An Analysis of the Algebraic Group Model

The algebraic group model (AGM), proposed by Fuchsbauer, Kiltz and Loss (CRYPTO 2018) has received huge attention. One of the most appealing properties of the AGM, is that, the hardness of security games in the generic group model (GGM) can be transferred via a generic reduction in the AGM. More concretely, for any two security games, G and H, if there exists a generic reduction from H to G in the AGM, and H is hard in the GGM, then G is also hard in the GGM. In this work, we analyze the relationship between the AGM and Shoup’s GGM (Eurocrypt 1997) and give evidence that: • hardness of security games in Shoup’s GGM cannot be transferred via a generic reduction in the AGM; • the AGM and Shoup’s GGM are incomparable.