## Abstract

A distributed computation in which nodes are connected by a partial communication graph is called *topology hiding* if it does not reveal information about the graph beyond what is revealed by the output of the function. Previous results have shown that topology-hiding computation protocols exist for graphs of constant degree and logarithmic diameter in the number of nodes (Moran–Orlov–Richelson, TCC’15; Hirt et al., Crypto’16) as well as for other graph families, such as cycles, trees, and low circumference graphs (Akavia–Moran, Eurocrypt’17), but the feasibility question for general graphs was open. In this work, we positively resolve the above open problem: we prove that topology-hiding computation is feasible for *all* graphs under either the decisional Diffie–Hellman or quadratic residuosity assumption. Our techniques employ random or deterministic walks to generate paths covering the graph, upon which we apply the Akavia–Moran topology-hiding broadcast for chain graphs (paths). To prevent topology information revealed by the random walk, we design multiple graph-covering sequences that, together, are locally identical to receiving at each round a message from each neighbor and sending back a processed message from some neighbor (in a randomly permuted order).

## Keywords

Secure Multiparty Computation Topology-Hiding computation Random walks Networks Broadcast## Notes

## References

- 1.A. Akavia, R. LaVigne, T. Moran, Topology-hiding computation on all graphs, in
*Advances in Cryptology—CRYPTO 2017—37th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 20–24, 2017, Proceedings, Part I*(2017), pp. 447–467Google Scholar - 2.A. Akavia, T. Moran, Topology-hiding computation beyond logarithmic diameter, in
*Advances in Cryptology—EUROCRYPT 2017—36th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Paris, France, April 30–May 4, 2017, Proceedings, Part III*(2017), pp. 609–637Google Scholar - 3.R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz, C. Rackoff, Random walks, universal traversal sequences, and the complexity of maze problems, in
*Proceedings of the 20th Annual Symposium on Foundations of Computer Science, SFCS ’79*(IEEE Computer Society, Washington, DC, USA, 1979), pp. 218–223Google Scholar - 4.M. Ball, E. Boyle, T. Malkin, T. Moran, Exploring the boundaries of topology-hiding computation, in
*Advances in Cryptology—EUROCRYPT 2018—37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29–May 3, 2018 Proceedings, Part III*(2018), pp. 294–325Google Scholar - 5.J. Balogh, B. Bollobs, M. Krivelevich, T. Mller, M. Walters. Hamilton cycles in random geometric graphs.
*Ann. Appl. Probab.***21**(3), 1053–1072 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 6.A. Beimel, A. Gabizon, Y. Ishai, E. Kushilevitz, S. Meldgaard, A. Paskin-Cherniavsky, Non-interactive secure multiparty computation, in J.A. Garay, R. Gennaro, editors,
*Advances in Cryptology—CRYPTO 2014—34th Annual Cryptology Conference, Santa Barbara, CA, USA, August 17–21, 2014, Proceedings, Part II*, Lecture Notes in Computer Science, vol. 8617 (Springer, 2014), pp. 387–404Google Scholar - 7.R. Canetti, Universally composable security: A new paradigm for cryptographic protocols, in
*FOCS*(IEEE Computer Society, 2001), pp. 136–145Google Scholar - 8.A.K. Chandra, P. Raghavan, W.L. Ruzzo, R. Smolensky, The electrical resistance of a graph captures its commute and cover times, in
*Proceedings of the Twenty-first Annual ACM Symposium on Theory of Computing*, STOC ’89 (ACM, New York, NY, USA, 1989), pp. 574–586Google Scholar - 9.N. Chandran, W. Chongchitmate, J.A. Garay, S. Goldwasser, R. Ostrovsky, V. Zikas, The hidden graph model: Communication locality and optimal resiliency with adaptive faults, in
*Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science*, ITCS ’15 (ACM, New York, NY, USA, 2015), pp. 153–162Google Scholar - 10.M. Clear, A. Hughes, H. Tewari,
*Homomorphic Encryption with Access Policies: Characterization and New Constructions*(Springer, Berlin, Heidelberg, 2013), pp. 61–87Google Scholar - 11.C. Cocks, An identity based encryption scheme based on quadratic residues, in
*Proceedings of the 8th IMA International Conference on Cryptography and Coding*(Springer-Verlag, London, UK, 2001), pp. 360–363Google Scholar - 12.D. Estrin, R. Govindan, J. Heidemann, S. Kumar, Next century challenges: Scalable coordination in sensor networks, in
*Proceedings of the 5th Annual ACM/IEEE International Conference on Mobile Computing and Networking*(ACM, 1999), pp. 263–270Google Scholar - 13.T. Friedrich, T. Sauerwald, A. Stauffer, Diameter and broadcast time of random geometric graphs in arbitrary dimensions.
