Tight bounds for unconditional authentication protocols in the manual channel and shared key models

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 < ∈ < 1 there exists a log* n-round protocol for authenticating n-bit messages, in which only 2 log(1/∈) + O(1) bits are manually authenticated, and any adversary (even computationally unbounded) has probability of at most ∈ to 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/∈) - 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/∈) - 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 essential (and sufficient) for the existence of protocols breaking the above lower bounds in the computational setting.