 a. Evaluating Galois Field Connections (5.108) at = a gives (5.111) Hence, one shift of a d-element LFSR with connections specified by ma(z), corresponds to multiplication of a field element R(n)(a) of GF(q) by the ele- ment a, the register contents before the shift being the coefficients of the representation of R(n)(a) in the basis 1, a, ad-1 and after the shift being the coefficients R(n+1)(a). Solving the recursion in the finite field indicates that the register contents after n shifts represent (5.112) Certainly, the period of the state sequence {sn} for this LFSR (see (5.35)) is the same as the smallest value of N such that an=1. (5.113) This smallest value of N satisfying (5.113) is called the exponent (or order) of a, and can be shown to be a divisor of the number of non-zero elements of GF(q), namely qd - 1 (see Appendix 5A.6). Furthermore, it has been shown that the multiplicative group of a finite field is cyclic; hence, each Galois field GF (qd) contains elements whose exponents are exactly qd - 1, these elements being called primitive elements of the Galois field's multi- plicative group. The minimum polynomials of these primitive elements are the primitive polynomials appearing as characteristic polynomials of m- sequences. 5.5.3 Determining the Period of Memory Cell Outputs While the multiplier viewpoint relates the period of certain LFSR state sequences to field element properties, it is not yet clear that the period of the state sequence {sm} is identical to the period of the output sequence {bn}. Referring to Figure 5.6, let R(z) denote the formal power series represen- tation of {r{(n-1)}, the sequence of contents of the i-th memory element in an arbitrary Galois-configured LFSR. (5.114) The output memory element has index 1 and, therefore, R;(2) (5.115) where Q(z) is the characteristic polynomial of the output sequence {bn}. The series representation of the sequence produced by the second memory ele- ment can be related to R1(z) by noting that the input to the first memory element obeys the r