In this paper, a convolution theorem which is analogous to the theorem for Fourier transform is shown among a certain type of polynomials. We establish a fast method of the multiplication in a special class of quotient rings of multivariate polynomials over q-element finite field GF(q). The polynomial which we treat is one of expressing forms of the multiple-valued logic function from the product of the semigroups in GF(q) to GF(q). Our results can be applied to the speedup of both software and hardware concerning multiple-valued Boolean logic.