Abstract
We derive the first adaptively secure \(\mathsf {IBE}\) and \(\mathsf {ABE}\) for t-CNF, and selectively secure \(\mathsf {ABE}\) for general circuits from lattices, with \(1-o(1)\) leakage rates, in the both relative leakage model and bounded retrieval model (\(\mathsf {BRM}\)).
To achieve this, we first identify a new fine-grained security notion for \(\mathsf {ABE}\) – partially adaptive/selective security, and instantiate this notion from \(\mathsf {LWE}\). Then, by using this notion, we design a new key compressing mechanism for identity-based/attributed-based weak hash proof system (\(\mathsf {IB}\)/\(\mathsf {AB}\)-\(\mathsf {wHPS}\)) for various policy classes, achieving (1) succinct secret keys and (2) adaptive/selective security matching the existing non-leakage resilient lattice-based designs. Using the existing connection between weak hash proof system and leakage resilient encryption, the succinct-key \(\mathsf {IB}\)/\(\mathsf {AB}\)-\(\mathsf {wHPS}\) can yield the desired leakage resilient \(\mathsf {IBE}\)/\(\mathsf {ABE}\) schemes with the optimal leakage rates in the relative leakage model. Finally, by further improving the prior analysis of the compatible locally computable extractors, we can achieve the optimal leakage rates in the \(\mathsf {BRM}\).
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Notes
- 1.
This is the dual class of t-\(\mathsf {CNF} \) where the function is an assignment x and attribute is a description of t-\(\mathsf {CNF} \). We use the dual class as we are working on Key-policy ABE while the prior work [38] worked on Ciphertext-policy ABE.
- 2.
Notice that in the above experiment \(\mathbf {Exp}_{\mathsf {ABE},\mathcal {A}}^{\mathsf{LR}}(\lambda ,\ell ,\omega )\), we allow the adversary to interleave key queries in Test Stage 1 and leakage queries in \(\omega \)-Leakage queries Stage, in an arbitrary way.
- 3.
For the case that \(\mathsf {sk}:=S= (S_1,\dots , S_m)\) is an \(m \times e\) block source as in [39], we define leakage functions \(f_i:\{0,1\}^*\rightarrow \{0,1\}^{\ell }\) independently for each block \(S_i\) with all \(i\in [m]\). We say \((f_1,\ldots ,f_m)\) are block leakage functions, if the min-entropy of \(S_i\) is still large enough even given leakage \((f_1(S_1),\ldots ,f_{i-1}(S_{i-1}))\) for any \(i\in [m]\). Clearly, when \(m=1\), this is the trivial case in Definition 2.2. Here, we call \(\frac{m\ell }{|\mathsf {sk} |}\) the block leakage rate of the corresponding scheme.
- 4.
We use a “dual” variant of the \(\mathsf {CNF} \) functions as we discussed in the introduction. The formal definition is presented in Sect. 2.1.
- 5.
Recall that the function s(f) denotes the size of the extra part of the secret key, excluding the description of the function.
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Acknowledgements
We would like to thank the reviewers of PKC 2022 for their insightful advices. Qiqi Lai is supported by the National Natural Science Foundation of China (62172266, 61802241, U2001205), the National Cryptography Development Foundation during the 13th Five-year Plan Period (MMJJ20180217), and the Fundamental Research Funds for the Central Universities (GK202103093).
Feng-Hao Liu and Zhedong Wang are supported by the NSF Career Award CNS-1942400.
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Lai, Q., Liu, FH., Wang, Z. (2022). Leakage-Resilient \(\mathsf {IBE}\)/\(\mathsf {ABE}\) with Optimal Leakage Rates from Lattices. In: Hanaoka, G., Shikata, J., Watanabe, Y. (eds) Public-Key Cryptography – PKC 2022. PKC 2022. Lecture Notes in Computer Science(), vol 13178. Springer, Cham. https://doi.org/10.1007/978-3-030-97131-1_8
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