We learn about two efficient sub-linear proximity proofs (dsIPPs): proofs that are very fast to generate, and can be used to prove limited assertions about large inputs. Proof execution is very fast in the sense that it only needs to read a small (less than linear) portion of the input. Limited proof verification is even faster (reading a much smaller part of the input). Similar to local validation scripts, limited validation means that a reliable down-time supplicant can make a verifier accept local input, but no verifier can trick a verifier into accepting remote input. Such proof systems can be used to prove (and verify) claims about the properties of a large input, even if the proverb cannot read everything (and the verifier is even more limited). We develop such a proof-of-concept scheme for any area that can be determined by a continuously learn-and-oblivious branch program (ROOBP). We also develop proof systems for the approximate weighting of Hamming inputs, and for relaxing bipartiteness in a bounded-degree graph model.
