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It is well known that the convolution of two signals is equivalent to the multiplication of two polynomials. Fast polynomial multiplication can be implemented by evaluating the two input polynomials at a collection of points, multiplying the polynomial evaluations, and then interpolating the results into the product polynomial. In 2004 and 2005, van der Hoeven introduced algorithms which efficiently evaluate and interpolate a polynomial at certain collections of n <; N points which are roots of xN - 1 where N is a power of two. A finite field of characteristic two does not have the necessary properties to use this algorithm. In this paper, we present a generalization of the van der Hoeven algorithms which can also be applied to the finite field case. The generalized algorithms also improve van der Hoeven's presentation to take advantage of a slightly more efficient factorization due to Crandall and Fagin. The evaluation and interpolation problems will be viewed in terms of polynomial rings, building upon previous work by Fiduccia, Martens, and Bernstein.