A participial phrase can modify a subject, an entire predicate, a direct or indirect object, or an entire sentence.
Yet people trying to protect their data and transmissions apply less caution than the goats do: they attempt to distinguish between friend and foe using a mere sequence of symbols.[The first participial phrase modifies the subject; the second participial phrase modifies the predicate.]
The procedure then appends a signature consisting of the distorted stamp and
the solution to the equation. [modifies direct object]
The motion of the damping tubes shears the viscous fluid, dissipating the
vibration energy of the structure. [modifies whole sentence]