Abstract
A self-bilinear map (SBM) is a bilinear map where source and target groups are identical. An SBM naturally yields a multilinear map, which has numerous applications in cryptography. In spite of its usefulness, there is known a strong negative result on the existence of an ideal SBM. On the other hand, Yamakawa et al. (CRYPTO’14) introduced the notion of a self-bilinear map with auxiliary information (AI-SBM), which is a weaker variant of SBM and constructed it based on the factoring assumption and an indistinguishability obfuscation (\(i\mathcal {O}\)). In their work, they proved that their AI-SBM satisfies the Auxiliary Information Multilinear Computational Diffie-Hellman (AI-MCDH) assumption, which is a natural analogue of the Multilinear Computational Diffie-Hellman (MCDH) assumption w.r.t. multilinear maps. Then they show that they can replace multilinear maps with AI-SBMs in some multilinear-map-based primitives that is proven secure under the MCDH assumption.
In this work, we further investigate what hardness assumptions hold w.r.t. their AI-SBM. Specifically, we introduce a new hardness assumption called the Auxiliary Information Generalized Multilinear Diffie-Hellman (AI-GMDH) assumption. The AI-GMDH is parameterized by some parameters and thus can be seen as a family of hardness assumptions. We give a sufficient condition of parameters for which the AI-GMDH assumption holds under the same assumption as in the previous work. Based on this result, we can easily prove the AI-SBM satisfies certain hardness assumptions including not only the AI-GMDH assumption but also more complicated assumptions. This enable us to convert a multilinear-map-based primitive that is proven secure under a complicated hardness assumption to AI-SBP-based (and thus the factoring and \(i\mathcal {O}\)-based) one. As an example, we convert Catalano et al.’s multilinear-map-based homomorphic signatures (CRYPTO’14) to AI-SBP-based ones.
The first author is supported by a JSPS Fellowship for Young Scientists.
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Notes
- 1.
Actually in [29], this assumption is called Multilinear Computational Diffie-Hellman with Auxiliary Information (MCDHAI) assumption. We rename the assumption for the consistency with other assumptions in this paper.
- 2.
- 3.
- 4.
Note that actually not all of these elements are needed to evaluate the map \(e_n\).
- 5.
There is flexibility for the definition of the canonical circuit. However, any definition works if the size of \(\tilde{C}_{N,x}\) is polynomially bounded in \(\lambda \) and |x|.
- 6.
The actual presentation is slightly modified due to the modification of the definition.
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Acknowledgment
We would like to thank the anonymous reviewers of ACISP 2016 and members of the study group “Shin-Akarui-Angou-Benkyou-Kai” for their helpful comments. This work was supported by CREST, JST and JSPS KAKENHI Grant Number 14J03467.
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Yamakawa, T., Hanaoka, G., Kunihiro, N. (2016). Generalized Hardness Assumption for Self-bilinear Map with Auxiliary Information. In: Liu, J., Steinfeld, R. (eds) Information Security and Privacy. ACISP 2016. Lecture Notes in Computer Science(), vol 9723. Springer, Cham. https://doi.org/10.1007/978-3-319-40367-0_17
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