Recently nonlinear feedback shift registers (NFSRs) have frequently been used as basic building blocks for stream ciphers. A major problem concerning NFSRs is to construct NFSRs which generate de Bruijn sequences, namely maximum period sequences. In this paper, we present a new necessary condition for NFSRs to generate de Bruijn sequences. The new condition can not be deduced from the previously proposed necessary conditions. It is shown that the number of NFSRs whose feedback functions satisfy all the previous necessary conditions but not the new one is very large.

In this paper, we put forward a new method to construct n-variable Boolean functions with optimal algebraic immunity based on the factorization of n. Computer investigations for small values of n indicate that a class of Boolean functions constructed by the new method has a very good nonlinearity and also a good behavior against fast algebraic attacks.