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Adaptively secure revocable hierarchical IBE from k-linear assumption

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Abstract

Revocable identity-based encryption (RIBE) is an extension of IBE with an efficient key revocation mechanism. Revocable hierarchical IBE (RHIBE) is its further extension with key delegation functionality. Although there are various adaptively secure pairing-based RIBE schemes, all known hierarchical analogues only satisfy selective security. In addition, the currently known most efficient adaptively secure RIBE and selectively secure RHIBE schemes rely on non-standard assumptions, which are referred to as the augmented DDH assumption and q-type assumptions, respectively. In this paper, we propose a simple but effective design methodology for RHIBE schemes. We provide a generic design framework for RHIBE based on an HIBE scheme with a few properties. Fortunately, several state-of-the-art pairing-based HIBE schemes have the properties. In addition, our construction preserves the sizes of master public keys, ciphertexts, and decryption keys, as well as the complexity assumptions of the underlying HIBE scheme. Thus, we obtain the first RHIBE schemes with adaptive security under the standard k-linear assumption. We prove adaptive security by developing a new proof technique for RHIBE. Due to the compactness-preserving construction, the proposed R(H)IBE schemes have similar efficiencies to the most efficient existing schemes.

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Notes

  1. To be precise, these works are prior to Katsumata et al. [24]; thus, they do not consider the stronger DKER.

  2. Note that we consider insider security as the minimum requirement. The definition and necessity of insider security is discussed in Sect. 1.3.

  3. Although the security of Seo and Emura’s original scheme is based on the q-weak bilinear Diffie-Hellman inversion assumption, that of the modified scheme is based on the q-bilinear Diffie-Hellman exponent assumption [45].

  4. Recall that the difference between the weaker and the stronger DKER only appears in the hierarchical setting. Thus, the approach proposed by Watanabe et al. does not provide a pathway to achieve the stronger DKER.

  5. In other words, each parent user \(\textsf {pa}(\mathtt {ID})\) updates \(\textsf {delk}_{\textsf {pa}(\mathtt {ID})} = ( \textsf {delk}_{\textsf {pa}(\mathtt {ID}), \theta } )_{\theta \in \mathcal {AN}_{\textsf {pa}(\mathtt {ID})}}\) by adding \(\textsf {delk}_{\textsf {pa}(\mathtt {ID}), \theta '}\) when they create \(\textsf {sk}_{\mathtt {ID}, \theta '}\) and/or \(\textsf {ku}_{\textsf {pa}(\mathtt {ID}), \mathtt {T}, \theta '}\) for \(\theta ' \notin \mathcal {AN}_{\textsf {pa}(\mathtt {ID})}. \) This procedure also updates \(\mathcal {AN}_{\textsf {pa}(\mathtt {ID})} \leftarrow \mathcal {AN}_{\textsf {pa}(\mathtt {ID})} \cup \{\theta '\}\).

  6. Note that there may be an adversary that does not receive \(\textsf {sk}_\mathtt {ID}\) for any \(\mathtt {ID}\in \textsf {prefix}^+(\mathtt {ID}^\star )\). Here, we ignore such adversaries and assume that an adversary always receives \(\textsf {sk}_\mathtt {ID}\) for some \(\mathtt {ID}\in \textsf {prefix}^+(\mathtt {ID}^\star )\).

  7. Otherwise, the adversary can create \(\textsf {dk}_{\mathtt {ID}^\star ,\mathtt {T}^\star }\) from \(\textsf {ku}_{\textsf {pa}(\mathtt {ID}^\star _{[\ell ]}), \mathtt {T}^\star , \theta }\) and \(\textsf {sk}_{\mathtt {ID}^\star _{[\ell ]}, \theta }\). Note that, according to the definition of \(\ell ^\star \), the adversary can receive \(\textsf {ku}_{\textsf {pa}(\mathtt {ID}^\star _{[ \ell ]}), \mathtt {T}^\star , \theta }\) even if it shares the same \(\theta \) with \(\textsf {sk}_{\mathtt {ID}^\star _{[\ell ]}, \theta }\) for \(\ell \in [\ell ^\star - 1]\) because the adversary does not know the secret keys.

  8. Note that the original \(\mathsf {JR}\) IBE, which is secure under the SXDH assumption, is not compatible with the reduction. Here, Watanabe et al. had to modify the scheme and rely on the non-standard variant of the DDH assumption.

  9. Note that we can avoid this issue by guessing \(\mathtt {T}^\star \) with reduction loss \(|\mathcal {T}|\) as the proofs of the above adaptively secure RIBE. However, our construction does not require the guess; thus, it avoids this issue. That is why the proposed RIBE scheme achieves tighter reduction than the \(\mathsf {WES}\) RIBE [52] and its variant [19], as shown in Table 2.

  10. If \(|\mathtt {ID}'| = L\), this step is skipped.

  11. Here, \(\textsf {sk}_{\mathtt {ID}}\) is the latest secret key, i.e., the result of Step (2).

  12. This check ensures that previously revoked identities remain revoked in the next time period.

  13. This is the condition of the stronger DKER. The condition of the weaker DKER is replaced by \((\mathtt {ID}, \mathtt {T}) \ne (\mathtt {ID}', \mathtt {T}^\star )\) for all \(\mathtt {ID}' \in \textsf {prefix}^+(\mathtt {ID}^\star )\).

  14. In this sense, our notion is more closely related to wicked IBE rather than wildcarded IBE because wildcards are associated with a ciphertext/secret key in wildcarded/wicked IBE. Nonetheless, we use “wildcarded” to express this intuitive property.

  15. Shacham define an HIBE secret key \(\textsf {H}.\textsf {sk}_\mathtt {ID}\) with limited delegation such that it can derive a secret key \(\textsf {H}.\textsf {sk}_{\mathtt {ID}'}\) of a suffix identity \(\mathtt {ID}'\) only when \(|\mathtt {ID}'|\) is less than or equal to limited bound \(|\mathtt {ID}'| = L' < L\).

  16. To be precise, we use Chen-Gay-Wee’s instantiation of a dual system group [12] as Chen-Wee’s HIBE scheme. In addition, the scheme proposed by Gong et al. satisfies prefix decryption restriction only if \(n \ge 2\), where n is a predetermined parameter to manage the efficiency trade-off in their scheme, and we note that this flexible parameter n is crucial. It is difficult to prove that other unbounded HIBE schemes [31, 33] have the prefix decryption restriction because they do not have such a parameter.

  17. The guess is not required to prove selective security: thus, the reduction loss differs by a factor O(Q).

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Acknowledgements

This work is supported by JST CREST Grant Number JPMJCR14D6, JSPS KAKENHI Grant Number JP17K12697, JP18H05289, and MEXT Leading Initiative for Excellent Young Researchers.

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Authors and Affiliations

  1. National Institute of Information and Communications Technology (NICT), 4-2-1 Nukui-Kitamachi, Koganei, Tokyo, 184-8795, Japan

    Keita Emura & Atsushi Takayasu

  2. The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, 182-8585, Japan

    Yohei Watanabe

  3. National Institute of Advanced Industrial Science and Technology (AIST), 2-4-7 Aomi, Koto-ku, Tokyo, 135-0064, Japan

    Yohei Watanabe

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  1. Keita Emura
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Correspondence to Atsushi Takayasu.

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Emura, K., Takayasu, A. & Watanabe, Y. Adaptively secure revocable hierarchical IBE from k-linear assumption. Des. Codes Cryptogr. 89, 1535–1574 (2021). https://doi.org/10.1007/s10623-021-00880-w

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