Approximate Lower Bound Arguments.

Suppose a prover, in possession of a large body of valuable evidence, wants to quickly convince a verifier by presenting only a small portion of the evidence. We define an Approximate Lower Bound Argument, or ALBA, which allows the prover to do just that: to succinctly prove knowledge of a large number of elements satisfying a predicate (or, more generally, elements of a sufficient total weight when a predicate is generalized to a weight function). The argument is approximate because there is a small gap between what the prover actually knows and what the verifier is convinced the prover knows. This gap enables very efficient schemes. We present noninteractive constructions of ALBA in the random oracle and Uniform Random String models and show that our proof sizes are nearly optimal. We also show how our constructions can be made particularly communication-efficient when the evidence is distributed among multiple provers working together, which is of practical importance when ALBA is applied to a decentralized setting. We demonstrate two very different applications of ALBAs: for large-scale decentralized signatures and for achieving universal composability in general-purpose succinct proof systems (SNARKs).