Interactively Secure Groups from Obfuscation

Agrikola, Thomas; Hofheinz, Dennis

doi:10.1007/978-3-319-76581-5_12

Cited by 6 publications

(18 citation statements)

References 35 publications

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Section: Follow-up Workmentioning

confidence: 99%

“…On the other hand, the work [2] aims to construct groups (or, rather, encoding schemes) that support stronger computational assumptions. Specifically, [2] construct encoding schemes in which even an adaptive variant of the so-called Uber assumption [6] holds. The price that [2] pay is that their encoding scheme has no extraction algorithm (i.e., no algorithm that takes an encoding and outputs a bit string that is unique for the encoded group element).…”

Section: Follow-up Workmentioning

confidence: 99%

“…Specifically, [2] construct encoding schemes in which even an adaptive variant of the so-called Uber assumption [6] holds. The price that [2] pay is that their encoding scheme has no extraction algorithm (i.e., no algorithm that takes an encoding and outputs a bit string that is unique for the encoded group element). Not only such an extraction algorithm is useful to compare elements, it can also be used to transform non-unique group elements to a unique common secret in a Diffie-Hellman key exchange protocol.…”

Section: Follow-up Workmentioning

confidence: 99%

“…In this setting, the only means to compare two group elements (given by possibly different encodings) is an explicit comparison algorithm that takes two encodings as input and outputs whether these encodings represent the same group element. ( [2] provide such a comparison algorithm.) The techniques that [2] use are again an extension of our techniques.…”

Section: Follow-up Workmentioning

confidence: 99%

“…We resolve this by showing a way of evaluating the κ-linear map on any elements of G 1 using only the powers g ω i 0 for 1 ≤ i ≤ κ, and vectors extracted from the accompanying ciphertexts, and then applying IO to the two circuits. 2 The second assumption we made is that we can indeed switch from h to h without being noticed. In other words, that the vectors x i , y i representing g a i can be replaced (without being noticed) with vectors h i whose second coordinate is always fixed.…”

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Multilinear Maps from Obfuscation

et al. 2020

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We provide constructions of multilinear groups equipped with natural hard problems from indistinguishability obfuscation, homomorphic encryption, and NIZKs. This complements known results on the constructions of indistinguishability obfuscators from multilinear maps in the reverse direction. We provide two distinct, but closely related constructions and show that multilinear analogues of the DDH assumption hold for them. Our first construction is symmetric and comes with a κ-linear map e : G κ −→ G T for prime-order groups G and G T. To establish the hardness of the κ-linear DDH problem, we rely on the existence of a base group for which the κ-strong DDH assumption holds. Our second construction is for the asymmetric setting, where e : G 1 ×• • •×G κ −→ G T for a collection of κ + 1 prime-order groups G i and G T , and relies only on the 1-strong DDH assumption in its base group. In both constructions, the linearity κ can be set to any arbitrary but a priori fixed polynomial value in the security parameter. We rely on a number of powerful tools in our constructions: probabilistic indistinguishability obfuscation, dual-mode NIZK proof systems (with perfect soundness, witness-indistinguishability, and zero knowledge), and additively homomorphic encryption for the group Z + N. At a high level, we enable "bootstrapping" multilinear assumptions from their simpler counterparts in standard cryptographic groups and show the equivalence of PIO and multilinear maps under the existence of the aforementioned primitives. * This paper has been handled by Ivan Bjerre Damgård as acting editor in chief and communicated by Alon Rosen.

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Section: Follow-up Workmentioning

confidence: 99%

Section: Follow-up Workmentioning

confidence: 99%

Section: Follow-up Workmentioning

confidence: 99%

Section: Follow-up Workmentioning

confidence: 99%

mentioning

confidence: 99%

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Multilinear Maps from Obfuscation

et al. 2020

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Dual-Mode NIZKs from Obfuscation

Hofheinz

Ursu

2019

Lecture Notes in Computer Science

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On Instantiating the Algebraic Group Model from Falsifiable Assumptions

Agrikola

Hofheinz

Kästner

2020

Lecture Notes in Computer Science

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We provide a standard-model implementation (of a relaxation) of the algebraic group model (AGM, [Fuchsbauer, Kiltz, Loss, CRYPTO 2018]). Specifically, we show that every algorithm that uses our group is algebraic, and hence “must know” a representation of its output group elements in terms of its input group elements. Here, “must know” means that a suitable extractor can extract such a representation efficiently. We stress that our implementation relies only on falsifiable assumptions in the standard model, and in particular does not use any knowledge assumptions. As a consequence, our group allows to transport a number of results obtained in the AGM into the standard model, under falsifiable assumptions. For instance, we show that in our group, several Diffie-Hellman-like assumptions (including computational Diffie-Hellman) are equivalent to the discrete logarithm assumption. Furthermore, we show that our group allows to prove the Schnorr signature scheme tightly secure in the random oracle model. Our construction relies on indistinguishability obfuscation, and hence should not be considered as a practical group itself. However, our results show that the AGM is a realistic computational model (since it can be instantiated in the standard model), and that results obtained in the AGM are also possible with standard-model groups.

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Interactively Secure Groups from Obfuscation

Cited by 6 publications

References 35 publications

Multilinear Maps from Obfuscation

Multilinear Maps from Obfuscation

Dual-Mode NIZKs from Obfuscation

On Instantiating the Algebraic Group Model from Falsifiable Assumptions

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