Primitive polynomials for robust scramblers and stream ciphers against reverse engineering

A linear feedback shift register (LFSR) is a basic component of a linear scrambler and a stream cipher for a communication system. And primitive polynomials are used as the feedback polynomials of the LFSRs. In a non-cooperative context, the reverse-engineering of a linear scrambler and a stream cipher includes recovering the feedback polynomials and the LFSR's initial states (which are the secret keys in the case of stream ciphers). The problem of recovering the secret keys of stream ciphers has been extensively studied. For example, an effective approach for recovering a secret key is known as the correlation attack in the literature. The problem of reconstructing the feedback polynomials of a stream cipher and a linear scrambler has been studied recently. Both recovering the LFSR initial states by the above-mentioned correlation attack and reconstructing the feedback polynomials are highly dependent on an assumption, that is, they require that the feedback polynomials have sparse multiples of moderate degrees. Hence, in order to build linear scramblers and stream ciphers that are robust against reverse engineering, we should use primitive polynomials which do not have sparse multiples of moderate degrees. In this paper, we study the existence of primitive polynomials which do not have sparse multiples of moderate degrees, and the density of such primitive polynomials among all primitive polynomials. Our results on the existence and density of such primitive polynomials are better than the previous results in the literature.