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

A standard model of nonlinear combiner generator for stream cipher system combines the outputs of several independent Linear Feedback Shift Register (LFSR) sequences using a nonlinear Boolean function to produce the key stream. Given such a model, cryptanalytic attacks have been proposed by finding out the sparse multiples of the connection polynomials corresponding to the LFSRs. In this direction recently a few works are published on t-nomial multiples of primitive polynomials. We here provide further results on degree distribution of the t-nomial multiples. However, finding out the sparse multiples of just a single primitive polynomial does not suffice. The exact cryptanalysis of the nonlinear combiner model depends on finding out sparse multiples of the products of primitive polynomials. We here make a detailed analysis on t-nomial multiples of products of primitive polynomials. We present new enumeration results for these multiples and provide some estimation on their degree distribution.