Is there a faster algorithm about this type of FFT
Alas, a multiplication by a rectangular window in the time domain is a convolution in the frequency domain.  Thus (assuming no other constraints on or knowledge about A or B), the DFT of A (or B) would have to be the convolution of the DFT of C with a twisty Sinc function, which is computationally far more expensive than 2 FFTs (except by fast convolution, which would require the same number of FFTs as your "ordinary" method).