Computational Complexity of Cyclotomic Fast Fourier Transforms over Characteristic-2 Fields

TL;DR

This paper provides the first asymptotic analysis of the computational complexities of cyclotomic FFTs over characteristic-2 fields, revealing their advantages and limitations in multiplicative and additive complexities.

Contribution

It derives bounds for the multiplicative and additive complexities of CFFTs, offering insights into their asymptotic behavior and practical efficiency.

Findings

CFFTs have the lowest multiplicative complexity among known algorithms.

Their additive complexity is asymptotically suboptimal.

CFFTs are most efficient for practical lengths despite additive complexity issues.

Abstract

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.