Knowledge-Preserving Interactive Coding

How can we encode a communication protocol between two parties to become resilient to adversarial errors on the communication channel? If we encode each message in the communication protocol with a "good" error-correcting code (ECC), the error rate of the encoded protocol becomes poor (namely O(1/m) where m is the number of communication rounds). Towards addressing this issue, Schulman (FOCS\´92, STOC\´93) introduced the notion of interactive coding. We argue that whereas the method of separately encoding each message with an ECC ensures that the encoded protocol carries the same amount of information as the original protocol, this may no longer be the case if using interactive coding. In particular, the encoded protocol may completely leak a player\´s private input, even if it would remain secret in the original protocol. Towards addressing this problem, we introduce the notion of knowledge-preserving interactive coding, where the interactive coding protocol is required to preserve the "knowledge" transmitted in the original protocol. Our main results are as follows: The method of separately applying ECCs to each message has essentially optimal error rate: No knowledge-preserving interactive coding scheme can have an error rate of 1/m, where m is the number of rounds in the original protocol; If restricting to computationally-bounded (polynomial-time) adversaries, then assuming the existence of one-way functions (resp. sub exponentially-hard one-way functions), for every ϵ > 0, there exists a knowledge-preserving interactive coding schemes with constant error rate and information rate n^-ϵ (resp. 1/polylog(n)) where n is the security parameter; additionally to achieve an error of even 1/m requires the existence of one-way functions; Finally, even if we restrict to computationally-bounded adversaries, knowledge-preserving interactive coding schemes with constant error rate can have an information rate of at most o(1 log n). This result- applies even to non-constructive interactive coding schemes.