Generic and Algebraic Computation Models: When AGM Proofs Transfer to the GGM

The Fuchsbauer, Kiltz, and Loss (Crypto 2018) claim that (some) hardness results in the algebraic group model imply the same hardness results in the generic group model was recently called into question by Katz, Zhang, and Zhou (Asiacrypt 2022). The latter gave an interpretation of the claim under which it is incorrect. We give an alternate interpretation under which it is correct, using natural frameworks for capturing generic and algebraic models for arbitrary algebraic structures. Most algebraic analyses in the literature can be captured by our frameworks, making the claim correct for them.