  Message Authentication Code (MAC) is a keyed function $f_K$ such that when
Alice, who shares the secret $K$ with Bob, sends $f_K(M)$ to the latter, Bob
will be assured of the integrity and authenticity of $M$. Traditionally, it is
assumed that the channel is noiseless. However, Maurer showed that in this case
an attacker can succeed with probability $2^{-\frac{H(K)}{\ell+1}}$ after
authenticating $\ell$ messages. In this paper, we consider the setting where
the channel is noisy. Specifically, Alice and Bob are connected by a discrete
memoryless channel (DMC) $W_1$ and a noiseless but insecure channel. In
addition, an attacker Oscar is connected with Alice through DMC $W_2$ and with
Bob through a noiseless channel. In this setting, we study the framework that
sends $M$ over the noiseless channel and the traditional MAC $f_K(M)$ over
channel $(W_1, W_2)$. We regard the noisy channel as an expensive resource and
define the authentication rate $\rho_{auth}$ as the ratio of message length to
the number $n$ of channel $W_1$ uses. The security of this framework depends on
the channel coding scheme for $f_K(M)$. A natural coding scheme is to use the
secrecy capacity achieving code of Csisz\'{a}r and K\"{o}rner. Intuitively,
this is also the optimal strategy. However, we propose a coding scheme that
achieves a higher $\rho_{auth}.$ Our crucial point for this is that in the
secrecy capacity setting, Bob needs to recover $f_K(M)$ while in our coding
scheme this is not necessary. How to detect the attack without recovering
$f_K(M)$ is the main contribution of this work. We achieve this through random
coding techniques.
