Abstract
In this work, we first present general methods to construct information rate-1 PKE that is \(\mathsf {KDM}^{(n)}\)-secure with respect to block-affine functions for any unbounded polynomial n. To achieve this, we propose a new notion of extractor that satisfies reusability, homomorphic, and security against correlated-source attacks, and show how to use this extractor to improve the information rate of the \(\mathsf {KDM}\)-secure PKE of Brakerski et al. (Eurocrypt 18). Then, we show how to amplify \(\mathsf {KDM}\) security from block-affine function class into general bounded size circuits via a variant of the technique of Applebaum (Eurocrypt 11), achieving better efficiency. Furthermore, we show how to generalize these approaches to the IBE setting.
Additionally, our PKE and IBE schemes are also leakage resilient, with leakage rates \(1-o(1)\) against a slightly smaller yet still general class – block leakage functions. We can instantiate the required building blocks from \(\mathsf {LWE}\) or \(\mathsf {DDH}\).
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- 1.
Information rate is defined as the message-to-ciphertext ratio when one encrypts sufficiently long plaintexts.
- 2.
When the secret key is stored in blocks, a block leakage function can leak individual blocks one after another, as long as the blocks still remain a block source.
- 3.
The extractor can extract uniform string (up to statistical distance \(\varepsilon \)) for any source with min-entropy k.
- 4.
Clearly, this notion cannot be achieved unconditionally, as an information-theoretic extractor requires (conditional) min-entropy from the source, which would be exhausted after a bounded number of extractions.
- 5.
Clearly, this notion is stronger than the reusable extractor, which can be viewed as a special case where \(\mathfrak {g}_i\)’s are all the identity function. Thus, this notion is only possible under computational assumptions.
- 6.
Here, t denotes the number of times the weak source being reused.
- 7.
Just as described in full version, block leakage means that each block of source is leaked by an independent function and remain enough entropy conditioned on leakage against other blocks.
- 8.
In [4], Applebaum leverages the abstract notion of randomized encoding to achieve \(\mathsf {KDM}\) amplification. Here, we directly amplify our scheme through using Garbled Circuits, which is a well-known instantiation of randomized encoding.
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Acknowledgements
We would like to thank the anonymous reviewers of PKC 2021 for their insightful advices. Qiqi Lai is supported by the National Key R&D Program of China (2017YFB0802000), the National Natural Science Foundation of China (61802241, U2001205, 61772326, 61802242), the Natural Science Basic Research Plan in Shaanxi Province of China (2019JQ-360), 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 an NSF Award CNS-1657040 and an NSF Career Award CNS-1942400. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsors.
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Lai, Q., Liu, FH., Wang, Z. (2021). Rate-1 Key-Dependent Message Security via Reusable Homomorphic Extractor Against Correlated-Source Attacks. In: Garay, J.A. (eds) Public-Key Cryptography – PKC 2021. PKC 2021. Lecture Notes in Computer Science(), vol 12710. Springer, Cham. https://doi.org/10.1007/978-3-030-75245-3_16
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