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Streaming Functional Encryption

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

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Abstract

We initiate the study of streaming functional encryption (sFE) which is designed for scenarios in which data arrives in a streaming manner and is computed on in an iterative manner as the stream arrives. Unlike in a standard functional encryption (FE) scheme, in an sFE scheme, we (1) do not require the entire data set to be known at encryption time and (2) allow for partial decryption given only a prefix of the input. More specifically, in an sFE scheme, we can sequentially encrypt each data point \(x_i\) in a stream of data \(x = x_1\ldots x_n\) as it arrives, without needing to wait for all n values. We can then generate function keys for streaming functions which are stateful functions that take as input a message \(x_i\) and a state \(\textsf{st}_i\) and output a value \(y_i\) and the next state \(\textsf{st}_{i+1}\). For any \(k \le n\), a user with a function key for a streaming function f can learn the first k output values \(y_1\ldots y_k\) where \((y_i, \textsf{st}_{i+1}) = f(x_i, \textsf{st}_i)\) and \(\textsf{st}_1 = \bot \) given only ciphertexts for the first k elements \(x_1\ldots x_k\).

In this work, we introduce the notion of sFE and show how to construct it from FE. In particular, we show how to achieve a secure sFE scheme for \(\mathsf {P/Poly}\) from a compact, secure FE scheme for \(\mathsf {P/Poly}\), where our security notion for sFE is similar to standard FE security except that we require all function queries to be made before the challenge ciphertext query. Furthermore, by combining our result with the FE construction of Jain, Lin, and Sahai (STOC, 2022), we show how to achieve a secure sFE scheme for \(\mathsf {P/Poly}\) from the polynomial hardness of well-studied assumptions.

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Notes

  1. 1.

    We assume that unless specified otherwise, all streaming functions have the same starting state \(\bot \) (or the all zero string) which is included in their state space.

  2. 2.

    The purpose of the encryption state is to tie elements of the stream together and must be kept secret in order to prevent mix and match attacks. Suppose that the encryption state either did not exist or was made public. Then an adversary given a function key for some streaming function f and ciphertexts \(\textsf{ct}_1,\dots ,\textsf{ct}_n\) for some stream \(x = x_1\dots x_n\) could learn the value of f on any extension of the stream. That is, the adversary could encrypt any values \(z = z_1\dots z_k\) to get ciphertexts \(\textsf{ct}_1',\dots ,\textsf{ct}_k'\), and then learn the value of f on any interpolation of x and z (e.g. \(x_1z_2z_3x_4z_k\)) by simply decrypting using the corresponding interpolation of \(\textsf{ct}_1\dots \textsf{ct}_n\) and \(\textsf{ct}_1'\dots \textsf{ct}_k'\) (e.g. \(\textsf{ct}_1\textsf{ct}_2'\textsf{ct}_3'\textsf{ct}_4\textsf{ct}_5'\)). This is much more power than we want the adversary to have. Note that when the encryption state is secret, then this bad behavior is not allowed as decryption will fail on any ciphertexts \(\textsf{ct}_i'\) not generated using the secret encryption state.

  3. 3.

    As with all streaming functions, we assume that \(\textsf{Dec}.\textsf{st}_1 = \bot \) by default.

  4. 4.

    We can actually achieve security even when the challenge stream messages depend on the ciphertexts given for the previous stream values. See Remark 4.7.

  5. 5.

    In other sections of this paper, we refer to this notion as strong-compactness since the usual notion of compactness found in the literature only requires the encryption algorithm to be independent of the function size. However, all existing transformations achieving compactness also yield strong compactness.

  6. 6.

    Our actual scheme adds additional branches of computation to \(G_{f,s}\) and \(H_{i,x_i,t_i}\) which are only used in the security proof.

  7. 7.

    We assume that unless specified otherwise, all streaming functions have the same starting state \(\bot \) (or the all zero string) which is included in their state space.

  8. 8.

    We also allow \(\textsf{Enc}, \textsf{Enc}\textsf{Setup}, \textsf{KeyGen}\), and \(\textsf{Dec}\) to additionally receive parameters \(1^\lambda , 1^{{\ell _{\mathcal F}}}, 1^{\ell _{\mathcal S}}, 1^{{\ell _{\mathcal X}}}, 1^{{\ell _{\mathcal Y}}}\) as input, but omit them from our notation for convenience.

  9. 9.

    As with all streaming functions, we assume that \(\textsf{Dec}.\textsf{st}_1 = \bot \) by default.

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Acknowledgments

This research was supported in part from a Simons Investigator Award, DARPA SIEVE award, NTT Research, NSF Frontier Award 1413955, BSF grant 2012378, a Xerox Faculty Research Award, a Google Faculty Research Award, and an Okawa Foundation Research Grant. This material is based upon work supported by the Defense Advanced Research Projects Agency through Award HR00112020024. The views expressed are those of the authors and do not reflect the official policy or position of any funding source or the U.S. Government.

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Authors and Affiliations

  1. Princeton University, Princeton, USA

    Jiaxin Guan

  2. UCLA, Los Angeles, USA

    Alexis Korb & Amit Sahai

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  1. Jiaxin Guan
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Correspondence to Jiaxin Guan or Alexis Korb .

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

    Helena Handschuh

  2. Brown University, Providence, RI, USA

    Anna Lysyanskaya

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Guan, J., Korb, A., Sahai, A. (2023). Streaming Functional Encryption. 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_14

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