Abstract
Garbled Circuits (GCs) represent fundamental and powerful tools in cryptography, and many variants of GCs have been considered since their introduction. An important property of the garbled circuits is that they can be evaluated securely if and only if exactly 1 key for each input wire is obtained: no less and no more. In this work we study the case when: 1) some of the wire-keys are missing, but we are still interested in computing the output of the garbled circuit and 2) the evaluator of the GC might have both keys for a constant number of wires. We start to study this question in terms of non-interactive multi-party computation (NIMPC) which is strongly connected with GCs. In this notion there is a fixed number of parties (n) that can get correlated information from a trusted setup. Then these parties can send an encoding of their input to an evaluator, which can compute the output of the function. Similarly to the notion of ad hoc secure computation proposed by Beimel et al. [ITCS 2016], we consider the case when less than n parties participate in the online phase, and in addition we let these parties colluding with the evaluator. We refer to this notion as Threshold NIMPC.
In addition, we show that when the number of parties participating in the online phase is a fixed threshold \(l\le n\) then it is possible to securely evaluate any \(l\)-input function. We build our result on top of a new secret-sharing scheme (which can be of independent interest) and on the results proposed by Benhamouda, Krawczyk and Rabin [Crypto 2017]. Our protocol can be used to compute any function in \(NC^1\) in the information-theoretic setting and any function in P assuming one-way functions.
As a second (and main) contribution, we consider a slightly different notion of security in which the number of parties that can participate in the online phase is not specified, and can be any number c above the threshold \(l\) (in this case the evaluator cannot collude with the other parties). We solve an open question left open by Beimel, Ishai and Kushilevitz [Eurocrypt 2017] showing how to build a secure protocol for the case when c is constant, under the Learning with Errors assumption.
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.
The authors of [7] propose inefficient constructions for general functions.
- 2.
The shares of the PoSS scheme need to be opportunely permuted to not give a trivial advantage to the adversary. We refer the reader to the technical part of the paper for more detail.
- 3.
We recall that we do not know any NIMPC protocol that is secure in this setting when the inputs of \(\varPi _2\) are bit strings unless from assuming iO.
- 4.
We discuss the size of \(\lambda \) later in the paper.
- 5.
In our work we assume that the HSS is additive.
- 6.
\(f|_{\overline{T}, {x_{\overline{T}}}}\) works as before, with the difference that it outputs \(\bot \) in the case where less than \(|K|<l\).
- 7.
The oracle outputs \(\bot \) if \(N<l\).
- 8.
In this informal description of the protocol we assume that the algorithm \(\mathsf {Reconstruct^{PoSS}}\) outputs \(\bot \) in the case that some of the input shares are ill formed (e.g., the input shares are the combination of different execution of the algorithm \(\mathsf {Share^{PoSS}}\)).
- 9.
This function is defined as the hash function that on input x outputs \(x \mod \lambda '\).
- 10.
Our construction would work for inputs of size \(\mathsf{poly}(\lambda )\), but to not overburden the notation we consider only inputs of size \(\lambda \) only.
References
Applebaum, B.: Garbled circuits as randomized encodings of functions: a primer. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 24, p. 67 (2017). https://eccc.weizmann.ac.il/report/2017/067
Applebaum, B., Ishai, Y., Kushilevitz, E.: How to garble arithmetic circuits. In: Ostrovsky, R. (ed.) 52nd Annual Symposium on Foundations of Computer Science, Palm Springs, CA, USA, 22–25 October 2011, pp. 120–129. IEEE Computer Society Press (2011). https://doi.org/10.1109/FOCS.2011.40
Barak, B., et al.: On the (im)possibility of obfuscating programs. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 1–18. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44647-8_1
Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols (extended abstract). In: 22nd Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, 14–16 May 1990, pp. 503–513. ACM Press (1990). https://doi.org/10.1145/100216.100287
Beimel, A., Gabizon, A., Ishai, Y., Kushilevitz, E.: Distribution design. In: Sudan, M. (ed.) ITCS 2016: 7th Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, 14–16 January 2016, pp. 81–92. Association for Computing Machinery (2016). https://doi.org/10.1145/2840728.2840759
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., Kushilevitz, E.: Ad hoc PSM protocols: secure computation without coordination. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part III. LNCS, vol. 10212, pp. 580–608. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56617-7_20
Beimel, A., Kushilevitz, E., Nissim, P.: The complexity of multiparty PSM protocols and related models. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part II. LNCS, vol. 10821, pp. 287–318. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78375-8_10
Benhamouda, F., Krawczyk, H., Rabin, T.: Robust non-interactive multiparty computation against constant-size collusion. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 391–419. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_13
Boyle, E., Gilboa, N., Ishai, Y., Lin, H., Tessaro, S.: Foundations of homomorphic secret sharing. In: Karlin, A.R. (ed.) ITCS 2018: 9th Innovations in Theoretical Computer Science Conference, Cambridge, MA, USA, 11–14 January 2018, vol. 94, pp. 21:1–21:21. LIPIcs (2018). https://doi.org/10.4230/LIPIcs.ITCS.2018.21
