The Uber-Knowledge Assumption: A Bridge to the AGM

Abstract

The generic-group model (GGM) and the algebraic-group model (AGM) have been immensely successful in proving the security of many classical and modern cryptosystems. These models, however, come coupled with standard-model uninstantiability results, raising the question whether the schemes analyzed under them can be based on firmer standard-model footing. We formulate the uber-knowledge (UK) assumption, a standard-model assumption that naturally extends the uber-assumption family to knowledge assumptions. We justify the soundness of the UK in both the bilinear GGM and bilinear AGM. Along the way we extend these models to incorporate hashing into groups, an adversarial capability that is available in many concrete groups. (In contrast to standard assumptions, hashing may affect the validity of knowledge assumptions.) These results, in turn, enable a modular approach to security in GGM and AGM. As example applications, we use the UK to prove knowledge-soundness of Groth16 and KZG polynomial commitments in the standard model, where for the former we reuse the existing AGM proof without hashing