [DG03]
    Damgård, I., Groth, J.: Non-interactive and reusable non-malleable commitment schemes. In: 35th Annual ACM Symposium on Theory of Computing, pp. 426–437. ACM Press, June 2003Google Scholar
  32. [DN00]
    Dwork, C., Naor, M.: Zaps and their applications. In: 41st Annual Symposium on Foundations of Computer Science, pp. 283–293. IEEE Computer Society Press, November 2000Google Scholar
  33. [DNRS99]
    Dwork, C., Naor, M., Reingold, O., Stockmeyer, L.J.: Magic functions. In: 40th Annual Symposium on Foundations of Computer Science, pp. 523–534. IEEE Computer Society Press, October 1999Google Scholar
  34. [DORS08]
    Dodis, Y., Ostrovsky, R., Reyzin, L., Smith, A.D.: Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. SIAM J. Comput. 38(1), 97–139 (2008)MathSciNetCrossRefGoogle Scholar
  35. [DRV12]
    Dodis, Y., Ristenpart, T., Vadhan, S.: Randomness condensers for efficiently samplable, seed-dependent sources. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 618–635. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28914-9_35CrossRefGoogle Scholar
  36. [Fis05]
    Fischlin, M.: Communication-efficient non-interactive proofs of knowledge with online extractors. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 152–168. Springer, Heidelberg (2005).  https://doi.org/10.1007/11535218_10CrossRefGoogle Scholar
  37. [FKI06]
    Furukawa, J., Kurosawa, K., Imai, H.: An efficient compiler from \(\Sigma \)-protocol to 2-move deniable zero-knowledge. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 46–57. Springer, Heidelberg (2006).  https://doi.org/10.1007/11787006_5CrossRefGoogle Scholar
  38. [FKMV12]
    Faust, S., Kohlweiss, M., Marson, G.A., Venturi, D.: On the non-malleability of the Fiat-Shamir transform. In: Galbraith, S., Nandi, M. (eds.) INDOCRYPT 2012. LNCS, vol. 7668, pp. 60–79. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-34931-7_5CrossRefGoogle Scholar
  39. [FLS90]
    Feige, U., Lapidot, D., Shamir, A.: Multiple non-interactive zero knowledge proofs based on a single random string (extended abstract). In: 31st Annual Symposium on Foundations of Computer Science, pp. 308–317. IEEE Computer Society Press, October 1990Google Scholar
  40. [FS87]
    Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987).  https://doi.org/10.1007/3-540-47721-7_12CrossRefGoogle Scholar
  41. [GCvD02]
    Gassend, B., Clarke, D.E., van Dijk, M., Devadas, S.: Silicon physical random functions. In: Atluri, V. (ed.) 9th ACM Conference on Computer and Communications Security, CCS 2002, pp. 148–160. ACM Press, November 2002Google Scholar
  42. [GGH+13]
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th Annual Symposium on Foundations of Computer Science, pp. 40–49. IEEE Computer Society Press, October 2013Google Scholar
  43. [GIS+10]
    Goyal, V., Ishai, Y., Sahai, A., Venkatesan, R., Wadia, A.: Founding cryptography on tamper-proof hardware tokens. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 308–326. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-11799-2_19CrossRefzbMATHGoogle Scholar
  44. [GK03]
    Goldwasser, S., Kalai, Y.T.: On the (in)security of the Fiat-Shamir paradigm. In: 44th Annual Symposium on Foundations of Computer Science, pp. 102–115. IEEE Computer Society Press, October 2003Google Scholar
  45. [GKR08]
    Goldwasser, S., Kalai, Y.T., Rothblum, G.N.: One-time programs. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 39–56. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85174-5_3CrossRefGoogle Scholar
  46. [GM84]
    Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984)MathSciNetCrossRefGoogle Scholar
  47. [GMO16]
    Giacomelli, I., Madsen, J., Orlandi, C.: Zkboo: faster zero-knowledge for boolean circuits. In: 25th USENIX Security Symposium, USENIX Security 16, Austin, TX, USA, 10–12 August 2016, pp. 1069–1083 (2016)Google Scholar
  48. [GMR89]
