Initial conditions notwithstanding, the basic structural information about a linear generator is contained in its characteristic polynomial,Q(z). Assuming
bn⫽bn⫺2⫹bn⫺5.
that the generated sequence {bn} consists of elements from GF(q), the poly- nomial Q(z) will then be over GF(q), i.e., have coefficients in GF(q).
However, its roots may not be in GF(q). Let’s assume that Q(z) can be fac- tored so that
(5.45) where each polynomial Qj(z) is irreducible over GF(q), and the linear span Lof {bn} is given by
(5.46) The term irreducibleover GF(q), when applied to a polynomial A(z) over GF(q), means that A(z) cannot be factored into a product of polynomials over GF(q). Hence, the factorization described in (5.45) is similar to the decomposition of an integer into prime factors.
The factorization (5.45) indicates that the rational representation (5.16) of the formal power series B(z) can be expanded by partial fractions.
(5.47)
Therefore, the sequence represented by B(z) also can be generated by sum- ming the component sequences represented by
(5.48) thej-th sequence having characteristic polynomial and initial condi- tion polynomial Pj(z). Let’s define the period of the j-th sequence to be Nj. Then the period of the composite sequence {bn} is given by
(5.49) where lcm( ) represents the least common multiple of the listed integers.
Example 5.6. The sequence of Examle 5.4 was represented by
(5.50)
whereP1(z) and P2(z) are polynomials of degrees at most 1 and 3, respec- tively. Solving for these polynomials gives the component generators
B1z2 ⫽ 1
z6⫹z5⫹z4⫹z3⫹z2⫹1⫽ P11z2
1z⫹122⫹ P21z2
z4⫹z3⫹1 N⫽lcm1N1,N2,p,NJ2,
QPjj1z2 Bj1z2 ⫽ Pj1z2
Qjpj1z2 , j⫽1, 2,p,J, B1z2 ⫽ P1z2
q
J j⫽1
Qjpj1z2 ⫽ a
J j⫽1
Pj1z2 Qjpj1z2 . L⫽ a
J
j⫽1pj deg Qj1z2. Q1z2 ⫽ q
J j⫽1
Qjpj1z2,
(5.51a) (5.51b) The first generator produces the period 2 sequence
10
while the second generator outputs the period 15 sequence 101011001000111.
Extending these sequences periodically and summing gives the period 30 sequence of Example 5.4. The Galois-configured, component-structured, LFSR shown in Figure 5.8 can be initialized to have an output identical to the generators of Figure 5.7.
5.3.2 Maximization of Period for a Fixed Memory Size
The fundamental memory constraint (5.5) indicates that the maximum period achievable by an q-ary generator with Mmemory elements is qM. This maximum period is reduced by 1 if the generator uses linear feedback, because the zero state is self-perpetuating, i.e., an LFSR initiated with zeros produces the uninteresting all-zeros sequence. Thus, the most efficient LFSR generators must cycle through all possible non-zero states before repeating, and the period Nof an LFSR’s state sequence is bounded by
(5.52) In Example 5.6, the four-stage component LFSR generator achieved its max- imum possible period of 15, while the two-stage generator did not; and the composite generator of six stages produced a period 30 sequence, nowhere near the upper bound of 63 dictated by (5.52).
NⱕqM⫺1.
B21z2⫽ z3⫹z2⫹z⫹1 z4⫹z3⫹1 . B11z2⫽ z
1z⫹122
Figure 5.8. An LFSR generator for the sequence of Examples 5.4 and 5.6.
Assuming a Q(z) of the form (5.45), applying the bound (5.52) to the com- ponent sequences, and using (5.49) gives
(5.53) It is clear from (5.53) that the fundamental bound (5.52) on N can be achieved if and only if all the above inequalities hold with equality.
Equivalently, considering each inequality in turn, equality holds in (5.53) if and only if
(a) the periods of the component sequences are relatively prime,
(b) each component sequence achieves the bound (5.52) on its period for its memory size, and
(c) the number Jof component sequences must be 1.
