Long convolutions using transforms over reducible extensions of fermat number rings

We establish a system of conjugate symmetries for transforms over extensions R of a ring S in cases where R has fast transforms, and where multiplication in R has a fast algorithm. We take R equal to an extension of S mod a completely reducible polynomial, and define a group of S-automorphisms of R which allows a system of conjugate symmetries for the transforms. The system is applied to construct algorithms for filtering long sequences over S. When S is the ring of integers modulo the t-th Fermat number, the system allows use of a multiplication-free transform over R to compute convolutions of length up to 2^2t+3without use of inefficient multidimensional mappings. Iteration of the method effectively removes the bound on the convolution length suffered by other FNT techniques.