  Deterministic K-Identification (DKI) for the binary symmetric channel (BSC)
is developed. A full characterization of the DKI capacity for such a channel,
with and without the Hamming weight constraint, is established. As a key
finding, we find that for deterministic encoding the number of identifiable
messages $K$ may grow exponentially with the codeword length $n$, i.e., $K =
2^{\kappa n}$, where $\kappa$ is the target identification rate. Furthermore,
the eligible region for $\kappa$ as a function of the channel statistics, i.e.,
the crossover probability, is determined.
