Abstract
We introduce new isogeny-related assumptions called Isog-DDH and Isog-DBDH assumptions. By using the assumptions, we reinforce security of several existing (hierarchical) identity-/attribute-based encryption (HIBE/ABE) schemes. While the existing schemes are proven from the standard DBDH assumption, our reinforced secure ones have two incomparable security proofs: one is proven from the DBDH as well and another is proven from the Isog-DDH assumption which is incomparable with DBDH. As a result, if either DBDH or Isog-DDH assumption holds, the proposed HIBE/ABE schemes are secure. For obtaining our (H)IBE secure in the standard model, we assign a unique (product) group called ID-group to each (H)ID, and introduce a new proof technique, i.e., ID-group partitioning by using isogenies as trapdoors.
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Notes
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Since logical equivalence \(A \rightarrow C \equiv \lnot A \vee C\) holds for truth variables A, B, C, it holds that \((A \vee B) \rightarrow C \equiv \lnot (A \vee B) \vee C \equiv (\lnot A \vee C) \wedge (\lnot B \vee C) \equiv (A \rightarrow C) \wedge (B \rightarrow C)\). We have \((A \wedge B) \rightarrow C \equiv (A \rightarrow C) \vee (B \rightarrow C)\) in a similar manner (in the case [28]). In our case, \(A=\) “DBDH assumptions holds”, \(B=\) “\(N\)-Isog-DDH assumptions hold”, \(C=\) “our schemes are secure”.
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Koshiba, T., Takashima, K. (2019). New Assumptions on Isogenous Pairing Groups with Applications to Attribute-Based Encryption. In: Lee, K. (eds) Information Security and Cryptology – ICISC 2018. ICISC 2018. Lecture Notes in Computer Science(), vol 11396. Springer, Cham. https://doi.org/10.1007/978-3-030-12146-4_1
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