Abstract
We construct functional encryption (FE) schemes for the orthogonality (OFE) relation where each ciphertext encrypts some vector \(\varvec{\mathsf {x}}\) and each decryption key, associated to some vector \(\varvec{\mathsf {y}}\), allows to determine if \(\varvec{\mathsf {x}}\) is orthogonal to \(\varvec{\mathsf {y}}\) or not. Motivated by compelling applications, we aim at schemes which are function hidding, i.e. \(\varvec{\mathsf {y}}\) is not leaked.
Our main contribution are two such schemes, both rooted in existing constructions of FE for inner products (IPFE), i.e., where decryption keys reveal the inner product of \(\varvec{\mathsf {x}}\) and \(\varvec{\mathsf {y}}\). The first construction builds upon the very efficient IPFE by Kim et al. (SCN 2018) but just like the original scheme its security holds in the generic group model (GGM). The second scheme builds on recent developments in the construction of efficient IPFE schemes in the standard model and extends the work of Wee (TCC 2017) in leveraging these results for the construction of FE for Boolean functions. Conceptually, both our constructions can be seen as further evidence that shutting down leakage from inner product values to only a single bit for the orthogonality relation can be done with little overhead, not only in the GGM, but also in the standard model.
We discuss potential applications of our constructions to secure databases and provide efficiency benchmarks. Our implementation shows that the first scheme is extremely fast and ready to be deployed in practical applications.
A. Soleimanian—Work done while visiting student at the Università di Catania, Italy.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Here \(\langle \cdot , \cdot \rangle \) denotes the inner-product.
- 2.
In other works this type of OFE has been referred to predicate-hiding attribute-hiding predicate-only predicate encryption, but we prefer the view that we are dealing with a particular case of functional encryption rather than a particular case of attribute-based or predicate encryption.
- 3.
A (small) caveat of Lin’s transform is that it only achieves weak function hiding. This is a relaxation of the FH notion that imposes some additional constraints on the key derivation queries that the adversary is allowed to ask. This restriction is not too severe as generic (yet efficient) transforms to fully fledged (strong) function hiding are known [16].
- 4.
Recall that in the public key setting, adaptive single message indistinguishability implies adaptive many message indistinguishability via a standard hybrid argument.
- 5.
This implies that our scheme requires an asymmetric Type-III pairing group.
- 6.
The compiler has been proposed in the IPFE setting, but trivially extends to the OFE setting.
- 7.
- 8.
- 9.
- 10.
- 11.
References
Abdalla, M., Bourse, F., De Caro, A., Pointcheval, D.: Simple functional encryption schemes for inner products. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 733–751. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46447-2_33
Abdalla, M., Bourse, F., Caro, A.D., Pointcheval, D.: Better security for functional encryption for inner product evaluations. IACR Cryptology ePrint Archive 2016, 11 (2016)
Abdalla, M., Catalano, D., Fiore, D., Gay, R., Ursu, B.: Multi-input functional encryption for inner products: function-hiding realizations and constructions without pairings. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 597–627. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_20
Abdalla, M., Gay, R., Raykova, M., Wee, H.: Multi-input inner-product functional encryption from pairings. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 601–626. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_21
Agrawal, S., Gorbunov, S., Vaikuntanathan, V., Wee, H.: Functional encryption: new perspectives and lower bounds. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 500–518. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_28
Agrawal, S., Libert, B., Stehlé, D.: Fully secure functional encryption for inner products, from standard assumptions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 333–362. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_12
Boneh, D., Lewi, K., Raykova, M., Sahai, A., Zhandry, M., Zimmerman, J.: Semantically secure order-revealing encryption: multi-input functional encryption without obfuscation. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 563–594. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_19
Boneh, D., Raghunathan, A., Segev, G.: Function-private subspace-membership encryption and its applications. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8269, pp. 255–275. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42033-7_14
Boneh, D., Sahai, A., Waters, B.: Functional encryption: definitions and challenges. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 253–273. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19571-6_16
Boneh, D., Waters, B.: Conjunctive, subset, and range queries on encrypted data. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 535–554. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70936-7_29
Escala, A., Herold, G., Kiltz, E., Ràfols, C., Villar, J.L.: An algebraic framework for diffie-hellman assumptions. J. Cryptol. 30(1), 242–288 (2017)
Katz, J., Sahai, A., Waters, B.: Predicate encryption supporting disjunctions, polynomial equations, and inner products. J. Cryptol. 26(2), 191–224 (2013)
Kawai, Y., Takashima, K.: Predicate- and attribute-hiding inner product encryption in a public key setting. In: Cao, Z., Zhang, F. (eds.) Pairing 2013. LNCS, vol. 8365, pp. 113–130. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04873-4_7
Kim, S., Lewi, K., Mandal, A., Montgomery, H.W., Roy, A., Wu, D.J.: Function-hiding inner product encryption is practical. IACR Cryptology ePrint Archive 2016, 440 (2016)
Lin, H.: Indistinguishability obfuscation from SXDH on 5-linear maps and locality-5 PRGs. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 599–629. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_20
Lin, H., Vaikuntanathan, V.: Indistinguishability obfuscation from DDH-like assumptions on constant-degree graded encodings. In: Proceedings of IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, pp. 11–20 (2016)
Okamoto, T., Takashima, K.: Adaptively attribute-hiding (hierarchical) inner product encryption. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 591–608. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_35
Okamoto, T., Takashima, K.: Efficient (hierarchical) inner-product encryption tightly reduced from the decisional linear assumption. IEICE Trans. 96–A(1), 42–52 (2013)
O’Neill, A.: Definitional issues in functional encryption. IACR Cryptology ePrint Archive 2010, 556 (2010)
Shen, E., Shi, E., Waters, B.: Predicate privacy in encryption systems. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 457–473. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00457-5_27
Shi, E., Bethencourt, J., Chan, T.H.H., Song, D., Perrig, A.: Multi-dimensional range query over encrypted data. In: Proceedings of the 2007 IEEE Symposium on Security and Privacy, SP 2007, pp. 350–364. IEEE Computer Society, Washington, DC, USA (2007). https://doi.org/10.1109/SP.2007.29
Wee, H.: Attribute-hiding predicate encryption in bilinear groups, revisited. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 206–233. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_8
Acknowledgements
This work was supported in part by Royal Society grant for international collaboration and by the European Union Horizon 2020 Research and Innovation Programme under grant agreement 780108 (FENTEC). The first author is financed by Project NanoSTIMA (NORTE-01-0145-FEDER-000016) through the North Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement and the ERDF.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Barbosa, M., Catalano, D., Soleimanian, A., Warinschi, B. (2019). Efficient Function-Hiding Functional Encryption: From Inner-Products to Orthogonality. In: Matsui, M. (eds) Topics in Cryptology – CT-RSA 2019. CT-RSA 2019. Lecture Notes in Computer Science(), vol 11405. Springer, Cham. https://doi.org/10.1007/978-3-030-12612-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-12612-4_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-12611-7
Online ISBN: 978-3-030-12612-4
eBook Packages: Computer ScienceComputer Science (R0)