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Multi-Client Functional Encryption for Linear Functions in the Standard Model from LWE

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Advances in Cryptology – ASIACRYPT 2019 (ASIACRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11923))

Abstract

Multi-client functional encryption (MCFE) allows \(\ell \) clients to encrypt ciphertexts \((\mathbf {C}_{t,1},\mathbf {C}_{t,2},\ldots ,\mathbf {C}_{t,\ell })\) under some label. Each client can encrypt his own data \(X_i\) for a label t using a private encryption key \(\mathsf {ek}_i\) issued by a trusted authority in such a way that, as long as all \(\mathbf {C}_{t,i}\) share the same label t, an evaluator endowed with a functional key \(\mathsf {dk}_f\) can evaluate \(f(X_1,X_2,\ldots ,X_\ell )\) without learning anything else on the underlying plaintexts \(X_i\). Functional decryption keys can be derived by the central authority using the master secret key. Under the Decision Diffie-Hellman assumption, Chotard et al. (Asiacrypt 2018) recently described an adaptively secure MCFE scheme for the evaluation of linear functions over the integers. They also gave a decentralized variant (DMCFE) of their scheme which does not rely on a centralized authority, but rather allows encryptors to issue functional secret keys in a distributed manner. While efficient, their constructions both rely on random oracles in their security analysis. In this paper, we build a standard-model MCFE scheme for the same functionality and prove it fully secure under adaptive corruptions. Our proof relies on the Learning-With-Errors (\(\mathsf {LWE}\)) assumption and does not require the random oracle model. We also provide a decentralized variant of our scheme, which we prove secure in the static corruption setting (but for adaptively chosen messages) under the \(\mathsf {LWE}\) assumption.

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Notes

  1. 1.

    While their decentralized scheme is only proved secure under static corruptions, its centralized version is proved secure under adaptive corruptions.

  2. 2.

    Introduced in [12], the notation \(\lfloor x \rfloor _p\) stands for the rounded value \(\lfloor (p/q) \cdot x \rfloor \in \mathbb {Z}_p\), where \( x \in \mathbb {Z}_q\), and \(p<q\).

  3. 3.

    The standard Leftover Hash Lemma cannot be applied since the source \(\lfloor \mathbf {A}(x^\star )^\top \cdot \mathbf {s} \rfloor _p\) is not guaranteed to be independent of the seed. A deterministic extractor based on k-wise independent functions [31] is thus needed in [55].

  4. 4.

    In [55], the multiplication of ciphertexts \(\{\mathbf {A}_{i,\tau [i]}\}_{i=1}^L\) was computed in a parallel fashion \(\mathbf {A}_{0} \cdot \prod _{i=1}^L \mathbf {G}^{-1}(\mathbf {A}_{i,\tau [i]})\) because their initial proof required the matrices \(\{ \mathbf {A}_{i,b}\}_{i,b}\) to be generated in such a way that \(\mathbf {G}^{-1}(\mathbf {A}_{i,b})\) was invertible over \(\mathbb {Z}_q\).

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Acknowledgements

Part of this work was funded by the French ANR ALAMBIC project (ANR-16-CE39-0006) and by BPI-France in the context of the national project RISQ (P141580). This work was also supported by the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701).

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Authors and Affiliations

  1. CNRS, Laboratoire LIP, Lyon, France

    Benoît Libert

  2. ENS de Lyon, Laboratoire LIP (U. Lyon, CNRS, ENSL, INRIA, UCBL), Lyon, France

    Benoît Libert & Radu Ţiţiu

  3. Bitdefender, Bucharest, Romania

    Radu Ţiţiu

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  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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Libert, B., Ţiţiu, R. (2019). Multi-Client Functional Encryption for Linear Functions in the Standard Model from LWE. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11923. Springer, Cham. https://doi.org/10.1007/978-3-030-34618-8_18

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