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Reducing Depth in Constrained PRFs: From Bit-Fixing to \(\mathbf {NC}^{1}\)

  • Nishanth Chandran
  • Srinivasan Raghuraman
  • Dhinakaran Vinayagamurthy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9615)

Abstract

The candidate construction of multilinear maps by Garg, Gentry, and Halevi (Eurocrypt 2013) has lead to an explosion of new cryptographic constructions ranging from attribute-based encryption (ABE) for arbitrary polynomial size circuits, to program obfuscation, and to constrained pseudorandom functions (PRFs). Many of these constructions require \(\kappa \)-linear maps for large \(\kappa \). In this work, we focus on the reduction of \(\kappa \) in certain constructions of access control primitives that are based on \(\kappa \)-linear maps; in particular, we consider the case of constrained PRFs and ABE. We construct the following objects:
  • A constrained PRF for arbitrary circuit predicates based on \((n+\ell _{\mathsf {OR}}-1)-\)linear maps (where n is the input length and \(\ell _{\mathsf {OR}}\) denotes the OR-depth of the circuit).

  • For circuits with a specific structure, we also show how to construct such PRFs based on \((n+\ell _{\mathsf {AND}}-1)-\)linear maps (where \(\ell _{\mathsf {AND}}\) denotes the AND-depth of the circuit).

We then give a black-box construction of a constrained PRF for \(\mathbf {NC}^{1}\) predicates, from any bit-fixing constrained PRF that fixes only one of the input bits to 1; we only require that the bit-fixing PRF have certain key homomorphic properties. This construction is of independent interest as it sheds light on the hardness of constructing constrained PRFs even for “simple” predicates such as bit-fixing predicates.

Instantiating this construction with the bit-fixing constrained PRF from Boneh and Waters (Asiacrypt 2013) gives us a constrained PRF for \(\mathbf {NC}^{1}\) predicates that is based only on n-linear maps, with no dependence on the predicate. In contrast, the previous constructions of constrained PRFs (Boneh and Waters, Asiacrypt 2013) required \((n+\ell +1)-\)linear maps for circuit predicates (where \(\ell \) is the total depth of the circuit) and n-linear maps even for bit-fixing predicates.

We also show how to extend our techniques to obtain a similar improvement in the case of ABE and construct ABE for arbitrary circuits based on \((\ell _{\mathsf {OR}}+1)-\)linear (respectively \((\ell _{\mathsf {AND}}+1)-\)linear) maps.

Keywords

Boolean Circuit Pseudorandom Function Real Game Output Wire Input Wire 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [BFP+15]
    Banerjee, A., Fuchsbauer, G., Peikert, C., Pietrzak, K., Stevens, S.: Key-homomorphic constrained pseudorandom functions. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part II. LNCS, vol. 9015, pp. 31–60. Springer, Heidelberg (2015)Google Scholar
  2. [BGI14]
    Boyle, E., Goldwasser, S., Ivan, I.: Functional signatures and pseudorandom functions. In: Public Key Cryptography, pp. 501–519 (2014)Google Scholar
  3. [BV15]
    Brakerski, Z., Vaikuntanathan, V.: Constrained key-homomorphic PRFs from standard lattice assumptions. Or: how to secretly embed a circuit in your PRF. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part II. LNCS, vol. 9015, pp. 1–30. Springer, Heidelberg (2015)Google Scholar
  4. [BW13]
    Boneh, D., Waters, B.: Constrained pseudorandom functions and their applications. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part II. LNCS, vol. 8270, pp. 280–300. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. [CHL+15]
    Cheon, J.H., Han, K., Lee, C., Ryu, H., Stehlé, D.: Cryptanalysis of the multilinear map over the integers. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part I. LNCS, vol. 9056, pp. 3–12. Springer, Heidelberg (2015)Google Scholar
  6. [CLR15]
    Cheon, J.H., Lee, C., Ryu, H.: Cryptanalysis of the new CLT multilinear maps. Cryptology ePrint Archive, Report 2015/934 (2015)Google Scholar
  7. [CLT13]
    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)CrossRefGoogle Scholar
  8. [CLT15]
    Coron, J.-S., Lepoint, T., Tibouchi, M.: New multilinear maps over the integers. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015, Part I. LNCS, vol. 9215, pp. 267–286. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  9. [CRV14]
    Chandran, N., Raghuraman, S., Vinayagamurthy, D.: Constrained pseudorandom functions: verifiable and delegatable. IACR Cryptology ePrint Archive, 2014:522 (2014)Google Scholar
  10. [CRV15]
    Chandran, N., Raghuraman, S., Vinayagamurthy, D.: Reducing depth in constrained PRFs: from bit-fixing to \(\rm NC^{1}\). IACR Cryptology ePrint Archive, 2015:829 (2015)Google Scholar
  11. [FKPR14]
    Fuchsbauer, G., Konstantinov, M., Pietrzak, K., Rao, V.: Adaptive security of constrained PRFs. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part II. LNCS, vol. 8874, pp. 82–101. Springer, Heidelberg (2014)Google Scholar
  12. [Fuc14]
    Fuchsbauer, G.: Constrained verifiable random functions. In: Abdalla, M., De Prisco, R. (eds.) SCN 2014. LNCS, vol. 8642, pp. 95–114. Springer, Heidelberg (2014)Google Scholar
  13. [GGH13a]
    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)CrossRefGoogle Scholar
  14. [GGH+13b]
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26–29 October, 2013, Berkeley, CA, USA, pp. 40–49 (2013)Google Scholar
  15. [GGH+13c]
    Garg, S., Gentry, C., Halevi, S., Sahai, A., Waters, B.: Attribute-based encryption for circuits from multilinear maps. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 479–499. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. [GGH15]
    Gentry, C., Gorbunov, S., Halevi, S.: Graph-induced multilinear maps from lattices. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part II. LNCS, vol. 9015, pp. 498–527. Springer, Heidelberg (2015)Google Scholar
  17. [GGM86]
    Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)CrossRefMathSciNetGoogle Scholar
  18. [GPSW06]
    Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: ACM Conference on Computer and Communications Security, pp. 89–98 (2006)Google Scholar
  19. [GVW13]
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Attribute-based encryption for circuits. In: STOC, pp. 545–554 (2013)Google Scholar
  20. [HJ15]
    Hu, Y., Jia, H.: Cryptanalysis of GGH map. IACR Cryptology ePrint Archive, 2015:301 (2015)Google Scholar
  21. [HKKW14]
    Hofheinz, D., Kamath, A., Koppula, V., Waters, B.: Adaptively secure constrained pseudorandom functions. IACR Cryptology ePrint Archive, 2014:720 (2014)Google Scholar
  22. [KPTZ13]
    Kiayias, A., Papadopoulos, S., Triandopoulos, N., Zacharias, T.: Delegatable pseudorandom functions and applications. In: ACM SIGSAC Conference on Computer and Communications Security, CCS 2013, Berlin, Germany, 4–8 November, 2013, pp. 669–684 (2013)Google Scholar
  23. [MF15]
    Minaud, B., Fouque, P.-A.: Cryptanalysis of the new multilinear map over the integers. Cryptology ePrint Archive, Report 2015/941 (2015)Google Scholar
  24. [SW05]
    Sahai, A., Waters, B.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  • Nishanth Chandran
    • 1
  • Srinivasan Raghuraman
    • 2
  • Dhinakaran Vinayagamurthy
    • 3
  1. 1.Microsoft ResearchBengaluruIndia
  2. 2.CSAIL, Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.University of WaterlooWaterlooCanada

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