Abstract
Verifiable decryption allows one to prove the correct decryption of encrypted data. When the encrypted data is derived from homomorphic evaluations in the context of fully homomorphic encryption (FHE), verifiable decryption will be very useful in cloud computing or cryptographic protocols, e.g., secure medical computation, cryptographically verifiable election, etc. In this paper, we consider the problem of proving the correct decryption of an FHE ciphertext. Namely, we are interested in zero-knowledge proofs of knowledge of triples \((m, \mathbf {s}, \mathbf {c})\) such that the message m is the correct decryption of a ciphertext \(\mathbf {c}\) for a secret key \(\mathbf {s}\). While analogous statements admit efficient zero-knowledge proof protocols in the discrete logarithm setting, they have never been addressed in FHE so far. We provide such verifiable decryption for Brakerski-Gentry-Vaikuntanathan (BGV) scheme, since this scheme was recognized as one of the most efficient leveled FHE schemes. Our solution is nearly “one shot”, in the sense that a single instance of the proof already has negligible soundness error, yielding compact proofs even for individual ciphertexts. Furthermore, to illustrate the applicability of verifiable decryption, we also give two example instantiations.
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References
Alperin-Sheriff, J., Peikert, C.: Faster bootstrapping with polynomial error. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 297–314. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_17
Baum, C., Damgård, I., Larsen, K.G., Nielsen, M.: How to prove knowledge of small secrets. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 478–498. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_17
Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: CCS 1993, Proceedings of the 1st ACM Conference on Computer and Communications Security, Fairfax, Virginia, USA, 3–5 November 1993, pp. 62–73 (1993)
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Innovations in Theoretical Computer Science 2012, Cambridge, MA, USA, 8–10 January 2012, pp. 309–325 (2012)
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, 22–25 October 2011, pp. 97–106 (2011)
Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 505–524. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_29
Camenisch, J., Shoup, V.: Practical verifiable encryption and decryption of discrete logarithms. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 126–144. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_8
Carr, C., Costache, A., Davies, G.T., Gjøsteen, K., Strand, M.: Zero-knowledge proof of decryption for the ciphertexts. Technical report, Cryptology ePrint Archive, Report 2018/026 (2018). https://eprint.iacr.org/2018/026
Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 3–33. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_1
Cohen, J.D., Fischer, M.J.: A robust and verifiable cryptographically secure election scheme (extended abstract). In: 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21–23 October 1985, pp. 372–382 (1985)
Cramer, R., Damgård, I.: On the amortized complexity of zero-knowledge protocols. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 177–191. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_11
Cramer, R., Damgård, I., Xing, C., Yuan, C.: Amortized complexity of zero-knowledge proofs revisited: achieving linear soundness slack. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 479–500. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_17
del Pino, R., Lyubashevsky, V.: Amortization with fewer equations for proving knowledge of small secrets. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10403, pp. 365–394. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63697-9_13
Ducas, L., Micciancio, D.: FHEW: bootstrapping homomorphic encryption in less than a second. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 617–640. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_24
Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. IACR Cryptology ePrint Archieve, 2012:144 (2012)
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_12
Gama, N., Nguyen, P.Q.: Predicting lattice reduction. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_3
Gentry, C.: A Fully Homomorphic Encryption Scheme. Stanford University (2009)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31–June 2 2009, pp. 169–178 (2009)
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
Gjøsteen, K., Strand, M.: A roadmap to fully homomorphic elections: stronger security, better verifiability. In: Brenner, M., et al. (eds.) FC 2017. LNCS, vol. 10323, pp. 404–418. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70278-0_25
Halevi, S., Shoup, V.: Bootstrapping for HElib. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 641–670. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_25
Hirt, M., Sako, K.: Efficient receipt-free voting based on homomorphic encryption. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 539–556. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45539-6_38
Khedr, A., Gulak, P.G.: Securemed: secure medical computation using gpu-accelerated homomorphic encryption scheme. IEEE J. Biomed. Health Inf. 22(2), 597–606 (2018)
Libert, B., Ling, S., Mouhartem, F., Nguyen, K., Wang, H.: Signature schemes with efficient protocols and dynamic group signatures from lattice assumptions. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 373–403. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53890-6_13
Libert, B., Ling, S., Nguyen, K., Wang, H.: Zero-knowledge arguments for lattice-based PRFs and applications to E-cash. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10626, pp. 304–335. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70700-6_11
López-Alt, A., Tromer, E., Vaikuntanathan, V.: On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, 19–22 May 2012, pp. 1219–1234 (2012)
Lyubashevsky, V.: Fiat-shamir with aborts: applications to lattice and factoring-based signatures. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 598–616. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10366-7_35
Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 738–755. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_43
Lyubashevsky, V.: Digital signatures based on the hardness of ideal lattice problems in all rings. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 196–214. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53890-6_7
Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006). https://doi.org/10.1007/11787006_13
Lyubashevsky, V., Neven, G.: One-shot verifiable encryption from lattices. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 293–323. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_11
Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_1
Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. J. ACM (JACM) 60(6), 43 (2013)
Micciancio, D., Peikert, C.: Trapdoors for lattices: simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_41
Peikert, C., Vaikuntanathan, V.: Noninteractive statistical zero-knowledge proofs for lattice problems. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 536–553. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85174-5_30
Peng, K., Aditya, R., Boyd, C., Dawson, E., Lee, B.: Multiplicative homomorphic E-voting. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 61–72. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30556-9_6
Rivest, R.L., Adleman, L., Dertouzos, M.L.: On data banks and privacy homomorphisms. Found. Secur. Comput. 4(11), 169–180 (1978)
Smart, N.P., Vercauteren, F.: Fully homomorphic encryption with relatively small key and ciphertext sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_25
Stern, J.: A new paradigm for public key identification. IEEE Trans. Inf. Theory 42(6), 1757–1768 (1996)
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
Acknowledgments
This research is supported in part by the National Basic Research Program of China (973 project, No. 2014CB340603) and the National Nature Science Foundation of China (Nos. 61672030 and 61272040). The authors would like to thank the anonymous reviewers for their detailed reviews and helpful comments.
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Luo, F., Wang, K. (2018). Verifiable Decryption for Fully Homomorphic Encryption. In: Chen, L., Manulis, M., Schneider, S. (eds) Information Security. ISC 2018. Lecture Notes in Computer Science(), vol 11060. Springer, Cham. https://doi.org/10.1007/978-3-319-99136-8_19
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