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989)MathSciNetCrossRefGoogle Scholar
  49. [GMY06]
    Garay, J.A., MacKenzie, P.D., Yang, K.: Strengthening zero-knowledge protocols using signatures. J. Cryptol. 19(2), 169–209 (2006)MathSciNetCrossRefGoogle Scholar
  50. [Gol01]
    Goldreich, O.: Foundations of Cryptography: Basic Techniques, vol. 1. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  51. [GOS06a]
    Groth, J., Ostrovsky, R., Sahai, A.: Non-interactive zaps and new techniques for NIZK. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 97–111. Springer, Heidelberg (2006).  https://doi.org/10.1007/11818175_6CrossRefGoogle Scholar
  52. [GOS06b]
    Groth, J., Ostrovsky, R., Sahai, A.: Perfect non-interactive zero knowledge for NP. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 339–358. Springer, Heidelberg (2006).  https://doi.org/10.1007/11761679_21CrossRefGoogle Scholar
  53. [GOSV14]
    Goyal, V., Ostrovsky, R., Scafuro, A., Visconti, I.: Black-box non-black-box zero knowledge. In: Shmoys, D.B. (ed.) 46th Annual ACM Symposium on Theory of Computing, pp. 515–524. ACM Press, May/June 2014Google Scholar
  54. [GS08]
    Groth, J., Sahai, A.: Efficient Non-interactive proof systems for bilinear groups. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 415–432. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78967-3_24CrossRefGoogle Scholar
  55. [HL08]
    Hazay, C., Lindell, Y.: Constructions of truly practical secure protocols using standardsmartcards. In: Ning, P., Syverson, P.F., Jha, S. (eds.) 15th ACM Conference on Computer and Communications Security, CCS 2008, pp. 491–500. ACM Press, October 2008Google Scholar
  56. [Kat07]
    Katz, J.: Universally composable multi-party computation using tamper-proof hardware. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 115–128. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-72540-4_7CrossRefGoogle Scholar
  57. [KRR16]
    Kalai, Y.T., Rothblum, G.N., Rothblum, R.D.: From obfuscation to the security of Fiat-Shamir for proofs. IACR Cryptology ePrint Archive 2016:303 (2016)Google Scholar
  58. [Lin06]
    Lindell, Y.: A simpler construction of CCA2-secure public-key encryption under general assumptions. J. Cryptol. 19(3), 359–377 (2006)MathSciNetCrossRefGoogle Scholar
  59. [Lin15]
    Lindell, Y.: An efficient transform from sigma protocols to NIZK with a CRS and Non-programmable random oracle. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9014, pp. 93–109. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46494-6_5CrossRefGoogle Scholar
  60. [Mau15]
    Maurer, U.: Zero-knowledge proofs of knowledge for group homomorphisms. Des. Codes Cryptogr. 77(2–3), 663–676 (2015)MathSciNetCrossRefGoogle Scholar
  61. [MP03]
    Micciancio, D., Petrank, E.: Simulatable commitments and efficient concurrent zero-knowledge. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 140–159. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-39200-9_9CrossRefGoogle Scholar
  62. [MV16]
    Mittelbach, A., Venturi, D.: Fiat–Shamir for highly sound protocols is instantiable. In: Zikas, V., De Prisco, R. (eds.) SCN 2016. LNCS, vol. 9841, pp. 198–215. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44618-9_11CrossRefzbMATHGoogle Scholar
  63. [NY90]
    Naor, M., Yung, M.: Public-key cryptosystems provably secure against chosen ciphertext attacks. In: 22nd Annual ACM Symposium on Theory of Computing, pp. 427–437. ACM Press, May 1990Google Scholar
  64. [OPV10]
    Ostrovsky, R., Pandey, O., Visconti, I.: Efficiency preserving transformations for concurrent non-malleable zero knowledge. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 535–552. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-11799-2_32CrossRefGoogle Scholar
  65. [OSVW13]
