We report on a multi-threaded implementation of Fast Fourier Transforms over generalized Fermat prime fields. This work extends a previous study realized on graphics processing units to multi-core processors. In this new context, we overcome the less fine control of hardware resources by successively using FFT in support of the multiplication in those fields. We obtain favorable speedup factors (up to 6.9x on a 6-core, 12 threads node, and 4.3x on a 4-core, 8 threads node) of our parallel implementation compared to the serial implementation for the overall application thanks to the low memory footprint and the sharp control of arithmetic instructions of our implementation of generalized Fermat prime fields.