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Programmable Distributed Point Functions

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13510))

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Abstract

A distributed point function (DPF) is a cryptographic primitive that enables compressed additive sharing of a secret unit vector across two or more parties. Despite growing ubiquity within applications and notable research efforts, the best 2-party DPF construction to date remains the tree-based construction from (Boyle et al., CCS’16), with no significantly new approaches since.

We present a new framework for 2-party DPF construction, which applies in the setting of feasible (polynomial-size) domains. This captures in particular all DPF applications in which the keys are expanded to the full domain. Our approach is motivated by a strengthened notion we put forth, of programmable DPF (PDPF): in which a short, input-independent “offline” key can be reused for sharing many point functions.

  • PDPF from OWF. We construct a PDPF for feasible domains from the minimal assumption that one-way functions exist, where the second “online” key size is polylogarithmic in the domain size N.

Our approach offers multiple new efficiency features and applications:

  • Privately puncturable PRFs. Our PDPF gives the first OWF-based privately puncturable PRFs (for feasible domains) with sublinear keys.

  • O(1)-round distributed DPF Gen. We obtain a (standard) DPF with polylog-size keys that admits an analog of Doerner-shelat (CCS’17) distributed key generation, requiring only O(1) rounds (versus \(\log N\)).

  • PCG with 1 short key. Compressing useful correlations for secure computation, where one key is of minimal size. This provides up to exponential communication savings in some application scenarios.

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Notes

  1. 1.

    Slightly more generally, \(f_{\alpha ,\beta }\) with \(f_{\alpha ,\beta }(\alpha ) = \beta \) for \(\beta \in \{0,1\}\).

  2. 2.

    Or rather, \(\lambda \) bits, where \(\lambda \) is the security parameter.

  3. 3.

    In fact, the naive construction, as mentioned in Sect. 1.1, can provide a negligible privacy error for small output groups. Nevertheless, in aggregation-type applications, over output group \(\mathbb {Z}\), we get a constant privacy error. See Remark 3 for more details.

  4. 4.

    DPFs with significantly worse key size \(N^\epsilon \) for constant \(\epsilon > 0\) can be built with lower depth \(\textsf{Gen}\), e.g. by “flattening” the tree structure of current best DPF constructions.

  5. 5.

    In this approach, to get statistical error of \(\epsilon \) we need to reduce the value of \(\epsilon \) in each of the \(\ell \) instances by a factor of \(\ell \). Since the computational cost per instance depends quadratically on \(1/\epsilon \), this results in a total slowdown (compared to the 1-bit baseline) of \(\ell \cdot \ell ^2=\ell ^3\).

  6. 6.

    To account for the fact that the payload could be \(\beta =0\), we actually introduce dummy bucket \(N+1\) to the PRF output space; removing a ball from this bucket means that all [N] buckets remain equal across parties.

  7. 7.

    Note that d is non-cryptographic. Concretely, for the case of Reed-Muller locally decodable codes, the mapping d corresponds to effectively generating Shamir secret shares of the input value \(\alpha \).

  8. 8.

    This secret sharing can be over \(\mathbb {Z}_N\), bitwise over \(\mathbb {Z}_2\), or otherwise, with insignificant effect for the given protocol. We describe w.r.t. shares over \(\mathbb {Z}_N\) for simplicity.

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Acknowledgements

We thank the CRYPTO reviewers for many useful comments and suggestions, including a simplification of the proof of Theorem 4. Elette Boyle was supported by AFOSR Award FA9550-21-1-0046, ERC Project HSS (852952), ERC Project NTSC (742754), and a Google Research Scholar Award. Niv Gilboa was supported by ISF grant 2951/20, ERC grant 876110, and a grant by the BGU Cyber Center. Yuval Ishai was supported by ERC Project NTSC (742754), BSF grant 2018393, and ISF grant 2774/20. Victor I. Kolobov was supported by ERC Project NTSC (742754) and ISF grant 2774/20.

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Authors and Affiliations

  1. IDC Herzliya, Herzliya, Israel

    Elette Boyle

  2. NTT Research, Sunnyvale, USA

    Elette Boyle

  3. Ben-Gurion University, Be’er Sheva, Israel

    Niv Gilboa

  4. Technion, Haifa, Israel

    Yuval Ishai & Victor I. Kolobov

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  1. Elette Boyle
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  4. Victor I. Kolobov
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Correspondence to Victor I. Kolobov .

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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Boyle, E., Gilboa, N., Ishai, Y., Kolobov, V.I. (2022). Programmable Distributed Point Functions. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13510. Springer, Cham. https://doi.org/10.1007/978-3-031-15985-5_5

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