Skip to main content
SpringerLink
Log in

The Pseudorandom Oracle Model and Ideal Obfuscation

  • Conference paper
  • First Online:
Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14084))

Included in the following conference series:

  • 1033 Accesses

Abstract

We introduce a new idealized model of hash functions, which we refer to as the pseudorandom oracle (Pr\(\mathcal {O}\)) model. Intuitively, it allows us to model cryptosystems that use the code of an ideal hash function in a non-black-box way. Formally, we model hash functions via a combination of a pseudorandom function (PRF) family and an ideal oracle. A user can initialize the hash function by choosing a PRF key k and mapping it to a public handle h using the oracle. Given the handle h and some input x, the oracle can also be called to evaluate the PRF at x with the corresponding key k. A user who chooses the PRF key k therefore has a complete description of the hash function and can use its code in non-black-box constructions, while an adversary, who just gets the handle h, only has black-box access to the hash function via the oracle.

As our main result, we show how to construct ideal obfuscation in the Pr\(\mathcal {O}\) model, starting from functional encryption (FE), which in turn can be based on well-studied polynomial hardness assumptions. In contrast, we know that ideal obfuscation cannot be instantiated in the basic random oracle model under any assumptions. We believe our result provides heuristic justification for the following: (1) most natural security goals implied by ideal obfuscation can be achieved in the real world; (2) obfuscation can be constructed from FE at polynomial security loss.

We also discuss how to interpret our result in the Pr\(\mathcal {O}\) model as a construction of ideal obfuscation using simple hardware tokens or as a way to bootstrap ideal obfuscation for PRFs to that for all functions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Ideal obfuscation is similar to virtual black-box (VBB) obfuscation [11], except that we consider it to be an idealized model rather than a security definition. In contrast, VBB was originally intended as a security definition, with some artificial choices (restricting adversaries to only 1-bit output) to rule out obvious counterexamples. Nevertheless, the main result of [11] shows that even with these restrictions, VBB security is unachievable in its full generality in the plain model.

  2. 2.

    We note that after the initial pre-print and prior to the publication of this work, DE-PIR, FHE for RAM [42], and ABE for TM/RAM [3, 33] are achieved in the standard model.

  3. 3.

    This is yet another reason why the Pr\(\mathcal {O}\) model and the ROM are morally equivalent. The ROM essentially says that good hash functions are “self-obfuscated PRFs” since having the full description of a hash function is no better than just having oracle access to a random function, which is also what the Pr\(\mathcal {O}\) model stipulates.

  4. 4.

    We assume that \({\textsf{SHA3}(k{\parallel }{\cdot })}\) is a PRF with k being the (secret) key, which is a very mild assumption for real-world hash functions.

  5. 5.

    This is similar to Shoup’s generic group model [49]. Alternatively, we can define the handle as a special symbol that cannot be operated on, like in Maurer’s GGM [43]. The two models are studied in the recent work of [51]. We choose Shoup’s flavor for its potential flexibility, although our construction is compatible with Maurer’s. However, in either flavor, per definition, h is independent of k, and the difference between practice and formalism still prevails.

  6. 6.

    The encryption time is \(|z|{\text {poly}}(\lambda )\), where z is the plaintext. This is independent of the functions for which secret keys are issued.

  7. 7.

    The required properties can be achieved by standard PRG extension techniques and indifferentiable domain extension of random oracles.

  8. 8.

    In retrospect, this notion is an interpolation between functional encryption and unary function-revealing encryption [37].

  9. 9.

    In a standard public-key FE, the scheme is set up for a master public/secret key pair not tied to f, and a key for f can be derived separately from the master secret key. Weak selective security means that the adversary chooses \(f,z_0,z_1\) independent of the master public key. Full adaptive security against unbounded collusion means that the adversary can choose \(z_0,z_1\) and arbitrarily many \(f_q\)’s after seeing the master public key and in an arbitrary interleaving manner.