Chandran, N., Goyal, V., Ostrovsky, R., Sahai, A.: Covert multi-party computation. In: 48th Annual Symposium on Foundations of Computer Science, Providence, RI, USA, 20–23 October 2007, pp. 238–248. IEEE Computer Society Press (2007). https://doi.org/10.1109/FOCS.2007.21
Dodis, Y., Halevi, S., Rothblum, R.D., Wichs, D.: Spooky Encryption and Its Applications. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part III. LNCS, vol. 9816, pp. 93–122. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_4
Feige, U., Kilian, J., Naor, M.: A minimal model for secure computation (extended abstract). In: 26th Annual ACM Symposium on Theory of Computing, Montréal, Québec, Canada, 23–25 May 1994, pp. 554–563. ACM Press (1994). https://doi.org/10.1145/195058.195408
Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: Reusable garbled circuits and succinct functional encryption. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th Annual ACM Symposium on Theory of Computing, Palo Alto, CA, USA, 1–4 June 2013, pp. 555–564. ACM Press (2013). https://doi.org/10.1145/2488608.2488678
Halevi, S., Ishai, Y., Jain, A., Komargodski, I., Sahai, A., Yogev, E.: Non-interactive multiparty computation without correlated randomness. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017, Part III. LNCS, vol. 10626, pp. 181–211. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70700-6_7
Halevi, S., Ishai, Y., Jain, A., Kushilevitz, E., Rabin, T.: Secure multiparty computation with general interaction patterns. In: Sudan, M. (ed.) Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, 14–16 January 2016, pp. 157–168. ACM (2016). https://doi.org/10.1145/2840728.2840760
Halevi, S., Lindell, Y., Pinkas, B.: Secure computation on the web: computing without simultaneous interaction. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 132–150. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_8
Hemenway, B., Jafargholi, Z., Ostrovsky, R., Scafuro, A., Wichs, D.: Adaptively Secure Garbled Circuits from One-Way Functions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part III. LNCS, vol. 9816, pp. 149–178. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_6
Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: 41st Annual Symposium on Foundations of Computer Science, Redondo Beach, CA, USA, 12–14 November 2000, pp. 294–304. IEEE Computer Society Press (2000). https://doi.org/10.1109/SFCS.2000.892118
Jafargholi, Z., Scafuro, A., Wichs, D.: Adaptively indistinguishable garbled circuits. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part II. LNCS, vol. 10678, pp. 40–71. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70503-3_2
Jafargholi, Z., Wichs, D.: Adaptive security of Yao’s garbled circuits. In: Hirt, M., Smith, A. (eds.) TCC 2016, Part I. LNCS, vol. 9985, pp. 433–458. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53641-4_17
Karchmer, M., Wigderson, A.: On span programs. In: Proceedings of Structures in Complexity Theory, pp. 102–111 (1993)
Kolesnikov, V.: Gate evaluation secret sharing and secure one-round two-party computation. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 136–155. Springer, Heidelberg (2005). https://doi.org/10.1007/11593447_8
Kolesnikov, V., Schneider, T.: Improved garbled circuit: free XOR gates and applications. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 486–498. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_40
Lindell, Y., Pinkas, B.: A proof of security of Yao’s protocol for two-party computation. J. Cryptol. 22(2), 161–188 (2008). https://doi.org/10.1007/s00145-008-9036-8
Naor, M., Pinkas, B., Sumner, R.: Privacy preserving auctions and mechanism design. In: Feldman, S.I., Wellman, M.P. (eds.) Proceedings of the First ACM Conference on Electronic Commerce (EC-99), Denver, CO, USA, 3–5 November 1999, pp. 129–139. ACM (1999). https://doi.org/10.1145/336992.337028
Yao, A.C.C.: How to generate and exchange secrets (extended abstract). In: 27th Annual Symposium on Foundations of Computer Science, Toronto, Ontario, Canada, 27–29, October 1986, pp. 162–167. IEEE Computer Society Press (1986). https://doi.org/10.1109/SFCS.1986.25
Acknowledgments
Vipul Goyal is supported in part by the NSF award 1916939, DARPA SIEVE program, a gift from Ripple, a DoE NETL award, a JP Morgan Faculty Fellowship, a PNC center for financial services innovation award, and a Cylab seed funding award. Rafail Ostrovsky is supported in part by DARPA under Cooperative Agreement No: HR0011-20-2-0025, NSF Grant CNS-2001096, US-Israel BSF grant 2015782, Google Faculty Award, JP Morgan Faculty Award, IBM Faculty Research Award, Xerox Faculty Research Award, OKAWA Foundation Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of DARPA, the Department of Defense, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright annotation therein. Michele Ciampi is supported by H2020 project PRIVILEDGE #780477 and the work is done in part while consulting for Stealth Software Technologies, Inc.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 International Association for Cryptologic Research
About this paper
Cite this paper
Ciampi, M., Goyal, V., Ostrovsky, R. (2021). Threshold Garbled Circuits and Ad Hoc Secure Computation. In: Canteaut, A., Standaert, FX. (eds) Advances in Cryptology – EUROCRYPT 2021. EUROCRYPT 2021. Lecture Notes in Computer Science(), vol 12698. Springer, Cham. https://doi.org/10.1007/978-3-030-77883-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-77883-5_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-77882-8
Online ISBN: 978-3-030-77883-5
eBook Packages: Computer ScienceComputer Science (R0)
