  The Pulsar Arecibo L-Band Feed Array (PALFA) survey, the most sensitive blind
search for radio pulsars yet conducted, is ongoing at the Arecibo Observatory
in Puerto Rico. The vast majority of the 180 pulsars discovered by PALFA have
spin periods shorter than 2 seconds. Pulsar surveys may miss long-period radio
pulsars due to the summing of a finite number of harmonic components in
conventional Fourier analyses (typically $\sim$16), or due to the strong effect
of red noise at low modulation frequencies. We address this reduction in
sensitivity by using a time-domain search technique: the Fast-Folding Algorithm
(FFA). We designed a program that implements a FFA-based search in the PALFA
processing pipeline, and tested the efficiency of the algorithm by performing
tests under both ideal, white noise conditions, as well as with real PALFA
observational data. In the two scenarios, we show that the time-domain
algorithm has the ability to outperform the FFT-based periodicity search
implemented in the survey. We perform simulations to compare the previously
reported PALFA sensitivity with that obtained using our new FFA implementation.
These simulations show that for a pulsar having a pulse duty cycle of roughly
3%, the performance of our FFA pipeline exceeds that of our FFT pipeline for
pulses with DM $\lesssim$ 40 pc cm$^{-3}$ and for periods as short as $\sim$500
ms, and that the survey sensitivity is improved by at least a factor of two for
periods $\gtrsim$ 6 sec. Discoveries from the implementation of the algorithm
in PALFA are also presented in this paper.
