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Generalized Hardness Assumption for Self-bilinear Map with Auxiliary Information

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Information Security and Privacy (ACISP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9723))

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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. 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. 2.

    Actually, the AI-GMDH assumption is also parametrized by an additional natural number M. We omit it here for making the intuition simpler. For more details, see Definition 3 followed by Remark 2.

  3. 3.

    This holds for example, if we assume the existence of a constant \(\delta \) such that all prime factors of \((P-1)(Q-1)/4\) is no less than \(\delta \ell _N\)-bit as in [22, 29]. Especially, strong RSA moduli (e.g., \(N=PQ=(2p+1)(2q+1)\) such that P, Q, p and q are distinct primes) suffice.

  4. 4.

    Note that actually not all of these elements are needed to evaluate the map \(e_n\).

  5. 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. 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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Authors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Takashi Yamakawa & Noboru Kunihiro

  2. National Institute of Advanced Industrial Science and Technology (AIST), Tokyo, Japan

    Takashi Yamakawa & Goichiro Hanaoka

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  1. Takashi Yamakawa
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  2. Goichiro Hanaoka
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Correspondence to Takashi Yamakawa .

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Editors and Affiliations

  1. Monash University , Melbourne, Victoria, Australia

    Joseph K. Liu

  2. Monash University , Melbourne, Victoria, Australia

    Ron Steinfeld

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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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  • DOI: https://doi.org/10.1007/978-3-319-40367-0_17

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