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Looking for a generalization of fast Fourier transform form for Gauss sums

I want to compute quickly compute a sum of the form $$\sum_{k=0}^{N}\sum_{l=0}^{M} e(g^{a^k*b^l})$$ Assume $a^N = b^M = 1$ modulo $q-1$. Where $e(x) = e^{2\pi ix /q}$. This is very similar to the ...
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