When the degrees of the component generator polynomials are large, then item (c) is not critical to the efficient use of memory in creating large periods.
5.3.3 Repeated Factors in the Characteristic Polynomial
Consider now the special case in which the characteristic polynomial of a sequence over GF(q) is the q-th power of a polynomial over GF(q).Theorem 5A.10 of Appendix 5A indicates that
(5.54) Hence, the coefficient list of Qq(z) is simply
where 0 . . . 0 represents a string of q⫺1 zeros. A shift register governed by this form of generator polynomial will produce qinterleaved sequences. That is, a symbol bnin the composite output sequence {bn} is linked recursively only to the prior symbols bn⫺mq,m⫽1, 2, 3, . . .
This line of reasoning can be iterated to yield the following result.
THEOREM5.2. Let N be the period of a sequence having characteristic poly- nomial Q(z) over GF(q). Then the period N⬘of the sequence having char- acteristic polynomial QP(z) is given by
(5.55) N¿⫽qIN,
q00p0q10p0q20p0qd⫺10p0qd, Qq1z2 ⫽ a a
d
i⫽0qd⫺izibq⫽ a
d
i⫽0qd⫺izqi. ⱕqL⫺1.
ⱕ q
J j⫽1
qpjdegQj1z2⫺1 Nⱕ q
J j⫽1
Nj
where the leaving exponent I is the unique integer satisfying the relation (5.56)
Assuming that the period of the sequence generated by Q(z) is maximal, i.e., is qd⫺1 where dis the degree of Q(z), it is apparent that interleaving is not an efficient way to achieve sequences with long periods. As an exam- ple, the achievable periods for a register of twenty binary storage elements are tabulated in Table 5.5.
5.3.4 M-Sequences
Linear-feedback shift registers with Lstages which produce the maximum possible period qL⫺1 do in fact exist, and the sequences which they pro- duce are called maximal LFSR sequencesor m-sequences.When the con- nection between shift register sequences over GF(q) and larger finite fields is established in Section 5.5, it will be apparent that irreducible polynomi- als of degree Lover GF(q) generate sequences whose periods must be divi- sors of qL ⫺ 1. Those special irreducible polynomials, which are the characteristic polynomials of m-sequences, are called primitive polynomials, and they exist for every degree over every finite field. Appendix 5B contains factorizations of 2L⫺ 1 and a table of primitive polynomials over GF(2).
Techniques described in Appendix 5A.8 can be applied to find other prim- itive/irreducible polynomials from those listed in Appendix 5B. It suffices at this point to note:
THEOREM 5.3. A linear generator of a given memory size produces a sequence of elements from GF(q), with the largest possible period if and only if its characteristic polynomial is primitive over GF(q).
Two sequences {bn} and {b⬘n}, each having period N, are called cyclically equivalentif there exists an integer tsuch that bn⫹t⫽bnœ for all n. A pair
qI⫺1 6 pⱕqI. Table 5.5
Maximum periods Nfor twenty-stage binary LFSRs with characteristic polynomials of the form Qp(z) over GF(2), where Q(z) has degree d.
degree power period
d p I N
20 1 0 1048575
10 2 1 2046
5 4 2 124
4 5 3 120
2 10 4 48
of sequences, which are not cyclically equivalent, are termed cyclically dis- tinct.Sincem-sequences are cyclically distinct if and only if their linear recur- sions differ, the number of cyclically distinct m-sequences over GF(q) with linear span L is equal to the number Np(L) of primitive polynomials of degreeLover GF(q), and is given by
(5.57) where the Jprime numbers pi,i⫽1, 2, . . . ,Jare determined from the prime decomposition of qL⫺1, i.e.,
(5.58) where ei,i ⫽ 1, . . . ,Jare positive integers. Table 5.6 illustrates the expo- nential nature of Np(L) for binary m-sequences.