References

  1. Agrawal, S., et al.: Secure computation from one-way noisy communication, or: anti-correlation via anti-concentration. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12826, pp. 124–154. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84245-1_5

    Chapter  Google Scholar 

  2. Ananth, P., Brakerski, Z., Segev, G., Vaikuntanathan, V.: From selective to adaptive security in functional encryption. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 657–677. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_32

    Chapter  MATH  Google Scholar 

  3. Ananth, P., Chung, K.M., Fan, X., Qian, L.: Collusion-resistant functional encryption for RAMs. In: Agrawal, S., Lin, D. (eds.) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol. 13791, pp. 160–194. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-22963-3_6

  4. Ananth, P., Jain, A.: Indistinguishability obfuscation from compact functional encryption. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 308–326. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_15

    Chapter  Google Scholar 

  5. Ananth, P., Jain, A., Sahai, A.: Indistinguishability obfuscation from functional encryption for simple functions. Cryptology ePrint Archive, Report 2015/730 (2015). https://eprint.iacr.org/2015/730

  6. Ananth, P., Sahai, A.: Functional encryption for turing machines. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9562, pp. 125–153. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49096-9_6

    Chapter  Google Scholar 

  7. Applebaum, B.: Bootstrapping obfuscators via fast pseudorandom functions. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8874, pp. 162–172. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45608-8_9

    Chapter  Google Scholar 

  8. Badrinarayanan, S., Ishai, Y., Khurana, D., Sahai, A., Wichs, D.: Refuting the dream XOR lemma via ideal obfuscation and resettable MPC. In: Dachman-Soled, D. (ed.) ITC 2023. LIPIcs, vol. 230, pp. 1–21. Schloss Dagstuhl (2022). https://doi.org/10.4230/LIPIcs.ITC.2022.10

  9. Barak, B.: How to go beyond the black-box simulation barrier. In: 42nd FOCS, pp. 106–115. IEEE Computer Society Press, October 2001. https://doi.org/10.1109/SFCS.2001.959885

  10. Barak, B., Bitansky, N., Canetti, R., Kalai, Y.T., Paneth, O., Sahai, A.: Obfuscation for evasive functions. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 26–51. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_2

    Chapter  Google Scholar 

  11. 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

    Chapter  Google Scholar 

  12. Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Denning, D.E., Pyle, R., Ganesan, R., Sandhu, R.S., Ashby, V. (eds.) ACM CCS 93, pp. 62–73. ACM Press, November 1993. https://doi.org/10.1145/168588.168596

  13. 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). https://doi.org/10.1007/978-3-642-14623-7_28

    Chapter  Google Scholar 

  14. Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for SNARKS and proof-carrying data. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, pp. 111–120. ACM Press, June 2013. https://doi.org/10.1145/2488608.2488623

  15. Bitansky, N., Canetti, R., Goldwasser, S., Halevi, S., Kalai, Y.T., Rothblum, G.N.: Program obfuscation with leaky hardware. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 722–739. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_39

    Chapter  Google Scholar 

  16. Bitansky, N., Nishimaki, R., Passelègue, A., Wichs, D.: From cryptomania to obfustopia through secret-key functional encryption. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 391–418. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_15

    Chapter  Google Scholar 

  17. Bitansky, N., Vaikuntanathan, V.: Indistinguishability obfuscation from functional encryption. In: Guruswami, V. (ed.) 56th FOCS, pp. 171–190. IEEE Computer Society Press, October 2015. https://doi.org/10.1109/FOCS.2015.20

  18. Boyle, E., Ishai, Y., Pass, R., Wootters, M.: Can we access a database both locally and privately? In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10678, pp. 662–693. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70503-3_22

    Chapter  Google Scholar 

  19. Canetti, R., Kalai, Y.T., Paneth, O.: On obfuscation with random oracles. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 456–467. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_18

    Chapter  Google Scholar 

  20. Canetti, R., Lin, H., Tessaro, S., Vaikuntanathan, V.: Obfuscation of probabilistic circuits and applications. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 468–497. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_19

