On quadratic m-sequences

Maximal sequences generated by linear feedback shift registers (FSRs), known as m-sequences, have been well-studied in the literature. These sequences have long periods, good statistical properties and two-valued autocorrelation functions. However, m-sequences are extremely vulnerable to a known plaintext attack. In order to overcome these weaknesses, nonlinearities have been introduced. We study nonlinear feedback functions by investigating quadratic functions. The quadratic span of a periodic binary sequence is the length of the shortest quadratic FSR that generates the sequence. This paper considers the question as to whether the sequence obtained from a DeBruijn sequence by dropping the all-zero state can now have quadratic span n. Such sequences are the quadratic analog of the linear m-sequences and present an attractive extremal case to explore further the structure of nonlinear FSRs