Abstract
We propose the concept of quasi-adaptive hash proof system (QAHPS), where the projection key is allowed to depend on the specific language for which hash values are computed. We formalize leakage-resilient(LR)-ardency for QAHPS by defining two statistical properties, including LR-\(\langle \mathscr {L}_0, \mathscr {L}_1 \rangle \)-universal and LR-\(\langle \mathscr {L}_0, \mathscr {L}_1 \rangle \)-key-switching.
We provide a generic approach to tightly leakage-resilient CCA (LR-CCA) secure public-key encryption (PKE) from LR-ardent QAHPS. Our approach is reminiscent of the seminal work of Cramer and Shoup (Eurocrypt’02), and employ three QAHPS schemes, one for generating a uniform string to hide the plaintext, and the other two for proving the well-formedness of the ciphertext. The LR-ardency of QAHPS makes possible the tight LR-CCA security. We give instantiations based on the standard k-Linear (k-LIN) assumptions over asymmetric and symmetric pairing groups, respectively, and obtain fully compact PKE with tight LR-CCA security. The security loss is \({{O}}(\log {Q_{{e}}})\) where \({Q_{{e}}}\) denotes the number of encryption queries. Specifically, our tightly LR-CCA secure PKE instantiation from SXDH has only 4 group elements in the public key and 7 group elements in the ciphertext, thus is the most efficient one.
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Notes
- 1.
Gay et al. [19] constructed the state-of-the-art tightly secure (structure-preserving) signature schemes, where the signature is comprised of 14 group elements. By applying the framework in [2, 25], this signature scheme can be transformed to a tightly secure SS-NIZK/tSE-NIZK whose proof contains around 40 group elements.
- 2.
The properties of “constrained soundness” and “extensibility” of QPS are needed for the tight IND-CCA security proof of the PKE proposed by Gay et al. [18]. We note that these two properties of their QPS are unlikely to hold when partial information about the secret key of QPS is leaked to adversary. See our full version [22] for more details. Thus it is reasonable to conjecture that their PKE is not LR-CCA secure.
- 3.
To the best of our knowledge, the PKE scheme in [2] is the only tightly LR-CCA secure one prior to our work.
- 4.
Here \(\mathcal {L}_{\rho _0}\) is from another language collection \(\mathscr {L}_0\) and only appears in the security proof. The same is true for \(\mathcal {L}_{\rho _1}\) and \(\mathscr {L}_1\), as shown later.
- 5.
Note that for the instance \(x^* \in \mathcal {L}_{\rho _0} \cup \mathcal {L}_{\rho _1}\) in challenge ciphertext, the bit indicating whether \(x^* \in \mathcal {L}_{\rho _0}\) or \(x^* \in \mathcal {L}_{\rho _1}\) is consistent with the \((i+1)\)-th bit of \({{ctr}}\), i.e., \(x^* \in \mathcal {L}_{\rho _0}\) if \({{ctr}}_{i+1} = 0\) and \(x^* \in \mathcal {L}_{\rho _1}\) if \({{ctr}}_{i+1} = 1\). But this might not be true for the instances \(x \in \mathcal {L}_{\rho _0} \cup \mathcal {L}_{\rho _1}\) in the decryption queries. This problem is circumvented by borrowing the trick from [18, 24]. We refer to the main body for details.
- 6.
Quasi-adaptiveness of HPS was discussed in [27]. Here we give a formal definition of QAHPS and build our novel LR-ardency notion over it.
- 7.
In fact, this condition can be weakened by only requiring \(\widehat{\varPi }\) and \(\widetilde{\varPi }\) to be subsets of an (additive) group.
- 8.
For technical reasons, the zero vector \([\mathbf {0}]_1\) (resp. \([\mathbf {0}]_2\)) must be excluded from \({\mathsf {span}( [\mathbf {A}_1]_1 )}\) and \(\mathbb {G}_1^{\ell }\) (resp. \({\mathsf {span}( [\mathbf {A}_2]_2 )}\) and \(\mathbb {G}_2^{\ell }\)). For the sake of simplicity, we forgo making this explicit in the sequel.
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Acknowledgments
We would like to thank the anonymous reviewers for their comments and suggestions. We are grateful to Dennis Hofheinz and Jiaxin Pan for helpful discussions and advices. Shuai Han, Shengli Liu and Lin Lyu are supported by the National Natural Science Foundation of China Grant (No. 61672346). Dawu Gu is supported by the National Natural Science Foundation of China Grant (No. U1636217) together with Program of Shanghai Academic Research Leader (16XD1401300). Shuai Han is also supported by the National Natural Science Foundation of China Grant (No. 61802255).
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Han, S., Liu, S., Lyu, L., Gu, D. (2019). Tight Leakage-Resilient CCA-Security from Quasi-Adaptive Hash Proof System. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_15
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