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2016 | OriginalPaper | Buchkapitel

Generalized Hardness Assumption for Self-bilinear Map with Auxiliary Information

verfasst von : Takashi Yamakawa, Goichiro Hanaoka, Noboru Kunihiro

Erschienen in: Information Security and Privacy

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel A New Attack on Three Variants of the RSA Cryptosystem
Nächstes Kapitel Deterministic Encoding into Twisted Edwards Curves
Fußnoten
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
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
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
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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Metadaten
Titel
Generalized Hardness Assumption for Self-bilinear Map with Auxiliary Information
verfasst von
Takashi Yamakawa
Goichiro Hanaoka
Noboru Kunihiro
Copyright-Jahr
2016
Buch
Information Security and Privacy

Print ISBN: 978-3-319-40366-3

Electronic ISBN: 978-3-319-40367-0

Copyright-Jahr: 2016

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