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