[Resource Topic] 2019/1018: Towards Instantiating the Algebraic Group Model

Welcome to the resource topic for 2019/1018

Towards Instantiating the Algebraic Group Model

Authors: Julia Kastner, Jiaxin Pan


The Generic Group Model (GGM) is one of the most important tools for analyzing the hardness of a cryptographic problem. Although a proof in the GGM provides a certain degree of confidence in the problem’s hardness, it is a rather strong and limited model, since it does not allow an algorithm to exploit any property of the group structure. To bridge the gap between the GGM and the Standard Model, Fuchsbauer, Kiltz, and Loss proposed a model, called the Algebraic Group Model (AGM, CRYPTO 2018). In the AGM, an adversary can take advantage of the group structure, but it needs to provide a representation of its group element outputs, which seems weaker than the GGM but stronger than the Standard Model. Due to this additional information we learn about the adversary, the AGM allows us to derive simple but meaningful security proofs. In this paper, we take the first step to bridge the gap between the AGM and the Standard Model. We instantiate the AGM under Standard Assumptions. More precisely, we construct two algebraic groups under the Knowledge of Exponent Assumption (KEA). In addition to the KEA, our first construction requires symmetric pairings, and our second construction needs an additively homomorphic Non-Interactive Zero-Knowledge (NIZK) argument system, which can be implemented by a standard variant of Diffie-Hellman Assumption in the asymmetric pairing setting. Furthermore, we show that both of our constructions provide cryptographic hardness which can be used to construct secure cryptosystems. We note that the KEA provably holds in the GGM. Our results show that, instead of instantiating the seemingly complex AGM directly, one can try to instantiate the GKEA under falsifiable assumptions in the Standard Model. Thus, our results can serve as a stepping stone towards instantiating the AGM under falsifiable assumptions.

ePrint: https://eprint.iacr.org/2019/1018

See all topics related to this paper.

Feel free to post resources that are related to this paper below.

Example resources include: implementations, explanation materials, talks, slides, links to previous discussions on other websites.

For more information, see the rules for Resource Topics .