A family of linear binary shift register circuits is described in terms of generator polynomialsg(x)which require fewer mod-2adders than those derived from the companion matrices. These circuits are derived from various combinations of the functionsa(x), b(x), andx^{d}ing(x) = 1 + a(x)x^{d}b(x)andc(D), e(D), andD^{m}of the reciprocal polynomialg^{\ast}(x) = f(D) = 1 + c(D)D^{m}e(D). Tables are provided for all simplifications thus achievable for primitive polynomial of degrees5through9, and for some nonprimitive polynomials.