On affine sub-families of Grain-like structures

Grain is one of eSTREAM hardware-oriented finalists. It contains a cascade connection of an 80-bit primitive linear feedback shift registers (\({{\mathrm{LFSR}}}\)) into an 80-bit nonlinear feedback shift register (\({{\mathrm{NFSR}}}\)). The variant Grain-128 has a cascade connection with both \({{\mathrm{LFSR}}}\) and \({{\mathrm{NFSR}}}\) of order 128. We consider Grain-like structures, i.e., the cascade connection of a primitive \({{\mathrm{LFSR}}}\) into an \({{\mathrm{NFSR}}}\) of the same order. It is easy to know that in such a structure, all the affine sub-families of the \({{\mathrm{NFSR}}}\) are also the affine sub-families of the cascade connection. We prove that if the degree of the characteristic function of the \({{\mathrm{NFSR}}}\) is bigger than 2, then affine sub-families of the cascade connection must also be affine sub-families of the \({{\mathrm{NFSR}}}\). The same result holds if the order of the primitive \({{\mathrm{LFSR}}}\) is bigger than the order of the \({{\mathrm{NFSR}}}\).