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ABE for DFA from LWE Against Bounded Collusions, Revisited

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Theory of Cryptography (TCC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13043))

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Abstract

We present a new public-key ABE for DFA based on the LWE assumption, achieving security against collusions of a-priori bounded size. Our scheme achieves ciphertext size \(\tilde{O}(\ell + B)\) for attributes of length \(\ell \) and collusion size B. Prior LWE-based schemes has either larger ciphertext size \(\tilde{O}(\ell \cdot B)\), or are limited to the secret-key setting. Along the way, we introduce a new technique for lattice trapdoor sampling, which we believe would be of independent interest. Finally, we present a simple candidate public-key ABE for DFA for the unbounded collusion setting.

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Notes

  1. 1.

    To facilitate comparison with Waters’ pairing-based scheme, we note that the terms corresponding to \(\mathbf {c}_i\) and \(\mathbf {k}_{u,\sigma }\) there-in are given by:

    $$(g_1^{s_{i-1}}, g_1^{s_{i-1} z + s_i w_{x_i}}, g_1^{s_i}), \;\; (g_2^{-\tilde{d}_{u}+zr}, g_2^r, g_2^{-\tilde{d}_{v}+w_{\sigma }r})$$

    where \(g_1,g_2\) are the respective generators the group \(\mathbb {G}_1,\mathbb {G}_2\) in a bilinear group \(e : \mathbb {G}_1 \times \mathbb {G}_2 \rightarrow \mathbb {G}_T\). We can then compute a pairing-product over these terms to derive \(e(g_1,g_2)^{s_{i-1} \tilde{d}_{u_{i-1}} -s_{i} \tilde{d}_{u_{i}}}\).

  2. 2.

    It seems plausible (with some considerable changes to the scheme and the proof) that we can replace \(\mathsf {cFE}\) for depth \(O(\log Q)\) circuits with an ABE for branching programs of size \(\textsf {poly}(Q)\). The latter can realized from LWE with a polynomial modulus-to-noise ratio [16, 17].

  3. 3.

    Following [22, Section 5.2], we choose each entry of \(\mathbf {R}\) to be 0 with probability 1/2, and \(\pm 1\) each with probability 1/4. This yields \(|\mathbf {R}| = 1\) and \(s_1(\mathbf {R}) = O(\sqrt{m})\) w.h.p. Moreover, \((\mathbf {A},\mathbf {A}\mathbf {R}) \approx _s \text{ uniform }\).

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Acknowledgments

I would like to thank Yilei Chen and Vinod Vaikuntanathan for illuminating discussions on lattice trapdoor sampling, as well as the reviewers for meticulous and constructive feedback.

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  1. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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  1. Hoeteck Wee
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Correspondence to Hoeteck Wee .

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  1. Georgetown University, Washington, WA, USA

    Kobbi Nissim

  2. The University of Texas at Austin, Austin, TX, USA

    Brent Waters

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Wee, H. (2021). ABE for DFA from LWE Against Bounded Collusions, Revisited. In: Nissim, K., Waters, B. (eds) Theory of Cryptography. TCC 2021. Lecture Notes in Computer Science(), vol 13043. Springer, Cham. https://doi.org/10.1007/978-3-030-90453-1_10

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