We address the message authentication problem in two seemingly different communication models. In the first model, the sender and receiver are connected by an insecure channel and by a low-bandwidth auxiliary channel, that enables the sender to manually authenticate one short message to the receiver (for example, by typing a short string or comparing two short strings). We consider this model in a setting where no computational assumptions are made, and prove that for any 0 < epsilon < 1 there exists a log* n-round protocol for authenticating n-bit messages, in which only 2 log(1/epsilon) + O(1) bits are manually authenticated, and any adversary (even computationally unbounded) has probability of at most epsilon cheat the receiver into accepting a fraudulent message. Moreover, we develop a proof technique showing that our protocol is essentially optimal by providing a lower bound of 2 log(1/epsilon) - 6 on the required length of the manually authenticated string.
The second model we consider is the traditional message authentication model. In this model the sender and the receiver share a short secret key; however, they are connected only by an insecure channel. Once again, we apply our proof technique, and prove a lower bound of 2 log(1/epsilon) - 2 on the required Shannon entropy of the shared key. This settles an open question posed by Gemmell and Naor (CRYPTO’93).
Finally, we prove that one-way functions are necessary (and sufficient) for the existence of protocols breaking the above lower bounds in the computational setting.
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