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CRYPTO 2019: Advances in Cryptology – CRYPTO 2019 pp 521–551Cite as

Adaptively Secure and Succinct Functional Encryption: Improving Security and Efficiency, Simultaneously

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11694))

Abstract

Functional encryption (FE) is advanced encryption that enables us to issue functional decryption keys where functions are hardwired. When we decrypt a ciphertext of a message m by a functional decryption key where a function f is hardwired, we can obtain f(m) and nothing else. We say FE is selectively or adaptively secure when target messages are chosen at the beginning or after function queries are sent, respectively. In the weakly-selective setting, function queries are also chosen at the beginning. We say FE is single-key/collusion-resistant when it is secure against adversaries that are given only-one/polynomially-many functional decryption keys, respectively. We say FE is sublinearly-succinct/succinct when the running time of an encryption algorithm is sublinear/poly-logarithmic in the function description size, respectively.

In this study, we propose a generic transformation from weakly-selectively secure, single-key, and sublinearly-succinct (we call “building block”) PKFE for circuits into adaptively secure, collusion-resistant, and succinct (we call “fully-equipped”) one for circuits. Our transformation relies on neither concrete assumptions such as learning with errors nor indistinguishability obfuscation (IO). This is the first generic construction of fully-equipped PKFE that does not rely on IO.

As side-benefits of our results, we obtain the following primitives from the building block PKFE for circuits: (1) laconic oblivious transfer (2) succinct garbling scheme for Turing machines (3) selectively secure, collusion-resistant, and succinct PKFE for Turing machines (4) low-overhead adaptively secure traitor tracing (5) key-dependent message secure and leakage-resilient public-key encryption. We also obtain a generic transformation from simulation-based adaptively secure garbling schemes that satisfy a natural decomposability property into adaptively indistinguishable garbling schemes whose online complexity does not depend on the output length.

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Notes

  1. 1.

    Of course, adversaries can send queries after they decided a pair of target messages.

  2. 2.

    See the subsequent paragraph for the reason of naming “obf-minimum”.

  3. 3.

    In the case of PKFE, \(\#\mathsf {ct}\) is trivially \(\mathsf {unb}\).

  4. 4.

    In the setting of SKFE, only an entity that has a master secret-key can generate ciphertexts. Thus, adversaries is allowed to send messages as queries and receives ciphertexts in its security game. When adversaries can send one/polynomially-many message(s), we say one/many-ciphertext SKFE.

  5. 5.

    In fact, there are subtle issues to transform a garbling scheme into a single-key and single-ciphertext SKFE (the opposite is easy). See the full version for more details.

  6. 6.

    Note that this is different from succinct garbling schemes [5, 12] since ours is for circuits while succinct garbling schemes are for TMs.

  7. 7.

    Jafargholi et al. wrote “It remains an open problem whether it is possible to show a more general transformation from garbled circuits with adaptive security (and maybe other natural properties) to garbled circuits with indistinguishability based adaptive security and online complexity independent of the output size.” [37].

  8. 8.

    Ananth and Lombardi present an LOT protocol based on IO [5].

  9. 9.

    The security level of our LOT is sufficient for their purpose.

  10. 10.

    Note that we cannot obtain an adaptively secure scheme in Corollary 1.2 since the succinct garbling for TMs by Ananth and Lombardi is not adaptively secure.

  11. 11.

    Note that their FE for TMs satisfies a stronger security notion called distributional indistinguishability than standard indistinguishability.

  12. 12.

    Cho et al.’s bootstrapping method is not sufficient for LOT whose security holds only when an adversary declares the challenge database before seeing CRS. Therefore, we cannot use the bootstrapping method of Cho et al. directly to make our selective-database (explained later) LOT updatable. However, we can use a minor variant of the bootstrapping method observed by Ananth and Lombardi [5] to bootstrap selective-database LOT into updatable one.

  13. 13.

    To achieve \(\frac{1}{2}\) compression in our construction, it is sufficient that the size of a master public-key is logarithmic in the length of identities. This requirement is more natural for IBE, and thus we assume only this mild condition in the actual construction.

  14. 14.

    We say that an SKFE scheme is function private if a decryption key does not reveal the associated function. As shown by Brakerski and Segev [17], we can generically add the function privacy to any SKFE scheme. Thus we do not care about function privacy in this overview.

  15. 15.

    Though Gorbunov et al. [33] presented their construction in the public key setting, the same construction works in the secret key setting.

  16. 16.

    Though Gorbunov et al. [33] does not use an abstraction as NCER, we observe that their construction can be seen like this.

  17. 17.

    Though Jafargholi and Wichs [38] showed that Yao’s garbling scheme is adaptively secure for certain class of circuits like \(\mathsf {NC}^1\), we do not know how to prove its adaptive security for all circuits.

  18. 18.

    Strictly speaking, the SKFE scheme achieves a security notion called key-adaptive security slightly weaker than the adaptive security, in which an adversary cannot make any encryption queries after making the key query. We note that this is sufficient for constructing an adaptively indistinguishable garbling scheme since the adaptive security of a garbling scheme only considers the case where a garbled input is generated after a garbled circuit is generated.

  19. 19.

    We can formally prove adaptive security of the somewhere adaptive garbling scheme by Garg et al. [26] by using specific properties of Yao’s selectively secure garbling scheme instead of using selective security in a black-box way.

  20. 20.

    We can always modify any IBE scheme so that it satisfies these two conditions by using PRF.

  21. 21.

    In fact, in our LOT protocol in Sect. 4, \(|\mathsf {crs}| = {\mathrm {poly}}(\lambda )\). However, it does not matter here since it is absorbed in \({\mathrm {poly}}(\log {|C|},\lambda )\) part.

  22. 22.

    Actually, the direct adaptation only achieves ciphertext-adaptive security where a decryption key must be queried before the challenge ciphertext is given to an adversary. This can be easily overcome by using one-time pad without sacrificing succinctness.

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Acknowledgments

The third author was supported by NTT Secure Platform Laboratories, JST OPERA JPMJOP1612, JST CREST JPMJCR14D6, JSPS KAKENHI JP16H01705, JP17H01695.

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Authors and Affiliations

  1. NTT Secure Platform Laboratories, Tokyo, Japan

    Fuyuki Kitagawa, Ryo Nishimaki & Takashi Yamakawa

  2. Tokyo Institute of Technology, Tokyo, Japan

    Keisuke Tanaka

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  1. Fuyuki Kitagawa
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Correspondence to Fuyuki Kitagawa or Ryo Nishimaki .

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

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Kitagawa, F., Nishimaki, R., Tanaka, K., Yamakawa, T. (2019). Adaptively Secure and Succinct Functional Encryption: Improving Security and Efficiency, Simultaneously. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11694. Springer, Cham. https://doi.org/10.1007/978-3-030-26954-8_17

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