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Compact Adaptively Secure ABE for \(\mathsf {NC^1}\) from k-Lin

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Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11476))

Abstract

We present compact attribute-based encryption (ABE) schemes for \(\mathsf {NC^1}\) that are adaptively secure under the k-Lin assumption with polynomial security loss. Our KP-ABE scheme achieves ciphertext size that is linear in the attribute length and independent of the policy size even in the many-use setting, and we achieve an analogous efficiency guarantee for CP-ABE. This resolves the central open problem posed by Lewko and Waters (CRYPTO 2011). Previous adaptively secure constructions either impose an attribute “one-use restriction” (or the ciphertext size grows with the policy size), or require q-type assumptions.

L. Kowalczyk—Supported in part by an NSF Graduate Research Fellowship DGE-16-44869; The Leona M. & Harry B. Helmsley Charitable Trust; ERC Project aSCEND (H2020 639554); the Defense Advanced Research Project Agency (DARPA) and Army Research Office (ARO) under Contract W911NF-15-C-0236; and NSF grants CNS-1445424, CNS-1552932 and CCF-1423306. Any opinions, findings and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the Defense Advanced Research Projects Agency, Army Research Office, the National Science Foundation, or the U.S. Government.

H. Wee—Supported in part by ERC Project aSCEND (H2020 639554).

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Notes

  1. 1.

    Note that there exist constructions of ABE for more general access policies like monotone span programs/Boolean formulas with threshold gates [17], and even polynomial-sized Boolean circuits [14, 16], but all such constructions sacrifice at least one of the properties (1)–(3).

  2. 2.

    Essentially, the dual system proof method provides guidance for transforming suitably-designed functional encryption schemes which are secure for one adversarial secret key request to the multi-key setting where multiple keys may be requested by the adversary. Our main technical contribution involves the analysis of the initial single-key-secure component, which we refer to later as our “Core 1-ABE” component.

  3. 3.

    Most directly by pushing all NOT gates to the input nodes of each circuit and using new attributes to represent the negation of each original attribute. It is likely that the efficiency hit introduced by this transformation can be removed through more advanced techniques à la [24, 29], but we leave this for future work.

  4. 4.

    Some works associate ciphertexts with a set \(S \subseteq [n]\) where [n] is referred to as the attribute universe, in which case \(\mathbf {x}\in \{0,1\}^n\) corresponds to the characteristic vector of S.

  5. 5.

    E.g.: \(k=1\) corresponds to security under the Symmetric External Diffie-Hellman Assumption (SXDH), and \(k=2\) corresponds to security under the Decisional Linear Assumption (DLIN).

  6. 6.

    Informally, \(\{\mathsf {H}^u\}\) describes the simulated games used in the security reduction, where the reduction guesses \(R'\) bits of information described by u about some choices z made by the adversary; these \(R'\) bits of information are described by \(h_\ell (z)\) in the \(\ell \)’th hybrid. In the \(\ell \)’th hybrid, the reduction guesses a \(u \in \{0,1\}^{R'}\) and simulates the game according to \(\mathsf {H}^u\) and hopes that the adversary will pick an z such that \(h_\ell (z) = u\); note that the adversary is not required to pick such an z. One way to think of \(\mathsf {H}^u\) is that the reduction is committed to u, but the adversary can do whatever it wants.

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Acknowledgments

We thank Allison Bishop, Sanjam Garg, Rocco Servedio, and Daniel Wichs for helpful discussions.

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Authors and Affiliations

  1. Columbia University, New York, USA

    Lucas Kowalczyk

  2. CNRS, ENS, PSL, Paris, France

    Hoeteck Wee

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  1. Technion, Haifa, Israel

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  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Kowalczyk, L., Wee, H. (2019). Compact Adaptively Secure ABE for \(\mathsf {NC^1}\) from k-Lin. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11476. Springer, Cham. https://doi.org/10.1007/978-3-030-17653-2_1

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