  Cyclotomic fast Fourier transforms (CFFTs) are efficient implementations of
discrete Fourier transforms over finite fields, which have widespread
applications in cryptography and error control codes. They are of great
interest because of their low multiplicative and overall complexities. However,
their advantages are shown by inspection in the literature, and there is no
asymptotic computational complexity analysis for CFFTs. Their high additive
complexity also incurs difficulties in hardware implementations. In this paper,
we derive the bounds for the multiplicative and additive complexities of CFFTs,
respectively. Our results confirm that CFFTs have the smallest multiplicative
complexities among all known algorithms while their additive complexities
render them asymptotically suboptimal. However, CFFTs remain valuable as they
have the smallest overall complexities for most practical lengths. Our additive
complexity analysis also leads to a structured addition network, which not only
has low complexity but also is suitable for hardware implementations.
