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Function-Hiding Inner Product Encryption Is Practical

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Book cover Security and Cryptography for Networks (SCN 2018)

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Abstract

In a functional encryption scheme, secret keys are associated with functions and ciphertexts are associated with messages. Given a secret key for a function f, and a ciphertext for a message x, a decryptor learns f(x) and nothing else about x. Inner product encryption is a special case of functional encryption where both secret keys and ciphertext are associated with vectors. The combination of a secret key for a vector \({\mathbf {x}}\) and a ciphertext for a vector \(\mathbf {y}\) reveal \(\langle {\mathbf {x}}, \mathbf {y}\rangle \) and nothing more about \(\mathbf {y}\). An inner product encryption scheme is function-hiding if the keys and ciphertexts reveal no additional information about both \({\mathbf {x}}\) and \(\mathbf {y}\) beyond their inner product.

In the last few years, there has been a flurry of works on the construction of function-hiding inner product encryption, starting with the work of Bishop, Jain, and Kowalczyk (Asiacrypt 2015) to the more recent work of Tomida, Abe, and Okamoto (ISC 2016). In this work, we focus on the practical applications of this primitive. First, we show that the parameter sizes and the run-time complexity of the state-of-the-art construction can be further reduced by another factor of 2, though we compromise by proving security in the generic group model. We then show that function privacy enables a number of applications in biometric authentication, nearest-neighbor search on encrypted data, and single-key two-input functional encryption for functions over small message spaces. Finally, we evaluate the practicality of our encryption scheme by implementing our function-hiding inner product encryption scheme. Using our construction, encryption and decryption operations for vectors of length 50 complete in a tenth of a second in a standard desktop environment.

The full version of this paper is available at https://eprint.iacr.org/2016/440.pdf.

K. Lewi—Work done while at Stanford University.

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Notes

  1. 1.

    Lin and Vaikuntanathan [47] subsequently showed how to generically boost function-hiding IPE schemes that satisfy this weaker notion of security to one that satisfies the full notion of security. Their generic transformation introduces a factor of 2 overhead in the length of the secret keys and ciphertexts.

  2. 2.

    This setting where the functionality f is fixed in advance is referred to as the single-key setting.

  3. 3.

    This limitation arises because the underlying FE scheme is only secure against “bounded collusions,” i.e., secure if the adversary makes a bounded number of key generation queries. After applying the single-input-to-multi-input transformation, this translates into an additional restriction on the number of ciphertexts the adversary can request.

  4. 4.

    Recently, Lewi and Wu [46] along with Joye and Passelègue [40] independently gave constructions of order-revealing encryption from one-way functions in the small-message-space setting. While the techniques of [46] can be further extended to work for any two-input functionalities, their construction necessarily reveals whether two ciphertexts encrypt the same underlying value. The construction we propose applies more generally to arbitrary two-input functionality without this limitation.

  5. 5.

    On certain classes of pairing curves, there are indeed non-generic attacks [44]. Nonetheless, there still remains a large class of pairing curves where there are no known non-generic attacks that perform significantly better than the generic ones.

  6. 6.

    Our open-source implementation is available at https://github.com/kevinlewi/fhipe.

  7. 7.

    There are many lower bounds [5, 21] for the types of functional encryption that can be achieved under a simulation-based definition in the standard model. These lower bounds do not apply in idealized models such as the generic group model. See Remark 6 for additional details.

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Acknowledgments

This work was supported by NSF, DARPA, the Simons foundation, a grant from ONR, and an NSF Graduate Research Fellowship. Opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA.

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  1. Stanford University, Stanford, USA

    Sam Kim & David J. Wu

  2. Facebook, Menlo Park, USA

    Kevin Lewi

  3. Fujitsu Laboratories of America, Sunnyvale, USA

    Avradip Mandal, Hart Montgomery & Arnab Roy

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Kim, S., Lewi, K., Mandal, A., Montgomery, H., Roy, A., Wu, D.J. (2018). Function-Hiding Inner Product Encryption Is Practical. In: Catalano, D., De Prisco, R. (eds) Security and Cryptography for Networks. SCN 2018. Lecture Notes in Computer Science(), vol 11035. Springer, Cham. https://doi.org/10.1007/978-3-319-98113-0_29

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