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Trapdoor Hash Functions and Their Applications

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Abstract

We introduce a new primitive, called trapdoor hash functions (TDH), which are hash functions \(\mathsf {H}: \{0,1\}^n \rightarrow \{0,1\}^\lambda \) with additional trapdoor function-like properties. Specifically, given an index \(i\in [n]\), TDHs allow for sampling an encoding key \(\mathsf {ek}\) (that hides i) along with a corresponding trapdoor. Furthermore, given \(\mathsf {H}(x)\), a hint value \(\mathsf {E}(\mathsf {ek},x)\), and the trapdoor corresponding to \(\mathsf {ek}\), the \(i^{th}\) bit of x can be efficiently recovered. In this setting, one of our main questions is: How small can the hint value \(\mathsf {E}(\mathsf {ek},x)\) be? We obtain constructions where the hint is only one bit long based on DDH, QR, DCR, or LWE.

This primitive opens a floodgate of applications for low-communication secure computation. We mainly focus on two-message protocols between a receiver and a sender, with private inputs x and y, resp., where the receiver should learn f(x, y). We wish to optimize the (download) rate of such protocols, namely the asymptotic ratio between the size of the output and the sender’s message. Using TDHs, we obtain:

  1. 1.

    The first protocols for (two-message) rate-1 string OT based on DDH, QR, or LWE. This has several useful consequences, such as:

    1. (a)

      The first constructions of PIR with communication cost poly-logarithmic in the database size based on DDH or QR. These protocols are in fact rate-1 when considering block PIR.

    2. (b)

      The first constructions of a semi-compact homomorphic encryption scheme for branching programs, where the encrypted output grows only with the program length, based on DDH or QR.

    3. (c)

      The first constructions of lossy trapdoor functions with input to output ratio approaching 1 based on DDH, QR or LWE.

    4. (d)

      The first constant-rate LWE-based construction of a 2-message “statistically sender-private” OT protocol in the plain model.

  2. 2.

    The first rate-1 protocols (under any assumption) for n parallel OTs and matrix-vector products from DDH, QR or LWE.

We further consider the setting where f evaluates a RAM program y with running time \(T\ll |x|\) on x. We obtain the first protocols with communication sublinear in the size of x, namely \(T\cdot \sqrt{|x|}\) or \(T\cdot \root 3 \of {|x|}\), based on DDH or, resp., pairings (and correlated-input secure hash functions).

G. Malavolta—Part of the work done while at Friedrich-Alexander-Universität Erlangen-Nürnberg.

T. Mour—Part of the work done while at Technion.

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Notes

  1. 1.

    It seems plausible that these gaps can be closed using indistinguishability obfuscation [25]. However, the focus of this work is on constructions that can be based on more traditional assumptions.

  2. 2.

    A simple “hybrid FHE” technique [28] can generically convert any FHE scheme into one whose encrypted (long) input is roughly as long as the input. However, no such generic technique is known for compressing the encrypted output.

  3. 3.

    Our protocols can be efficiently extended to provide security against malicious parties (under the same assumptions) using sublinear arguments [46]. This increases the number of rounds, but does not affect the asymptotic communication.

  4. 4.

    The work of Cho et al. [16] on laconic OT gives a batch OT protocol where \(|\mathsf {msg}_1|\) is independent of |x|. This generalizes to arbitrary functions f; however, even in the simple case of batch OT the download rate is sub-constant. The same applies to the more recent work of Quach et al. [52] on laconic function evaluation.

  5. 5.

    In this work, we consider this question in the more stringent two-message setting. However, we note that no other protocols with rate \(> \frac{1}{2}\) are known even when additional rounds of communication are allowed.

  6. 6.

    The notion of trapdoor hash functions is inspired by the closely related notion of hash encryption [13, 21, 22, 26] and somewhere statistically binding hash functions [36, 43, 49].

  7. 7.

    Our protocols achieve asymptotic download rate 1, which is clearly optimal. However, the (additive) difference between the sender’s message length and the output length grows with the security parameter \(\lambda \). In the full version we show that that this gap is necessary even in the more liberal setting of secure computation with preprocessing.

  8. 8.

    More specifically, as our group-based constructions are black-box in the underlying group, we can count the communication complexity in terms of the number of group elements, which in our case is \(\log n\cdot \mathsf {poly}(\lambda )\), where n is the size of the database and \(\lambda \) is the security parameter.

  9. 9.

    For this application, we insist on the two-message setting. Allowing O(T) rounds of interaction, similar protocols can be based on any single-server PIR scheme [46].

  10. 10.

    We can ensure that both sender and receiver run in strict polynomial time by introducing a suitable polynomial upper bound for the number of trials in the exhaustive search step of \(\mathsf {Dist}(\cdot )\). For small erasure probabilities, a near-quadratic improvement in the running time can be obtained via the recent optimal “distributed discrete log” algorithm of Dinur, Keller, and Klein [19].

  11. 11.

    In fact, string OT can be thought of as a special case of batch OT, where all the choice bits \(i_j\) are equal.

  12. 12.

    In fact they require an OT protocol with a strong notion of sender privacy, which is satisfied by all of our constructions.

  13. 13.

    for the SXDH-based construction we further assume the existence of correlated-input secure hash for group-induced correlations over bilinear groups [3, 32].

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Acknowledgments

We thank Craig Gentry, Shai Halevi, Srinath Setty, and Vinod Vaikuntanathan for helpful discussions and pointers.

S. Garg supported by DARPA/ARL SAFEWARE Award W911NF15C0210, AFOSR Award FA9550-15-1-0274, AFOSR YIP Award, DARPA and SPAWAR under contract N66001-15-C-4065, a Hellman Award and research grants by the Okawa Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). The views expressed are those of the author and do not reflect the official policy or position of the funding agencies.

Y. Ishai supported by ERC Project NTSC (742754), ISF grant 1709/14, NSF-BSF grant 2015782, and a grant from the Ministry of Science and Technology, Israel and Department of Science and Technology, Government of India.

G. Malavolta supported by a gift from Ripple, a gift from DoS Networks, a grant from Northrop Grumman, a Cylab seed funding award, and a JP Morgan Faculty Fellowship.

T. Mour supported by BSF grant 2012378, and NSF-BSF grant 2015782.

R. Ostrovsky supported by NSF grant 1619348, BSF grant 2015782, DARPA SafeWare subcontract to Galois Inc., DARPA SPAWAR contract N66001-15-C-4065, JP Morgan Faculty Research Award, OKAWA Foundation Research Award, IBM Faculty Research Award, Xerox Faculty Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. The views expressed are those of the authors and do not reflect position of the Department of Defense or the U.S. Government.

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Authors and Affiliations

  1. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Nico Döttling

  2. UC Berkeley, Berkeley, CA, USA

    Sanjam Garg

  3. Technion, Haifa, Israel

    Yuval Ishai

  4. Carengie Mellon University, Pittsburgh, PA, USA

    Giulio Malavolta

  5. Weizmann Institute of Science, Rehovot, Israel

    Tamer Mour

  6. UCLA, Los Angeles, CA, USA

    Rafail Ostrovsky

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  1. Nico Döttling
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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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Döttling, N., Garg, S., Ishai, Y., Malavolta, G., Mour, T., Ostrovsky, R. (2019). Trapdoor Hash Functions and Their Applications. 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_1

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