    Chapter  MATH  Google Scholar 

  21. Canetti, R., Raghuraman, S., Richelson, S., Vaikuntanathan, V.: Chosen-ciphertext secure fully homomorphic encryption. In: Fehr, S. (ed.) PKC 2017. LNCS, vol. 10175, pp. 213–240. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54388-7_8

    Chapter  Google Scholar 

  22. De Caro, A., Iovino, V., Jain, A., O’Neill, A., Paneth, O., Persiano, G.: On the achievability of simulation-based security for functional encryption. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 519–535. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_29

    Chapter  Google Scholar 

  23. Döttling, N., Mie, T., Müller-Quade, J., Nilges, T.: Basing obfuscation on simple tamper-proof hardware assumptions. Cryptology ePrint Archive, Report 2011/675 (2011). https://eprint.iacr.org/2011/675

  24. Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_12

    Chapter  Google Scholar 

  25. Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th FOCS, pp. 40–49. IEEE Computer Society Press, October 2013. https://doi.org/10.1109/FOCS.2013.13

  26. Garg, S., Srinivasan, A.: Single-key to multi-key functional encryption with polynomial loss. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 419–442. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_16

    Chapter  Google Scholar 

  27. Goldwasser, S., Kalai, Y.T.: On the (in)security of the Fiat-Shamir paradigm. In: 44th FOCS, pp. 102–115. IEEE Computer Society Press, October 2003. https://doi.org/10.1109/SFCS.2003.1238185

  28. Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: How to run Turing machines on encrypted data. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 536–553. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_30

    Chapter  Google Scholar 

  29. 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). https://doi.org/10.1007/978-3-540-70936-7_11

    Chapter  Google Scholar 

  30. Hamlin, A., Holmgren, J., Weiss, M., Wichs, D.: On the plausibility of fully homomorphic encryption for RAMs. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 589–619. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_21

    Chapter  Google Scholar 

  31. Ishai, Y., Korb, A., Lou, P., Sahai, A.: Beyond the Csiszár-korner bound: best-possible wiretap coding via obfuscation. In: Dodis, Y., Shrimpton, T. (eds.) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol. 13508. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15979-4_20

  32. Ishai, Y., Pandey, O., Sahai, A.: Public-coin differing-inputs obfuscation and its applications. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 668–697. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_26

    Chapter  Google Scholar 

  33. Jain, A., Lin, H., Luo, J.: On the optimal succinctness and efficiency of functional encryption and attribute-based encryption. In: Hazay, C., Stam, M. (eds.) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol. 14006, pp. 479–510. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-30620-4_16

  34. Jain, A., Lin, H., Luo, J., Wichs, D.: The pseudorandom oracle model and ideal obfuscation. Cryptology ePrint Archive, Report 2022/1204 (2022). https://eprint.iacr.org/2022/1204

  35. Jain, A., Lin, H., Sahai, A.: Indistinguishability obfuscation from well-founded assumptions. In: Khuller, S., Williams, V.V. (eds.) 53rd ACM STOC, pp. 60–73. ACM Press, June 2021. https://doi.org/10.1145/3406325.3451093

  36. Jain, A., Lin, H., Sahai, A.: Indistinguishability obfuscation from LPN over \(\mathbb{F} _p\), DLIN, and PRGs in \({NC}^0\). In: Dunkelman, O., Dziembowski, S. (eds.) Advances in Cryptology –EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol. 13275, pp. 670–699. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-06944-4_23

  37. Joye, M., Passelègue, A.: Function-revealing encryption. In: Catalano, D., De Prisco, R. (eds.) SCN 2018. LNCS, vol. 11035, pp. 527–543. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98113-0_28

    Chapter  MATH  Google Scholar 

  38. Kitagawa, F., Nishimaki, R., Tanaka, K.: Obfustopia built on secret-key functional encryption. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 603–648. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78375-8_20

    Chapter  Google Scholar 

  39. Kitagawa, F., Nishimaki, R., Tanaka, K., Yamakawa, T.: Adaptively secure and succinct functional encryption: improving security and efficiency, simultaneously. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 521–551. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_17

