Further Results on Multiples of Primitive Polynomials and Their Products over GF(2)

Recently the problem of analysing the multiples of primitive polynomials and their products has received a lot of attention. These primitive polynomials are basically the connection polynomials of the LFSRs (Linear Feedback Shift Registers) used in the stream cipher system. Analysis of sparse multiples of a primitive polynomial or product of primitive polynomials helps in identifying the robustness of the stream ciphers based on nonlinear combiner model. In this paper we first prove some important results related to the degree of the multiples. Earlier these results were only observed for small examples. Proving these results clearly identify the statistical behavior related to the degree of multiples of primitive polynomials or their products. Further we discuss a randomized algorithm for finding sparse multiples of primitive polynomials and their products. Our results clearly identify the time memory trade off for finding such multiples.