1. A Galois Field (GF) circuit production apparatus for fabricating a GF(2 m ) polynomial in a circuit, comprising: an input device for receiving a GF(2 m ) polynomial for which a circuit is desired; a processor and memory for producing a netlist representing the GF(2 m ) polynomial; and a Galois Field circuit fabricator for fabricating a circuit from a netlist that has chains of adders and chains of multipliers; wherein the processor and memory are configured to decompose a Galois Polynomial Decision Diagram (GPDD) and to factorize a first netlist corresponding to the GF(2 m ) polynomial, determining factors common to terms of the GF(2 m ) polynomial by passing through the chains of multipliers following the chains of adders and extracting the determined factors common to terms of the GF(2 m ) polynomial to generate a factorized netlist; wherein the circuit fabricator operably fabricates a circuit based on the factorized netlist produced by the processor; and wherein there is an optimization algorithm stored in the memory and the processor is configured to run the optimization algorithm on the factorized netlist to generate a second netlist and wherein the circuit fabricator operably fabricates a circuit based on the second netlist produced by the processor.