Shift Register Sequences

Abstract

In Chapter 8, we already encountered linear feedback shift registers in connection with the hardware implementation of arithmetics in GF(2n): In the present chapter, we will study the sequences produced by such devices—that is, shift register sequences—in detail. As we shall see, these sequences are just the solutions of linear recurrence relations over GF(2): In view of this connection, we will consider shift register sequences over F = GF(q) for general q, and initially even over arbitrary fields F. Even though shift registers (as hardware devices) are of little practical interest for other fields but GF(2), they are a useful concept for visualizing linear recurrences over general fields. In the first two sections, we present some fundamental results which are valid over arbitrary fields F, including characterizations of shift register sequences and results about (ultimately) periodic sequences. After that, we shall restrict ourselves to the case of finite fields; in particular, we will consider shift register sequences associated with irreducible polynomials in Sect. 9.3. In the subsequent two sections, we discuss two applications, namely the construction of pseudo-random sequences and of some cyclic difference sets. We then return to the general theory by considering the notions of the linear complexity and, more strongly, the linear complexity profile of an arbitrary sequence with entries from GF(q). We will describe methods for determining these quantities, in particular the Berlekamp-Massey algorithm. Finally, in Sect. 9.9, we give a further application and study the so-called GMW-sequences, a class of periodic binary sequences which combine good randomness properties with a comparatively high linear complexity.