Polynomial Sequence (in octal) 1001011001111100011011101010000 111110111000101011010000110010 1110100010010101100001110011011 1001001100001011010100011101111 1110110011100001101010010001011 1000010101110110001111100110100 shown in Table 5.10. The characteristic polynomials of sequences con- structed by the decimation process can be found directly by the techniques of Section 5.2.5, or by calculation of mo(z) from ma(z) as described in Appendix 5A.8. 5.6 NON-LINEAR FEED-FORWARD LOGIC Two related disadvantages of m-sequences and other LFSR sequences have already been discussed: (1) Simple linear analysis will allow an observer to predict the output of an L-stage LFSR from an observation of 2L consecu- tive output symbols. (2) The shift-and-add property of m-sequences allows an observer to produce different "time-advanced" versions of an m-sequence while observing the sequence, although the amount of shift cannot be pre- dicted without further analysis. In either case, the observer, possessing no a priori information about the nature of the SS code generator, has the poten- tial to predict the generator output and use this information to read the mes- sage, or to jam or spoof the intended receiver. As will be demonstrated in this section, vulnerability to this type of countermeasure can be significantly reduced by the inclusion of non-linear feed-forward logic (NLFFL) operat- ing on the contents of the m-sequence generator's shift register, to produce an SS code sequence having high linear span. 5.6.1 A Powers-of-a Representation Theorem The mathematical analysis of NLFFL effects requires use of the represen- tation of a sequence {bn} of GF(q) elements with period qM - 1, in terms of the qM - - 1 sequences {1} (the all ones sequence), {an}, {a2n} {a(9M-2)n}, where a is a primitive element of GF(qM). Such a representation of {bn} must be unique since the basis sequences in the representation are linearly inde- pendent. This can be proven by noting that if there exist coefficients Pseudonoise Generators ao, , aqM- not