Spooky Interaction and its Discontents: Compilers for Succinct Two-Message Argument Systems.

We are interested in constructing short two-message arguments for various languages, where the complexity of the verifier is small e.g. linear in the input size, or even sublinear if the input is coded appropriately. In 2000 Aiello et al. suggested the tantalizing possibility of obtaining such arguments for all of NP. These have proved elusive, despite extensive efforts. Our work builds on the compiler of Kalai and Raz, which takes as input an interactive proof system consisting of several rounds and produces a two-message argument system. The proof of soundness of their compiler relies on superpolynomial hardness assumptions. In this work we obtain a succinct two-message argument system for any language in NC, where the verifier's work is linear or even polylogarithmic. Soundness relies on any standard polynomially hard private information retrieval scheme or fully homomorphic encryption scheme. This is the first non trivial two-message succinct argument system that is based on a standard polynomial-time hardness assumption. We obtain this result by proving that the compiler is sound under standard polynomial hardness assumptions if the verifier in the original protocol runs in logarithmic space and public coins. We obtain our two-message argument by applying the compiler to an interactive proof protocol of Goldwasser, Kalai and Rothblum. On the other hand, we prove that under standard assumptions there is a sound interactive proof protocol that, when run through the compiler, results in a protocol that is not sound.