    Ostrovsky, R., Scafuro, A., Visconti, I., Wadia, A.: Universally composable secure computation with (Malicious) physically uncloneable functions. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 702–718. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_41CrossRefGoogle Scholar
  66. [Pas03]
    Pass, R.: On deniability in the common reference string and random oracle model. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 316–337. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-45146-4_19CrossRefGoogle Scholar
  67. [Pas13]
    Pass, R.: Unprovable security of perfect NIZK and non-interactive non-malleable commitments. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 334–354. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36594-2_19CrossRefzbMATHGoogle Scholar
  68. [PRTG02]
    Pappu, R., Recht, B., Taylor, J., Gershenfeld, N.: Physical one-way functions. Science 297(5589), 2026–2030 (2002)CrossRefGoogle Scholar
  69. [PS00]
    Pointcheval, D., Stern, J.: Security arguments for digital signatures and blind signatures. J. Cryptol. 13(3), 361–396 (2000)CrossRefGoogle Scholar
  70. [Ps05]
    Pass, R., Shelat, A.: Unconditional characterizations of non-interactive zero-knowledge. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 118–134. Springer, Heidelberg (2005).  https://doi.org/10.1007/11535218_8CrossRefGoogle Scholar
  71. [PsV06]
    Pass, R., Shelat, A., Vaikuntanathan, V.: Construction of a non-malleable encryption scheme from any semantically secure one. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 271–289. Springer, Heidelberg (2006).  https://doi.org/10.1007/11818175_16CrossRefGoogle Scholar
  72. [RS92]
    Rackoff, C., Simon, D.R.: Non-interactive zero-knowledge proof of knowledge and chosen ciphertext attack. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 433–444. Springer, Heidelberg (1992).  https://doi.org/10.1007/3-540-46766-1_35CrossRefGoogle Scholar
  73. [RSA78]
    Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signature and public-key cryptosystems. Commun. Assoc. Comput. Mach. 21(2), 120–126 (1978)MathSciNetzbMATHGoogle Scholar
  74. [RvD13]
    Rührmair, U., van Dijk, M.: PUFs in security protocols: Attack models and security evaluations. In: 2013 IEEE Symposium on Security and Privacy, pp. 286–300. IEEE Computer Society Press, May 2013Google Scholar
  75. [Sah99]
    Sahai, A.: Non-malleable non-interactive zero knowledge and adaptive chosen-ciphertext security. In: 40th Annual Symposium on Foundations of Computer Science, pp. 543–553. IEEE Computer Society Press, October 1999Google Scholar
  76. [SG02]
    Shoup, V., Gennaro, R.: Securing threshold cryptosystems against chosen ciphertext attack. J. Cryptol. 15(2), 75–96 (2002)MathSciNetCrossRefGoogle Scholar
  77. [TSS+05]
    Tuyls, P., Škorić, B., Stallinga, S., Akkermans, A.H.M., Ophey, W.: Information-theoretic security analysis of physical uncloneable functions. In: Patrick, A.S., Yung, M. (eds.) FC 2005. LNCS, vol. 3570, pp. 141–155. Springer, Heidelberg (2005).  https://doi.org/10.1007/11507840_15CrossRefzbMATHGoogle Scholar
  78. [VV09]
    Ventre, C., Visconti, I.: Co-sound zero-knowledge with public keys. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 287–304. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02384-2_18CrossRefGoogle Scholar
  79. [YZ06]
    Yung, M., Zhao, Y.: Interactive zero-knowledge with restricted random oracles. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 21–40. Springer, Heidelberg (2006).  https://doi.org/10.1007/11681878_2CrossRefGoogle Scholar
  80. [YZ07]
    Yung, M., Zhao, Y.: Generic and practical resettable zero-knowledge in the bare public-key model. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 129–147. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-72540-4_8CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of LuxembourgLuxembourg CityLuxembourg
  2. 2.DIEMUniversity of SalernoFiscianoItaly

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