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This correspondence describes a method for achieving synthesis of finite state algorithms by the use of a set of logic elements that execute field operations from the Galois field GF[pn]. The method begins with a definition of the algorithm to be synthesized in a completely specified finite state flow table form. A polynomial expansion of this flow table function is derived. A canonical sequential circuit corresponding to this polynomial expansion is defined. Subsequently, the given algorithm is synthesized using the canonical circuit by specification of a number of arbitrary constants in the canonical circuit. A mechanical method for deriving constants used in the canonical circuits is given. Finally, some estimates on complexity of the given circuit structure are stated assuming the most fundamental logic element structures.