    Chapter  Google Scholar 

  40. Li, B., Micciancio, D.: Compactness vs collusion resistance in functional encryption. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 443–468. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_17

    Chapter  Google Scholar 

  41. Lin, H., Pass, R., Seth, K., Telang, S.: Indistinguishability obfuscation with non-trivial efficiency. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016. LNCS, vol. 9615, pp. 447–462. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49387-8_17

    Chapter  Google Scholar 

  42. Lin, W.K., Mook, E., Wichs, D.: Doubly efficient private information retrieval and fully homomorphic RAM computation from ring LWE. In: Saha, B., Servedio, R.A. (eds.) 55th ACM STOC, pp. 595–608. ACM Press (Jun 2023). https://doi.org/10.1145/3564246.3585175

  43. 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). https://doi.org/10.1007/11586821_1

    Chapter  MATH  Google Scholar 

  44. Maurer, U., Renner, R., Holenstein, C.: Indifferentiability, impossibility results on reductions, and applications to the random oracle methodology. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 21–39. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24638-1_2

    Chapter  Google Scholar 

  45. Nayak, K., et al.: HOP: hardware makes obfuscation practical. In: NDSS 2017. The Internet Society (Feb/Mar 2017). https://doi.org/10.14722/ndss.2017.23349

  46. Nishimaki, R.: Personal communication (2022)

    Google Scholar 

  47. Pippenger, N., Fischer, M.J.: Relations among complexity measures. J. ACM 26(2), 361–381 (1979). https://doi.org/10.1145/322123.322138

    Article  MathSciNet  MATH  Google Scholar 

  48. Sahai, A., Waters, B.: How to use indistinguishability obfuscation: deniable encryption, and more. In: Shmoys, D.B. (ed.) 46th ACM STOC, pp. 475–484. ACM Press (May/Jun 2014). https://doi.org/10.1145/2591796.2591825

  49. 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). https://doi.org/10.1007/3-540-69053-0_18

    Chapter  Google Scholar 

  50. Valiant, P.: Incrementally verifiable computation or proofs of knowledge imply time/space efficiency. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 1–18. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78524-8_1

    Chapter  MATH  Google Scholar 

  51. Zhandry, M.: To label, or not to label (in generic groups). In: Dodis, Y., Shrimpton, T. (eds.) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol. 13509, , pp. 66–96. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15982-4_3

Download references

Acknowledgments

Aayush Jain was supported by gifts from CyLab of CMU and Google. Huijia Lin and Ji Luo were supported by NSF CNS-1936825 (CAREER), CNS-2026774, a JP Morgan AI Research Award, a Cisco Research Award, and a Simons Collaboration on the Theory of Algorithmic Fairness. Daniel Wichs was supported by NSF CNS-1750795, CNS-2055510, and the JP Morgan Faculty Research Award. The authors thank the anonymous reviewers for their valuable feedback.

Author information

Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, USA

    Aayush Jain

  2. University of Washington, Seattle, USA

    Huijia Lin & Ji Luo

  3. Northeastern University, Boston, USA

    Daniel Wichs

  4. NTT Research, Sunnyvale, USA

    Daniel Wichs

Authors
  1. Aayush Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huijia Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ji Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Daniel Wichs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Aayush Jain , Huijia Lin , Ji Luo or Daniel Wichs .

Editor information

Editors and Affiliations

  1. Rambus Inc., San Jose, CA, USA

    Helena Handschuh

  2. Brown University, Providence, RI, USA

    Anna Lysyanskaya

Rights and permissions

Reprints and permissions

Copyright information

© 2023 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Jain, A., Lin, H., Luo, J., Wichs, D. (2023). The Pseudorandom Oracle Model and Ideal Obfuscation. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14084. Springer, Cham. https://doi.org/10.1007/978-3-031-38551-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-38551-3_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-38550-6

  • Online ISBN: 978-3-031-38551-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation