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2020 | OriginalPaper | Buchkapitel

Compact Adaptively Secure ABE from k-Lin: Beyond \(\mathsf {NC}^1\) and Towards \(\mathsf {NL}\)

verfasst von : Huijia Lin, Ji Luo

Erschienen in: Advances in Cryptology – EUROCRYPT 2020

Verlag: Springer International Publishing

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Abstract

We present a new general framework for constructing compact and adaptively secure attribute-based encryption (ABE) schemes from k-Lin in asymmetric bilinear pairing groups. Previously, the only construction [Kowalczyk and Wee, Eurocrypt ’19] that simultaneously achieves compactness and adaptive security from static assumptions supports policies represented by Boolean formulae. Our framework enables supporting more expressive policies represented by arithmetic branching programs.

Our framework extends to ABE for policies represented by uniform models of computation such as Turing machines. Such policies enjoy the feature of being applicable to attributes of arbitrary lengths. We obtain the first compact adaptively secure ABE for deterministic and non-deterministic finite automata (DFA and NFA) from k-Lin, previously unknown from any static assumptions. Beyond finite automata, we obtain the first ABE for large classes of uniform computation, captured by deterministic and non-deterministic logspace Turing machines (the complexity classes \(\mathsf {L}\) and \(\mathsf {NL}\)) based on k-Lin. Our ABE scheme has compact secret keys of size linear in the description size of the Turing machine M. The ciphertext size grows linearly in the input length, but also linearly in the time complexity, and exponentially in the space complexity. Irrespective of compactness, we stress that our scheme is the first that supports large classes of Turing machines based solely on standard assumptions. In comparison, previous ABE for general Turing machines all rely on strong primitives related to indistinguishability obfuscation.

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Vorheriges Kapitel Signatures from Sequential-OR Proofs

Nächstes Kapitel Adaptively Secure ABE for DFA from k-Lin and More

Some works call x a set of attributes, and each bit or component of x an attribute. We treat the attribute as a single vector.

More precisely, they construct ABE for read-once branching programs. For general branching programs, one can duplicate each component in the attribute for the number of times it is accessed [43]. As such, the ciphertext size grows linearly with the size of the branching program.

An ABP is specified by a directed graph, with edges weighted by affine functions of the input. The size of an ABP is measured by the number of vertices (instead of edges) in the graph.

Message-hiding encodings [42] are a weaker variant of randomized encodings that allow encoding a public computation f, x with a secret message m such that the encoding reveals m if and only if \(f(x)=1\). Such encodings are succinct if the time to encode is much smaller than the running time of the computation. A pair of ABE secret key for predicate f and ciphertext for attribute x and message m is a message-hiding encoding.

DFA and NFA both characterize regular languages, yet a DFA recognizing a language could have exponentially more states than an NFA recognizing the same language. In this work, by ABE for DFA/NFA, we mean ABE schemes that run in time polynomial in the description size of the finite automata.

As mentioned in Footnote 2, read-once restriction can be circumvented by duplicating attribute components at the cost of losing ciphertext compactness.

Some works use “inner-product encryption” (IPE) to refer to IPFE [17, 25, 49, 50] and some others [24, 39, 52‐55] use it for inner-product predicate encryption.

We can also handle policies of the form \(f_{=0}\) so that \(\mu \) is revealed if and only if \(f(\mathbf{x}) = 0\). For simplicity, we focus on one case in this overview.

For simplicity, in this overview, we assume there is only one accepting state.

This means \(\mathsf {RevSamp}\) is deterministic, and we can reversely sample \(\ell _\text {init}\) in the exponent and when the randomness is not uniform, which is important for our construction.

Our scheme readily extends to be based on k-Lin. However, that makes the scheme more complex to present. We choose to present this scheme using SXDH in this paper.

The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government.

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Titel: Compact Adaptively Secure ABE from k-Lin: Beyond and Towards
verfasst von: Huijia Lin
Ji Luo
Verlag: Springer International Publishing
Buch: Advances in Cryptology – EUROCRYPT 2020
Print ISBN: 978-3-030-45726-6

Electronic ISBN: 978-3-030-45727-3

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-45727-3_9

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