A fundamental mathematical tool used in the further investigation of PN generators is a particular linear mapping from a finite field onto a subfield.
This mapping, called the trace function,will now be reviewed, and an explicit expression for the elements of an m-sequence will be constructed.
Let GF(q) be any finite field contained within a larger field GF(qd). Then the trace polynomial from GF(qd) to GF(q) is defined as
(5.121) The trace (function) in GF(q) of an element a in GF(qd) is defined as , i.e., the trace polynomial evaluated at a. The values of qand dare often obvious in the context of a particular application, in which case the cumbersome superscript and subscript are dropped from the trace notation.
The trace function has the following useful properties which are proved in Appendix 5A.9.
Property T-1.
When ais in GF(qd), then is in GF(q).
Property T-2.
All roots of an irreducible polynomial ma(z) over GF(q), with root ain GF(qd), have the same trace, i.e.,
(5.122) Trqqd1aqi2 Trqqd1a2 for all i.
Trqqd1a2 Trqqd1a2
Trqqd1z2 ¢ a
d1 i0
zqi Trqqd1z2
Galois Field Connections 301
Figure 5.9. A Galois LFSR producing sequences with differing periods. Single peri- ods are underlined.
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Property T-3.
The trace function is linear. That is, for aand bin GF(q), and aand bin GF(qd),
(5.123) Property T-4.
For each choice of bin GF(q) there are qd1elements ain GF(qd) for which (5.124) Property T-5.
If GF(q) (GF(qk) (GF(qd), then
(5.125) for all ain GF(qd).
Those familiar with finite fields will recognize that when the minimum polynomial ma(z) of aover GF(q) has degree d, then
are its distinct roots, is the coefficient of zd1in ma(z), and the trace is therefore in GF(q). However, when ma(z) has degree k,k d(kmust divide d), then the roots of ma(z) appear d/ktimes in (5.121) and
is .
Once a primitive element ahas been specified by choosing its minimum polynomial ma(z) of degree d over GF(q), then the trace of any non-zero element akin GF(q) can be evaluated directly.
(5.126) The substitution of zjfor zin the trace polynomial allows the evaluation of the trace of ajby substitution of a, and the reduction of the resulting poly- nomial mod ma(z) eliminates terms which will be zero upon evaluation at a. The result on the right side of (5.126) is an element of (see (5.103)), while Property T-2 implies that the quantity on the left must be in GF(q).
Since the elements of corresponding to GF(q) elements are in fact those same elements, i.e., only x0in (5.104) is non-zero in this case, the substitu- tion of zaon the right side of (5.126) is not necessary.
Example 5.8. Let abe a root of the primitive polynomial z5z21. The root aand its conjugates (other roots of the same primitive polynomial), namely a2, a4, a8, and a16, are elements of GF(32). Other elements of GF(32) can also be grouped into root sets. The powers on acorresponding to a root set must be relatively prime to the order of ato insure that the corresponding polynomial is primitive. In this case primitivity is guarantee for all degree 5 irreducible polynomials because the order of a, namely 31,
sd
sd Trqqd1aj2 Trqqd1zj2mod ma1z2.
1d>k2Trqqk1a2 Trqqd1a2
Trqqd1a2
a, aq, aq2,p, aqd1 Trqqd1a2 Trqqk1Trqqkd1a2 2,
Trqqd1a2 b.
Trqqd1aabb2 aTrqqd1a2bTrqqd1b2.
302 Pseudonoise Generators
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is prime. Each set of root powers is called a cyclotomic coset(see Appendix 5A.8). Property T-2 indicates that all the roots of the same irreducible poly- nomial have the same trace value; hence, trace values can be computed by (5.126) and associated with cyclotomic cosets. This is illustrated in Table 5.9 for this example.
The following theorem provides a convenient representation for certain LFSR sequences having irreducible characteristic polynomials. Here, and in discussions to follow, the superscripts and subscripts on the trace function will be omitted when their values are evident.
THEOREM 5.6. Let GF(qd) be the smallest field containing the element a, and let ma(z) be the degree d minimum polynomial of aover GF(q). Then the sequence n being the sequence index, has characteristic polynomial ma(z).
Proof. Consider a d-cell Galois LFSR producing an output sequence hav- ing characteristic polynomial ma(z). Combining the representation (5.22) for the remainder polynomial in terms of memory contents, with the field ele- ment interpretation of (5.112) gives
(5.127) where
(5.128) with aand bin GF(qd).Applying the trace function with Property T-3 to both sides of (5.127) and noting that the cell sequences are elements of GF(q), gives
(5.129) Tr1an2 a
L i1
Tr1baLi2ri1n2. b 3P1a2 41 anbR1n21a2ba
L i1
ri1n2aLi 5Trqqd1an2 6,
Galois Field Connections 303
Table 5.9
Cyclotomic cosets and associated trace function values when a is a root of z5z21 over GF(2).
Cyclotomic coset elements x
1, 2, 4, 8, 16 0
3, 6, 12, 24, 17 1
5, 10, 20, 9, 18 1
7, 14, 28, 25, 19 0
11, 22, 13, 26, 21 1
15, 30, 29, 27, 23 0
0 1
Tr2321ax2 http://jntu.blog.com
Corollary 5.1 indicates that the Lmemory cell sequences {ri(n)},i1,,L, all have the same characteristic polynomial ma(z); therefore, the linear com- bination of those sequences specified by (5.129), namely the sequence {Tr(an)}, has the same characteristic polynomial.
The following corollary provides an explicit and compact mathematical representation of the elements of an m-sequence, which will be used in sev- eral forthcoming analyses.
COROLLARY5.2. If {bn} is an m-sequence over GF(q)with the minimum polynomial ma(z)as its characteristic polynomial, then there exists a non- zero element gin GF(qd)such that
(5.130) Proof. By Theorem 5.6, the same recursion which generates {bn} must also produce {Tr(an)}, as well as any shift thereof. Since {bn} is an m-sequence, its characteristic polynomial ma(z) is primitive, and the corresponding LFSR supports only two cyclically distinct state sequences, namely the per- petual zero-state sequence and the state sequence of {bn}. Property T-4 indi- cates that Tr(an) must be non-zero for some n; hence, the state sequence corresponding to the trace sequence cannot be the zero-state sequence.
Therefore, the trace sequence or some shift of it must be the m-sequence.
That is,
(5.131) for some m, and setting gamcompletes the proof.