Further Results on Affine Sub-Families of NFSR Sequences

Nonlinear feedback shift registers (NFSRs) have been widely used in hardware-oriented stream ciphers. Whether a family of NFSR sequences includes an affine sub-family of sequences is a fundamental problem for NFSRs. Let f be the characteristic function of an NFSR whose algebraic degree is d. The previous necessary condition on affine sub-families of NFSR sequences given by Zhang et al. [IEEE Trans. Inf. Theory, 65(2), 2019] provides a set of possible affine NFSRs defined by the variables appearing in the terms with the maximum degree d in f, which leads to the fastest algorithm so far for finding affine sub-families. In this paper, a new necessary condition for the existence of an affine sub-family in a family of NFSR sequences is proposed. The new necessary condition is further concerned with the algebraic relations between the terms with the maximum degree d in f, not only the variables involved in them, and so yields a smaller space of possible affine sub-families and less computation complexity for a large number of NFSRs.