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Convolutions over residue classes of quadratic integers

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2 Author(s)

A Fourier-like transform is defined over a ring of quadratic integers modulo a prime number q in the quadratic field R(\sqrt {m}) , where m is a square-free integer. If q is a Fermat prime, one can utilize the fast Fourier transform (FFT) algorithm over the resulting finite fields to yield fast convolutions of quadratic integer sequences in R(\sqrt {m}) . The theory is also extended to a direct sum of such finite fields. From these results, it is shown that Fourier-like transforms can also be defined over the quadratic integers in R( \sqrt {m}) modulo a nonprime Fermat number.

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Information Theory, IEEE Transactions on  (Volume:22 ,  Issue: 4 )