  We consider the problem of guessing a random, finite-alphabet, secret
$n$-vector, where the guesses are transmitted via a noisy channel. We provide a
single-letter formula for the best achievable exponential growth rate of the
$\rho$--th moment of the number of guesses, as a function of $n$. This formula
exhibits a fairly clear insight concerning the penalty due to the noise. We
describe two different randomized schemes that achieve the optimal guessing
exponent. One of them is fully universal in the sense of being independent of
source (that governs the vector to be guessed), the channel (that corrupts the
guesses), and the moment power $\rho$. Interestingly, it turns out that, in
general, the optimal guessing exponent function exhibits a phase transition
when it is examined either as a function of the channel parameters, or as a
function of $\rho$: as long as the channel is not too distant (in a certain
sense to be defined precisely) from the identity channel (i.e., the clean
channel), or equivalently, as long $\rho$ is larger than a certain critical
value, $\rho_{\mbox{\tiny c}}$, there is no penalty at all in the guessing
exponent, compared to the case of noiseless guessing.
