, Volume 1, Issue 3, pp 235-256

Existence and efficient construction of fast Fourier transforms on supersolvable groups

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Abstract

The linear complexityL K(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,L K(G):= min{L K(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|2log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:L K(G)≤8.5|G|log|G| for any supersolvable groupG andL K(G)≤1.5|G|log|G| for any 2-groupG.