We provide tight upper and lower bounds on the noise resilience of interactive communication over noisy channels with feedback. In this setting, we show that the maximal fraction of noise that any robust protocol can resist is 1 3. Additionally, we provide a simple and efficient robust protocol that succeeds as long as the fraction of noise is at most 1 3 - . Surprisingly, both bounds hold regardless of whether the parties send bits or symbols from an arbitrarily large alphabet. We also consider interactive communication over erasure channels. We provide a protocol that matches the optimal tolerable erasure rate of 1 2 - of previous protocols (, CRYPTO '13) but operates in a much simpler and more efficient way. Our protocol works with an alphabet of size 4, in contrast to prior protocols in which the alphabet size grows as epsilon goes to zero. Building on the above algorithm with a fixed alphabet size, we are able to devise a protocol for binary erasure channels that tolerates erasure rates of up to 1 3 - .
For erasure channels, a tight bound of @math on the erasure rate of robust protocols was given in @cite_11 . For the case of adaptive protocols, @cite_6 provided a coding scheme with a that resists a relative erasure rate of up to @math in a setting that allows parties to remain silent in an adaptive way. The case where the parties share a memoryless erasure channel with a noiseless feedback was considered by Schulman @cite_7 who showed that for any function @math , the communication complexity of solving @math in that setting equals the distributional complexity of @math (over noiseless channels), up to a factor of the channel's capacity.
@cite_7 @cite_6 @cite_11
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Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol spl pi be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless in each direction. We study the following question: if in fact the channel is noisy, what is the effect upon the number of transmissions needed in order to solve the computation problem reliably? Technologically this concern is motivated by the increasing importance of communication as a resource in computing, and by the tradeoff in communications equipment between bandwidth, reliability, and expense. We treat a model with random channel noise. We describe a deterministic method for simulating noiseless-channel protocols on noisy channels, with only a constant slowdown. This is an analog for general, interactive protocols of Shannon's coding theorem, which deals only with data transmission, i.e., one-way protocols. We cannot use Shannon's block coding method because the bits exchanged in the protocol are determined only one at a time, dynamically, in the course of the interaction. Instead, we describe a simulation protocol using a new kind of code, explicit tree codes. How much adversarial noise can protocols for interactive communication tolerate? This question was examined by Braverman and Rao (IEEE Trans. Inf. Theory, 2014) for the case of "robust" protocols, where each party sends messages only in fixed and predetermined rounds. We consider a new class of non-robust protocols for Interactive Communication, which we call adaptive protocols. Such protocols adapt structurally to the noise induced by the channel in the sense that both the order of speaking, and the length of the protocol may vary depending on observed noise. We define models that capture adaptive protocols and study upper and lower bounds on the permissible noise rate in these models. When the length of the protocol may adaptively change according to the noise, we demonstrate a protocol that tolerates noise rates up to @math . When the order of speaking may adaptively change as well, we demonstrate a protocol that tolerates noise rates up to @math . Hence, adaptivity circumvents an impossibility result of @math on the fraction of tolerable noise (Braverman and Rao, 2014). Error correction and message authentication are well studied in the literature, and various efficient solutions have been suggested and analyzed. This is however not the case for data streams in which the message is very long, possibly infinite, and not known in advance to the sender. Trivial solutions for error-correcting and authenticating data streams either suffer from a long delay at the receiver’s end or cannot perform well when the communication channel is noisy.