Leakage-resilient \(\textsf {IBE} \)/\(\textsf {ABE} \)  with optimal leakage rates from lattices

We derive the first adaptively secure identity-based encryption ( \(\textsf {IBE} \) $\mathsf{IBE}$ ) and attribute-based encryption ( \(\textsf {ABE} \) $\mathsf{ABE}$ ) for t-conjunctive normal form formula (t-CNF), and selectively secure \(\textsf {ABE} \) $\mathsf{ABE}$ for general circuits from lattices, with \(1-o(1)\) $1-o(1)$ leakage rates, in the both relative leakage model and bounded retrieval model ( \(\textsf {BRM} \) $\mathsf{BRM}$ ). To achieve this, we first identify a new fine-grained security notion for \(\textsf {ABE} \) $\mathsf{ABE}$ —partially adaptive/selective security, and instantiate this notion from the learning with errors ( \(\textsf {LWE} \) $\mathsf{LWE}$ ) assumption. Then, by using this notion, we design a new key compressing mechanism for identity-based/attributed-based weak hash proof system ( \(\textsf {IB} \) $\mathsf{IB}$ / \(\textsf {AB} \) $\mathsf{AB}$ - \(\textsf {wHPS} \) $\mathsf{wHPS}$ ) 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 \(\textsf {IB} \) $\mathsf{IB}$ / \(\textsf {AB} \) $\mathsf{AB}$ - \(\textsf {wHPS} \) $\mathsf{wHPS}$  can yield the desired leakage resilient \(\textsf {IBE} \) $\mathsf{IBE}$ / \(\textsf {ABE} \) $\mathsf{ABE}$ 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 \(\textsf {BRM} \) $\mathsf{BRM}$ .