Hard Languages in NP ∩ coNP and NIZK Proofs from Unstructured Hardness

Explore a 49-minute lecture on the existence of unstructured hard languages in NP ∩ coNP and the construction of non-interactive zero-knowledge (NIZK) proofs. Delve into the first evidence supporting the existence of these languages, addressing Bennett and Gill's long-standing question about P separation from NP ∩ coNP relative to a random oracle. Examine the conditional separation based on UP not being contained in RP, and discover how this relates to injective one-way functions. Learn about a novel NIZK proof construction for NP with a uniformly random reference string, derived from a special type of hash function implied by an unstructured random oracle. Gain insights from speakers Amit Sahai and Alexis Korb from UCLA as they present their findings on minimal complexity assumptions for cryptography at the Simons Institute.