Generating Good Span n Sequences Using Orthogonal Functions in Nonlinear Feedback Shift Registers

Abstract

A binary span n sequence generated by an n-stage nonlinear feedback shift register (NLFSR) is in a one-to-one correspondence with a de Bruijn sequence and has the following randomness properties: period 2 n − 1, balance, and ideal n-tuple distribution. A span n sequence may have a high linear span. However, how to find a nonlinear feedback function that generates such a sequence constitutes a long-standing challenging problem for about 5 decades since Golomb’s pioneering book, Shift Register Sequences, published in the middle of the 1960s. In hopes of finding good span n sequences with large linear span, in this chapter we study the generation of span n sequences using orthogonal functions in polynomial representation as nonlinear feedback in a nonlinear feedback shift register. Our empirical study shows that the success probability of obtaining a span n sequence in this technique is better than that of obtaining a span n sequence in a random span n sequence generation method. Moreover, we analyze the linear span of new span n sequences, and the linear span of a new sequence lies between 2 n − 2 − 3n (near optimal) and 2 n − 2 (optimal). Two conjectures on the linear span of new sequences are presented and are valid for n ≤ 20.