Verifying and decoding in constant depth

We develop a general approach for improving the efficiency of a computationally bounded receiver interacting with a powerful and possibly malicious sender. The key idea we use is that of delegating some of the receiver's computation to the (potentially malicious) sender. This idea was recently introduced by Goldwasser et al. [14] in the area of program checking. A classic example of such a sender-receiver setting is interactive proof systems. By taking the sender to be a (potentially malicious) prover and the receiver to be a verifier, we show that (p-prover) interactive proofs with k rounds of interaction are equivalent to (p-prover) interactive proofs with k + O(1) rounds, where the verifier is in NC0. That is, each round of the verifier's computation can be implemented in constant parallel time. As a corollary, we obtain interactive proof systems, with (optimally) constant soundness, for languages in AM and NEXP, where the verifier runs in constant parallel-time. Another, less immediate sender-receiver setting arises in considering error correcting codes. By taking the sender to be a (potentially corrupted) codeword and the receiver to be a decoder, we obtain explicit families of codes that are locally (list-)decodable by constant-depth circuits of size polylogarithmic in the length of the codeword. Using the tight connection between locally list-decodable codes and average-case complexity, we obtain a new, more efficient, worst-case to average-case reduction for languages in EXP.