Abstract
We present a new general-purpose obfuscator for all polynomial size circuits. The obfuscator uses graded encoding schemes, a generalization of multilinear maps. We prove that the obfuscator exposes no more information than the program’s black-box functionality, and achieves virtual black-box security, in the generic graded encoded scheme model. This proof is under the Bounded Speedup Hypothesis (BSH, a plausible worst-case complexity-theoretic assumption related to the Exponential Time Hypothesis), in addition to standard cryptographic assumptions. We also prove that it satisfies the notion of indistinguishability obfuscation without without relying on BSH (in the same generic model and under standard cryptographic assumptions).
Very recently, Garg et al. (FOCS 2013) used graded encoding schemes to present a candidate obfuscator for indistinguishability obfuscation. They posed the problem of constructing a provably secure indistinguishability obfuscator in the generic graded encoding scheme model. Our obfuscator resolves this problem (indeed, under BSH it achieves the stronger notion of virtual black box security, which is our focus in this work).
Our construction is different from that of Garg et al., but is inspired by it, in particular by their use of permutation branching programs. We obtain our obfuscator by developing techniques used to obfuscate d-CNF formulas (ITCS 2014), and applying them to permutation branching programs. This yields an obfuscator for the complexity class \(\mathcal{NC}^1\). We then use homomorphic encryption to obtain an obfuscator for any polynomialsize circuit.
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References
Adida, B., Wikström, D.: How to shuffle in public. In: TCC 2007. LNCS, vol. 4392, pp. 555–574. Springer, Heidelberg (2007)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in nc0. SIAM J. Comput. 36(4), 845–888 (2006)
Babai, L.: Trading group theory for randomness. In: STOC, pp. 421–429 (1985)
Barak, B., Garg, S., Kalai, Y.T., Paneth, O., Sahai, A.: Protecting obfuscation against algebraic attacks. Cryptology ePrint Archive, Report 2013/631 (2013), http://eprint.iacr.org/
Barak, B., Goldreich, O., Impagliazzo, R., Rudich, S., Sahai, A., Vadhan, S.P., Yang, K.: On the (im)possibility of obfuscating programs. J. ACM 59(2), 6 (2012); Preliminary version in Kilian, J. (ed.): CRYPTO 2001. LNCS, vol. 2139. Springer, Heidelberg (2001)
Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. In: Hartmanis, J. (ed.) STOC, pp. 1–5. ACM (1986); Full version in [7]
Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. J. Comput. Syst. Sci. 38(1), 150–164 (1989)
Bitansky, N., Canetti, R.: On strong simulation and composable point obfuscation. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 520–537. Springer, Heidelberg (2010)
Boneh, D., Silverberg, A.: Applications of multilinear forms to cryptography. IACR Cryptology ePrint Archive 2002, 80 (2002)
Brakerski, Z., Rothblum, G.N.: Obfuscating conjunctions. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 416–434. Springer, Heidelberg (2013), http://eprint.iacr.org/2013/471
Brakerski, Z., Rothblum, G.N.: Black-box obfuscation for d-CNFs. Cryptology ePrint Archive (2013). Extended abstract in ITCS 2014
Brakerski, Z., Rothblum, G.N.: Virtual black-box obfuscation for all circuits via generic graded encoding. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 1–25. Springer, Heidelberg (2014)
Canetti, R.: Towards realizing random oracles: Hash functions that hide all partial information. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 455–469. Springer, Heidelberg (1997)
Canetti, R., Dakdouk, R.R.: Obfuscating point functions with multibit output. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 489–508. Springer, Heidelberg (2008)
Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited (preliminary version). In: Vitter, J.S. (ed.) STOC, pp. 209–218. ACM (1998); Full version in [16]
Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. J. ACM 51(4), 557–594 (2004)
Canetti, R., Micciancio, D., Reingold, O.: Perfectly one-way probabilistic hash functions (preliminary version). In: Vitter, J.S. (ed.) STOC, pp. 131–140. ACM (1998)
Canetti, R., Rothblum, G.N., Varia, M.: Obfuscation of hyperplane membership. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 72–89. Springer, Heidelberg (2010)
Canetti, R., Vaikuntanathan, V.: Obfuscating branching programs using black-box pseudo-free groups. Cryptology ePrint Archive, Report 2013/500 (2013), http://eprint.iacr.org/
Coron, J.-S., Lepoint, T., Tibouchi, M.: Practical multilinear maps over the integers. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 476–493. Springer, Heidelberg (2013)
Dodis, Y., Smith, A.: Correcting errors without leaking partial information. In: Gabow, H.N., Fagin, R. (eds.) STOC, pp. 654–663. ACM (2005)
Feige, U., Kilian, J., Naor, M.: A minimal model for secure computation (extended abstract). In: STOC, pp. 554–563 (1994)
Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013)
Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. Cryptology ePrint Archive, Report 2013/451 (2013); Extended abstract in FOCS 2013
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) STOC, pp. 169–178. ACM (2009)
Goldwasser, S., Kalai, Y.T.: On the impossibility of obfuscation with auxiliary input. In: FOCS, pp. 553–562 (2005)
Goldwasser, S., Rothblum, G.N.: On best-possible obfuscation. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 194–213. Springer, Heidelberg (2007)
Hofheinz, D., Malone-Lee, J., Stam, M.: Obfuscation for cryptographic purposes. J. Cryptology 23(1), 121–168 (2010)
Hohenberger, S., Rothblum, G.N., Shelat, A., Vaikuntanathan, V.: Securely obfuscating re-encryption. J. Cryptology 24(4), 694–719 (2011)
Impagliazzo, R., Paturi, R.: Complexity of k-sat. In: IEEE Conference on Computational Complexity, pp. 237–240. IEEE Computer Society (1999)
Kilian, J.: Founding cryptography on oblivious transfer. In: Simon, J. (ed.) STOC, pp. 20–31. ACM (1988)
Lynn, B.Y.S., Prabhakaran, M., Sahai, A.: Positive results and techniques for obfuscation. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 20–39. Springer, Heidelberg (2004)
Maurer, U.: Abstract models of computation in cryptography. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 1–12. Springer, Heidelberg (2005)
Naor, M.: On cryptographic assumptions and challenges. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003)
Rivest, R., Adleman, L., Dertouzos, M.: On data banks and privacy homomorphisms. In: Foundations of Secure Computation, pp. 169–177. Academic Press (1978)
Rothblum, R.D.: On the circular security of bit-encryption. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 579–598. Springer, Heidelberg (2013)
Sahai, A., Waters, B.: How to use indistinguishability obfuscation: Deniable encryption, and more. Cryptology ePrint Archive, Report 2013/454 (2013), http://eprint.iacr.org/
Shoup, V.: Lower bounds for discrete logarithms and related problems. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 256–266. Springer, Heidelberg (1997)
Wee, H.: On obfuscating point functions. In: Gabow, H.N., Fagin, R. (eds.) STOC, pp. 523–532. ACM (2005)
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Brakerski, Z., Rothblum, G.N. (2014). Virtual Black-Box Obfuscation for All Circuits via Generic Graded Encoding. In: Lindell, Y. (eds) Theory of Cryptography. TCC 2014. Lecture Notes in Computer Science, vol 8349. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54242-8_1
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