NDLI: An Algebraic Framework for Diffie

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An Algebraic Framework for Diffie–Hellman Assumptions

Content Provider	Springer Nature Link
Author	Escala, Alex Herold, Gottfried Kiltz, Eike Ràfols, Carla Villar, Jorge
Copyright Year	2015
Abstract	We put forward a new algebraic framework to generalize and analyze Diffie–Hellman like decisional assumptions which allows us to argue about security and applications by considering only algebraic properties. Our $$\mathcal {D}_{\ell ,k}\text{- }\textsf {MDDH}$$ Assumption states that it is hard to decide whether a vector in $$\mathbb {G}^\ell $$ is linearly dependent of the columns of some matrix in $$\mathbb {G}^{\ell \times k}$$ sampled according to distribution $$\mathcal {D}_{\ell ,k}$$ . It covers known assumptions such as $$\textsf {DDH},\, 2\text{- }\textsf {Lin}$$ (Linear Assumption) and $$k\text{- }\textsf {Lin}$$ (the k-Linear Assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m-linear groups to the irreducibility of certain polynomials which describe the output of $$\mathcal {D}_{\ell ,k}$$ . We use the hardness results to find new distributions for which the $$\mathcal {D}_{\ell ,k}\text{- }\textsf {MDDH}$$ Assumption holds generically in m-linear groups. In particular, our new assumptions $$2\text{- }\textsf {SCasc}$$ and $$2\text{- }\textsf {ILin}$$ are generically hard in bilinear groups and, compared to $$2\text{- }\textsf {Lin}$$ , have shorter description size, which is a relevant parameter for efficiency in many applications. These results support using our new assumptions as natural replacements for the $$2\text{- }\textsf {Lin}$$ assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any $$\textsf {MDDH}$$ Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash proof systems, pseudo-random functions, and Groth–Sahai NIZK and NIWI proofs. As an independent contribution, we give more efficient NIZK and NIWI proofs for membership in a subgroup of $$\mathbb {G}^\ell $$ . The results imply very significant efficiency improvements for a large number of schemes.
Starting Page	242
Ending Page	288
Page Count	47
File Format	PDF
ISSN	09332790
Journal	Journal of Cryptology
Volume Number	30
Issue Number	1
e-ISSN	14321378
Language	English
Publisher	Springer US
Publisher Date	2015-10-22
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Diffie–Hellman assumption Generic hardness Groth–Sahai proofs Hash proof systems Public-key encryption Coding and Information Theory Computational Mathematics and Numerical Analysis Combinatorics Probability Theory and Stochastic Processes Communications Engineering, Networks
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Science Applications Software

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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