User:Cobn/sandbox/Fermat Number Transform

A number theoretic transform (NTT) is a generalization of the fast fourier transform. Instead of $$ W_{N}= e^{-j\frac{2\pi}{N}}$$ in the field of complex numbers being used, an nth primitive root of unity over a quotient ring $\mathbb{Z} / p\mathbb{Z}$   is utilized, with p prime.

In a number theoretic transform, the kernel satisfies $W_{N}^{N} = 1 (\text{mod p})$.

The NTT over a ring $$\mathbb{Z} / m\mathbb{Z}$$ (m not prime) is still useful if an nth order primitve root exits. One such case is the Fermat Number Transform (FNT) with composite modulus $m = 2^k+1$.

The tth Fermat number is given by