We develop a toolbox for modular construction and analysis of succinct, non-interactive commitments and vector commitments in the random oracle model, while guaranteeing universally composable security. To demonstrate its power, we use the toolbox to construct and analyze a modular variant of the Kilian-Micali ZK-SNARK. Along the way we also propose a new UC formulation of a global random oracle, that avoids a weakness in existing formulations and also enables expressing more nuanced, session-specific abstractions. We hope that this toolbox will be useful for building secure applications in settings where both succinctness and non-interactivity are key.