Abstract
This paper proposes t-secure homomorphic secret sharing schemes for low degree polynomials. Homomorphic secret sharing is a cryptographic technique to outsource the computation to a set of servers while restricting some subsets of servers from learning the secret inputs. Prior to our work, at Asiacrypt 2018, Lai, Malavolta, and Schröder proposed a 1-secure scheme for computing polynomial functions. They also alluded to t-secure schemes without giving explicit constructions; constructing such schemes would require solving set cover problems, which are generally NP-hard. Moreover, the resulting implicit schemes would require a large number of servers. In this paper, we provide a constructive solution for threshold-t structures by combining homomorphic encryption with the classic secret sharing scheme for general access structure by Ito, Saito, and Nishizeki. Our scheme also quantitatively improves the number of required servers from \(O(t^2)\) to O(t), compared to the implicit scheme of Lai et al. We also suggest several ideas for future research directions.
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References
Applebaum, B., Ishai, Y., Kushilevitz, E.: Computationally private randomizing polynomials and their applications. Comput. Complex. 15(2), 115–162 (2006)
Attrapadung, N., Hanaoka, G., Mitsunari, S., Sakai, Y., Shimizu, K., Teruya, T.: Efficient two-level homomorphic encryption in prime-order bilinear groups and a fast implementation in WebAssembly. In: Asia Conference on Computer and Communications Security, pp. 685–697 (2018)
Babai, L., Kimmel, P.G., Lokam, S.V.: Simultaneous messages vs. communication. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 361–372. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-59042-0_88
Barkol, O., Ishai, Y., Weinreb, E.: On d-multiplicative secret sharing. J. Cryptol. 23(4), 580–593 (2010). https://doi.org/10.1007/s00145-010-9056-z
Beaver, D.: Efficient multiparty protocols using circuit randomization. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 420–432. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-46766-1_34
Beimel, A., Gabizon, A., Ishai, Y., Kushilevitz, E., Meldgaard, S., Paskin-Cherniavsky, A.: Non-interactive secure multiparty computation. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 387–404. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_22
Beimel, A., Ishai, Y.: Information-theoretic private information retrieval: a unified construction. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 912–926. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-48224-5_74
Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30576-7_18
Boyle, E.: Recent advances in function and homomorphic secret sharing - (invited talk). In: International Conference on Cryptology in India, pp. 1–26 (2017)
Boyle, E., Gilboa, N., Ishai, Y.: Function secret sharing. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 337–367. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_12
Boyle, E., Gilboa, N., Ishai, Y.: Breaking the circuit size barrier for secure computation under DDH. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. Breaking the circuit size barrier for secure computation under DDH, vol. 9814, pp. 509–539. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_19
Boyle, E., Gilboa, N., Ishai, Y., Lin, H., Tessaro, S.: Foundations of homomorphic secret sharing. In: Innovations in Theoretical Computer Science Conference (2018)
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: IEEE Symposium on Foundations of Computer Science, pp. 97–106 (2011)
Canetti, R.: Universally composable security: a new paradigm for cryptographic protocols. In: IEEE Symposium on Foundations of Computer Science, pp. 136–145 (2001)
Catalano, D., Fiore, D.: Using linearly-homomorphic encryption to evaluate degree-2 functions on encrypted data. In: ACM SIGSAC Conference on Computer and Communications Security, pp. 1518–1529 (2015)
Chabanne, H., de Wargny, A., Milgram, J., Morel, C., Prouff, E.: Privacy-preserving classification on deep neural network. IACR Cryptology ePrint Archive (2017)
Dodis, Y., Halevi, S., Rothblum, R.D., Wichs, D.: Spooky encryption and its applications. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 93–122. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_4
ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inf. Theor. 31, 469–472 (1985)
Fiore, D., Gennaro, R., Pastro, V.: Efficiently verifiable computation on encrypted data. In: ACM SIGSAC Conference on Computer and Communications Security, pp. 844–855 (2014)
Freeman, D.M.: Converting pairing-based cryptosystems from composite-order groups to prime-order groups. In: Gilbert, H. (ed.) EUROCRYPT 2010. Converting pairing-based cryptosystems from composite-order groups to prime-order groups, vol. 6110, pp. 44–61. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_3
Gennaro, R., Rabin, M.O., Rabin, T.: Simplified VSS and fast-track multiparty computations with applications to threshold cryptography. In: ACM Symposium on Principles of Distributed Computing, pp. 101–111 (1998)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: ACM Symposium on Theory of Computing, 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. Homomorphic encryption from learning with errors: Conceptually-simpler, asymptotically-faster, attribute-based, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_5
Gilad-Bachrach, R., Dowlin, N., Laine, K., Lauter, K., Naehrig, M., Wernsing, J.: Cryptonets: applying neural networks to encrypted data with high throughput and accuracy. In: International Conference on Machine Learning, pp. 201–210 (2016)
Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: ACM Symposium on Theory of Computing, pp. 218–229 (1987)
Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984)
Ito, M., Saito, A., Nishizeki, T.: Secret sharing schemes realizing general access structure. In: IEEE Global Telecommunication Conference, pp. 99–102 (1987)
Jain, A., Rasmussen, P.M.R., Sahai, A.: Threshold fully homomorphic encryption. IACR Cryptology ePrint Archive (2017)
Kamara, S., Mohassel, P., Raykova, M.: Outsourcing multi-party computation. IACR Cryptology ePrint Archive (2011)
Kamara, S., Mohassel, P., Riva, B.: Salus: a system for server-aided secure function evaluation. In: ACM Conference on Computer and Communications Security, pp. 797–808 (2012)
Lai, R.W.F., Malavolta, G., Schröder, D.: Homomorphic secret sharing for low degree polynomials. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11274, pp. 279–309. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03332-3_11
López-Alt, A., Tromer, E., Vaikuntanathan, V.: On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: ACM Symposium on Theory of Computing, pp. 1219–1234 (2012)
Martins, P., Sousa, L., Mariano, A.: A survey on fully homomorphic encryption: an engineering perspective. ACM Comput. Surv. 50(6), 1–33 (2017)
Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48910-X_16
Pedersen, T.P.: Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-46766-1_9
Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)
van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. In: Annual International Conference on the Theory and Applications of Cryptographic Techniques, pp. 24–43 (2010)
Acknowledgment
Nuttapong Attrapadung was partly supported by JST CREST Grant Number JPMJCR19F6, and by JSPS KAKENHI Kiban-A Grant Number 19H01109.
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Phalakarn, K., Suppakitpaisarn, V., Attrapadung, N., Matsuura, K. (2020). Constructive t-secure Homomorphic Secret Sharing for Low Degree Polynomials. In: Bhargavan, K., Oswald, E., Prabhakaran, M. (eds) Progress in Cryptology – INDOCRYPT 2020. INDOCRYPT 2020. Lecture Notes in Computer Science(), vol 12578. Springer, Cham. https://doi.org/10.1007/978-3-030-65277-7_34
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