Abstract
We study the natural problem of secure n-party computation (in the computationally unbounded attack model) of circuits over an arbitrary finite non-Abelian group (G,⋅), which we call G-circuits. Besides its intrinsic interest, this problem is also motivating by a completeness result of Barrington, stating that such protocols can be applied for general secure computation of arbitrary functions. For flexibility, we are interested in protocols which only require black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our investigations focus on the passive adversarial model, where up to t of the n participating parties are corrupted.
Our results are as follows. We initiate a novel approach for the construction of black-box protocols for G-circuits based on k-of-k threshold secret-sharing schemes, which are efficiently implementable over any black-box (non-Abelian) group G. We reduce the problem of constructing such protocols to a combinatorial coloring problem in planar graphs. We then give three constructions for such colorings. Our first approach leads to a protocol with optimal resilience t<n/2, but it requires exponential communication complexity \(O({\binom{2 t+1}{t}}^{2} \cdot N_{g})\) group elements and round complexity \(O(\binom{2 t + 1}{t} \cdot N_{g})\), for a G-circuit of size N g . Nonetheless, using this coloring recursively, we obtain another protocol to t-privately compute G-circuits with communication complexity \(\mathcal{P}\mathit{oly}(n)\cdot N_{g}\) for any t∈O(n 1−ϵ) where ϵ is any positive constant. For our third protocol, there is a probability δ (which can be made arbitrarily small) for the coloring to be flawed in term of security, in contrast to the first two techniques, where the colorings are always secure (we call this protocol probabilistic, and those earlier protocols deterministic). This third protocol achieves optimal resilience t<n/2. It has communication complexity O(n 5.056(n+log δ −1)2⋅N g ) and the number of rounds is O(n 2.528⋅(n+log δ −1)⋅N g ).
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References
N. Alon, J.H. Spencer, The Probabilistic Method (Wiley-Interscience, New York, 2000)
J. Bar-Ilan, D. Beaver, Non-cryptographic fault-tolerant computing in a constant number of rounds of interaction, in 8th Annual ACM Symposium on Principles of Distributed Computing (ACM, New York, 1989), pp. 201–209
D.A. Barrington, Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1, in 18th Annual ACM Symposium on Theory of Computing (ACM, New York, 1986), pp. 1–5
M. Ben-Or, S. Goldwasser, A. Wigderson, Completeness theorems for non-cryptographic fault-tolerant distributed computation, in 20th Annual ACM Symposium on Theory of Computing (ACM, New York, 1988), pp. 1–10
J. Benaloh, Secret sharing homomorphisms: keeping shares of a secret, in Advances in Cryptology: Crypto’86. Lecture Notes in Computer Science, vol. 263 (Springer, Berlin, 1987), pp. 251–260
B. Bollobàs, O. Riordan, Percolation (Cambridge University Press, Cambridge, 2006)
D. Chaum, C. Crépeau, I. Damgård, Multi-party unconditionally secure protocols, in 20th Annual ACM Symposium on Theory of Computing (ACM, New York, 1988), pp. 11–19
R. Cramer, S. Fehr, Y. Ishai, E. Kushilevitz, Efficient multi-party computation over rings, in Advances in Cryptology: Eurocrypt’03. Lecture Notes in Computer Science, vol. 2656 (Springer, Berlin, 2003), pp. 596–613
I. Damgård, J.B. Nielsen, Scalable and unconditionally secure multiparty computation, in Advances in Cryptology—Crypto’07. Lecture Notes in Computer Science, vol. 4622 (Springer, Berlin, 2007), pp. 572–590
