Computing Complex Convolutions Using Surrogate Fields

Abstract

In this contribution we show how representations of finite fields as Gaussian integers modulo a Gaussian prime can be used for computing complex convolutions. The main idea is to switch from the representation of finite fields as Gaussian integers to the representation of a finite field as integers modulo a prime p and to perform the convolution modulo p. The transformation from the representation of the finite field as Gaussian integers modulo a Gaussian prime to the representation as integers modulo p is done via a simple formula which stems from the extended Euclidean algorithm. The transformation back can easily be done using the complex modulo function.