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., $K = 2^{κn}$, where $κ$ is the target identification rate. Furthermore, the eligible region for $κ$ as a function of the channel statistics, i.e., the crossover probability, is determined.