Results on multiples of primitive polynomials and their products over GF(2)

Linear feedback shift registers (LFSR) are important building blocks in stream cipher cryptosystems. To be cryptographically secure, the connection polynomials of the LFSRs need to be primitive over GF(2). Moreover, the polynomials should have high weight and they should not have sparse multiples at low or moderate degree. Here we provide results on t-nomial multiples of primitive polynomials and their products. We present results for counting t-nomial multiples and also analyse the statistical distribution of their degrees. The results in this paper helps in deciding what kind of primitive polynomial should be chosen and which should be discarded in terms of cryptographic applications. Further the results involve important theoretical identities in terms of t-nomial multiples which were not known earlier.