*Algorithmica***67**(1), 65–88 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 14.O. Goldreich,
*Foundations of Cryptography: Basic Applications*(Cambridge University Press, New York, NY, USA, 2004), vol. 2CrossRefzbMATHGoogle Scholar - 15.O. Goldreich,
*Foundations of Cryptography: Basic Applications*(Cambridge University Press, New York, NY, USA, 2004), vol. 2CrossRefzbMATHGoogle Scholar - 16.O. Goldreich, S. Micali, A. Wigderson, How to play any mental game, in
*Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing*, STOC ’87 (ACM, New York, NY, USA, 1987), pp. 218–229Google Scholar - 17.S. Goldwasser, S.D. Gordon, V. Goyal, A. Jain, J. Katz, F. Liu, A. Sahai, E. Shi, H. Zhou, Multi-input functional encryption, in P.Q. Nguyen, E. Oswald, editors,
*Advances in Cryptology—EUROCRYPT 2014—33rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Copenhagen, Denmark, May 11–15, 2014. Proceedings*, Lecture Notes in Computer Science, vol. 8441 (Springer, 2014), pp. 578–602Google Scholar - 18.S.D. Gordon, T. Malkin, M. Rosulek, H. Wee, Multi-party computation of polynomials and branching programs without simultaneous interaction, in
*Advances in Cryptology—EUROCRYPT 2013, 32nd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Athens, Greece, May 26–30, 2013. Proceedings*(2013), pp. 575–591Google Scholar - 19.S. Halevi, Y. Ishai, A. Jain, E. Kushilevitz, T. Rabin, Secure multiparty computation with general interaction patterns, in
*Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science*, ITCS ’16 (ACM, New York, NY, USA, 2016), pp. 157–168Google Scholar - 20.S. Halevi, Y. Lindell, B. Pinkas, Secure computation on the web: Computing without simultaneous interaction, in Rogaway [34], pp. 132–150Google Scholar
- 21.M. Hinkelmann, A. Jakoby, Communications in unknown networks: Preserving the secret of topology.
*Theor. Comput. Sci.*,**384**(2–3), 184–200 (2007). Structural Information and Communication Complexity (SIROCCO 2005).Google Scholar - 22.M. Hirt, U. Maurer, D. Tschudi, V. Zikas, Network-hiding communication and applications to multi-party protocols, in
*Advances in Cryptology—CRYPTO 2016—36th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 14–18, 2016, Proceedings, Part II*(2016), pp. 335–365Google Scholar - 23.M. Joye, Identity-based cryptosystems and quadratic residuosity, in
*Public-Key Cryptography—PKC 2016—19th IACR International Conference on Practice and Theory in Public-Key Cryptography, Taipei, Taiwan, March 6–9, 2016, Proceedings, Part I*(2016), pp. 225–254Google Scholar - 24.J.D. Kahn, N. Linial, N. Nisan, M.E. Saks, On the cover time of random walks on graphs.
*J. Theor. Probab.*,**2**(1), 121–128 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 25.M. Koucky,
*On Traversal Sequences, Exploration Sequences and Completeness of Kolmogorov Random Strings*. Ph.D. thesis, New Brunswick, NJ, USA, AAI3092958 (2003).Google Scholar - 26.R. LaVigne, Simple homomorphisms of cocks IBE and applications.
*IACR Cryptol. ePrint Arch.*,**2016**, 1150 (2016)Google Scholar - 27.R. LaVigne, C.L. Zhang, U. Maurer, T. Moran, M. Mularczyk, D. Tschudi, Topology-hiding computation beyond semi-honest adversaries, in
*Theory of Cryptography—16th International Conference, TCC 2018, Panaji, India, November 11–14, 2018, Proceedings, Part II*(2018), pp. 3–35Google Scholar - 28.M. Mitzenmacher, E. Upfal,
*Probability and Computing—Randomized Algorithms and Probabilistic Analysis*(Cambridge University Press, Cambridge, 2005)CrossRefzbMATHGoogle Scholar - 29.T. Moran, I. Orlov, S. Richelson, Topology-hiding computation, in Y. Dodis, J. B. Nielsen, editors,
*TCC 2015*, Lecture Notes in Computer Science, vol. 9014 (Springer, 2015), pp. 169–198Google Scholar - 30.M. Penrose,
*Random geometric graphs*, vol. 5 (Oxford University Press, 2003)Google Scholar - 31.G.J. Pottie, W.J. Kaiser, Wireless integrated network sensors.
*Commun. ACM*,**43**(5), 51–58 (2000)CrossRefGoogle Scholar - 32.O. Regev, On lattices, learning with errors, random linear codes, and cryptography, in
*Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 22–24, 2005*(2005), pp. 84–93Google Scholar - 33.O. Reingold, Undirected connectivity in log-space.
*J. ACM*,**55**(4), 17:1–17:24 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 34.P. Rogaway, editor.
*Advances in Cryptology—CRYPTO 2011—31st Annual Cryptology Conference, Santa Barbara, CA, USA, August 14–18, 2011. Proceedings*, Lecture Notes in Computer Science, vol. 6841 (Springer, 2011)Google Scholar - 35.A.C.-C. Yao, How to generate and exchange secrets, in
*Proceedings of the 27th Annual Symposium on Foundations of Computer Science*, SFCS ’86 (IEEE Computer Society, Washington, DC, USA, 1986), pp. 162–167Google Scholar