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Obfuscating Simple Functionalities from Knowledge Assumptions

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Public-Key Cryptography – PKC 2019 (PKC 2019)

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This paper shows how to obfuscate several simple functionalities from a new Knowledge of OrthogonALity Assumption (KOALA) in cyclic groups which is shown to hold in the Generic Group Model. Specifically, we give simpler and stronger security proofs for obfuscation schemes for point functions, general-output point functions and pattern matching with wildcards. We also revisit the work of Bishop et al. (CRYPTO 2018) on obfuscating the pattern matching with wildcards functionality. We improve upon the construction and the analysis in several ways:

  • attacks and stronger guarantees: We show that the construction achieves virtual black-box security for a simulator that runs in time roughly \(2^{n/2}\), as well as distributional security for larger classes of distributions. We give attacks that show that our results are tight.

  • weaker assumptions: We prove security under KOALA.

  • better efficiency: We also provide a construction that outputs \(n+1\) instead of 2n group elements.

We obtain our results by first obfuscating a simpler “big subset functionality”, for which we establish full virtual black-box security; this yields a simpler and more modular analysis for pattern matching. Finally, we extend our distinguishing attacks to a large class of simple linear-in-the-exponent schemes.

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  1. 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).

    Chapter  Google Scholar 

  2. Bartusek, J., Lepoint, T., Ma, F., Zhandry, M.: New techniques for obfuscating conjunctions. Cryptology ePrint Archive, Report 2018/936 (2018).

  3. Bishop, A., Kowalczyk, L., Malkin, T., Pastro, V., Raykova, M., Shi, K.: A simple obfuscation scheme for pattern-matching with wildcards. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 731–752. Springer, Cham (2018).

    Chapter  Google Scholar 

  4. 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).

    Chapter  Google Scholar 

  5. 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).

    Chapter  Google Scholar 

  6. Brakerski, Z., Vaikuntanathan, V., Wee, H., Wichs, D.: Obfuscating conjunctions under entropic ring LWE. In: ITCS, pp. 147–156. ACM (2016)

    Google Scholar 

  7. Canetti, R.: Towards realizing random oracles: hash functions that hide all partial information. In: Kaliski, B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 455–469. Springer, Heidelberg (1997).

    Chapter  Google Scholar 

  8. 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).

    Chapter  Google Scholar 

  9. 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).

    Chapter  MATH  Google Scholar 

  10. Damgård, I.: Towards practical public key systems secure against chosen ciphertext attacks. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 445–456. Springer, Heidelberg (1992).

    Chapter  Google Scholar 

  11. Fuchsbauer, G., Kiltz, E., Loss, J.: The algebraic group model and its applications. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 33–62. Springer, Cham (2018).

    Chapter  Google Scholar 

  12. Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. SIAM J. Comput. 45(3), 882–929 (2016)

    Article  MathSciNet  Google Scholar 

  13. Goyal, R., Koppula, V., Waters, B.: Lockable obfuscation. In: FOCS, pp. 612–621 (2017)

    Google Scholar 

  14. Hada, S.: Zero-knowledge and code obfuscation. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 443–457. Springer, Heidelberg (2000).

    Chapter  Google Scholar 

  15. Lynn, B., 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).

    Chapter  Google Scholar 

  16. Wee, H.: On obfuscating point functions. In: STOC, pp. 523–532. ACM (2005)

    Google Scholar 

  17. Wichs, D., Zirdelis, G.: Obfuscating compute-and-compare programs under LWE. In: FOCS, pp. 600–611 (2017)

    Google Scholar 

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This work started at ENS over the summer; we thank Luke Kowalczyk for telling us about [3], as well as Michel Abdalla, Georg Fuchsbauer and Hendrik Waldner for helpful discussions. This work was supported in part by the Research Council KU Leuven: C16/15/058, C14/18/067 and STG/17/019. In addition, this work was supported by the European Commission through the Horizon 2020 research and innovation programme under grant agreement H2020-DS-LEIT-2017-780108 FENTEC, by the Flemish Government through FWO SBO project SNIPPET and by the IF/C1 on Cryptanalysis of post-quantum cryptography. Ward Beullens is funded by an FWO fellowship. Hoeteck Wee is supported by ERC Project aSCEND (H2020 639554).

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Beullens, W., Wee, H. (2019). Obfuscating Simple Functionalities from Knowledge Assumptions. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11443. Springer, Cham.

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