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New Techniques for Efficient Trapdoor Functions and Applications

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Book cover Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11478))

Abstract

We develop techniques for constructing trapdoor functions (TDFs) with short image size and advanced security properties. Our approach builds on the recent framework of Garg and Hajiabadi [CRYPTO 2018]. As applications of our techniques, we obtain

  • The first construction of deterministic-encryption schemes for block-source inputs (both for the CPA and CCA cases) based on the Computational Diffie-Hellman (CDH) assumption. Moreover, by applying our efficiency-enhancing techniques, we obtain CDH-based schemes with ciphertext size linear in plaintext size.

  • The first construction of lossy TDFs based on the Decisional Diffie-Hellman (DDH) assumption with image size linear in input size, while retaining the lossiness rate of [Peikert-Waters STOC 2008].

Prior to our work, all constructions of deterministic encryption based even on the stronger DDH assumption incurred a quadratic gap between the ciphertext and plaintext sizes. Moreover, all DDH-based constructions of lossy TDFs had image size quadratic in the input size.

At a high level, we break the previous quadratic barriers by introducing a novel technique for encoding input bits via hardcore output bits with the use of erasure-resilient codes. All previous schemes used group elements for encoding input bits, resulting in quadratic expansions.

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Notes

  1. 1.

    We note that building a TDF providing mere one-wayness with linear-size images is simple: if \(\mathsf {TDF}.\mathsf {F} (\mathsf {ik}, \cdot )\) maps n-bit inputs to \(n^c\)-bit outputs, define \(\mathsf {TDF}.\mathsf {F} '(\mathsf {ik}, \mathsf {x}|| \mathsf {x}')\), where \(|\mathsf {x}| = n\) and \(|\mathsf {x}'| = n^c\), as \(\mathsf {TDF}.\mathsf {F} (\mathsf {ik}, \mathsf {x}) || \mathsf {x}'\). Although this transformation results in TDFs with linear-image size, it destroy more advanced properties such as CCA2 security, deterministic-encryption security and the lossiness rate.

  2. 2.

    This is the indistinguishability-based, single-message version of their notion, which as they show, is equivalent to the multiple-message version both for the indistinguishability- and simulation-based definitions.

  3. 3.

    We note that this lossiness property is weaker than the one of [PW08, PW11], but it can be realized under CDH. We will later show efficient DDH-based instantiations of lossiness in the sense of [PW08, PW11].

  4. 4.

    We mention that the transformation of [RS09] results in CCA-secure PKE schemes which use randomness, but this can be avoided by using the techniques of [KMO10] to get CCA2-secure TDFs.

  5. 5.

    We have not yet optimized nor tried to get some upper bounds on the constants.

  6. 6.

    \(\mathsf {ct} \) is assumed to contain (i, b).

  7. 7.

    The choices of the constants were made as above so to have slackness in proofs—they have not been optimized for efficiency.

  8. 8.

    For simplicity assume \(g_1 \ne g_{1,b}^r\), hence we will not have a hung situation.

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Acknowledgements

We thank Xiao Liang for suggesting a simplification to Construction 3. We are also grateful to the anonymous reviewers for their comments. Research supported in part from 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), and a Google PhD fellowship. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies.

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Authors and Affiliations

  1. University of California, Berkeley, USA

    Sanjam Garg & Mohammad Hajiabadi

  2. Département informatique de l’ENS, École normale supérieure, CNRS, PSL University, 75005, Paris, France

    Romain Gay

  3. INRIA, Paris, France

    Romain Gay

  4. University of Virginia, Charlottesville, USA

    Mohammad Hajiabadi

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  1. Sanjam Garg
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Correspondence to Sanjam Garg .

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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Garg, S., Gay, R., Hajiabadi, M. (2019). New Techniques for Efficient Trapdoor Functions and Applications. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11478. Springer, Cham. https://doi.org/10.1007/978-3-030-17659-4_2

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