I. Damgård, M. Fitzi, E. Kiltz, J.B. Nielsen, T. Toft, Unconditionally secure constant-rounds multi-party computation for equality, comparison, bits and commitments, in 3rd Theory of Cryptography Conference. Lecture Notes in Computer Science, vol. 3876 (Springer, Berlin, 2006), pp. 285–304
I. Damgård, Y. Ishai, M. Krøigaard, Perfectly secure multiparty computation and the computational overhead of cryptography, in Advances in Cryptology—Eurocrypt’10. Lecture Notes in Computer Science, vol. 6110 (Springer, Berlin, 2010), pp. 445–465
Y. Desmedt, Y. Wang, M. Burmester, A complete characterization of tolerable adversary structures for secure point-to-point transmissions, in 16th International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol. 3827 (Springer, Berlin, 2005), pp. 277–287
Y. Desmedt, J. Pieprzyk, R. Steinfeld, H. Wang, On secure multi-party computation in black-box groups, in Advances in Cryptology—Crypto’07. Lecture Notes in Computer Science, vol. 4622 (Springer, Berlin, 2007), pp. 591–612
R. Diestel, Graph Theory, 2nd edn. Graduate Texts in Mathematics (Springer, Berlin, 2000)
W. Diffie, M.E. Hellman, New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976)
T. El Gamal, A public-key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inf. Theory 31(4), 469–472 (1985)
Y. Frankel, Y. Desmedt, M. Burmester, Non-existence of homomorphic general sharing schemes for some key spaces (extended abstract), in Advances in Cryptology—Crypto’92. Lecture Notes in Computer Science, vol. 740 (Springer, Berlin, 1993), pp. 549–557
O. Goldreich, Foundations of Cryptography: Basic Applications, vol. II (Cambridge University Press, Cambridge, 2004)
J.M. Hammersley, Percolation processes: lower bounds for the critical probability. Ann. Math. Stat. 28(3), 790–795 (1957)
M. Hirt, U. Maurer, Complete characterization of adversaries tolerable in secure multi-party computation (extended abstract), in 16th Annual ACM Symposium on Principles of Distributed Computing (ACM, New York, 1997), pp. 25–34
H. Kesten, Percolation Theory for Mathematicians (Birkhäuser, Basel, 1982)
S.S. Magliveras, D.R. Stinson, T. van Trung, New approaches to designing public key cryptosystems using one-way functions and trapdoors in finite groups. J. Cryptol. 15(4), 285–297 (2002)
A. Myasnikov, V. Shpilrain, A. Ushakov, Group-Based Cryptography. Advanced Courses in Mathematics—CRM Barcelona (Birkhäuser, Basel, 2008)
S.-H. Paeng, K.-C. Ha, J.H. Kim, S. Chee, C. Park, New public key cryptosystem using finite non Abelian groups, in Advances in Cryptology—Crypto’01. Lecture Notes in Computer Science, vol. 2139 (Springer, Berlin, 2001), pp. 470–485
R.L. Rivest, A. Shamir, L.M. Adleman, A method for obtaining digital signatures and public key cryptosystems. Commun. ACM 21(2), 120–126 (1978)
A. Shamir, How to share a secret. Commun. ACM 22(11), 612–613 (1979)
S. Smirnov, W. Werner, Critical exponents for two-dimensional percolation. Math. Res. Lett. 8, 729–744 (2001)
X. Sun, A.C.-C. Yao, C. Tartary, Graph design for secure multiparty computation over non-Abelian groups, in Advances in Cryptology—Asiacrypt’08. Lecture Notes in Computer Science, vol. 5350 (Springer, Berlin, 2008), pp. 37–53
N.R. Wagner, M.R. Magyarik, A public key encryption scheme based on the word problem, in Advances in Cryptology—Crypto’84. Lecture Notes in Computer Science, vol. 196 (Springer, Berlin, 1985), pp. 19–36
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Desmedt, Y., Pieprzyk, J., Steinfeld, R. et al. Graph Coloring Applied to Secure Computation in Non-Abelian Groups. J Cryptol 25, 557–600 (2012). https://doi.org/10.1007/s00145-011-9104-3
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DOI: https://doi.org/10.1007/s00145-011-9104-3