1. A Galois Field circuit production apparatus for fabricating a polynomial over GF(2 m ) in a circuit, comprising: an input device for allowing a specification relating to a desired polynomial over GF(2 m ) to be entered; a processor and memory for producing an improved netlist defined over GF; and a Galois Field circuit fabricator for fabricating a circuit from an improved netlist; wherein the processor and memory are configured to factorize a netlist defined over GF and corresponding to an input specification, determining common factors by passing through chains of multipliers following chains of adders; and wherein the circuit fabricator operably fabricates a circuit based on the improved netlist produced by the processor.