Theory of transformation groups of polynomials over GF(2) with applications to linear shift register sequences

Theory of transformation groups of polynomials over GF(2) with applications to linear shift register sequences

~NFORMA~ON 87 SCIENCES Theory of Transformation Groups of Polynomials Over GF(2) with Applications to Linear Shift Register Sequences SOLOMON W. GO...

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~NFORMA~ON

87

SCIENCES

Theory of Transformation Groups of Polynomials Over GF(2) with Applications to Linear Shift Register Sequences SOLOMON W. GOLOMB? Department of Electrical Engineering, University of Southern Ca[ifornia, Los Angeles, California

ABSTRACT The group of unimodular transformations on the roots of polynomials over GF(2) is considered, and those polynomials with symmetries in the unimodular group are identified. The cross-correlation function between two maximum-length linear shift register sequences of the same degree is shown to be computable as an explicit linear transformation, in matrix form, on either one of the sequences, regarded as a vector. The underlying vector space is the “cyclotomic algebra,” generated by the cyclotomic cosets, or by what Gauss termed the “periods” of the cyclotomic equation. A variety of numerical examples are worked in detail.

1. THE ALGEBRAIC CLOSURE OF GF(2)

We may summarize the elementary theory of polynomials over GF(2) by describing the field B, which denotes the algebraic closure of GF(2). The elements of B, in addition to the zero-element, 0, are all rth roots of unity. More specifically, B contains, as distinct elements, exactly 4(r) primitive rth roots of unity for each odd r, r = 1, 3, 5,7,9, 11, , . ., in addition to 0. The multiplicative group of B is reasonably represented as a unit circle, which is a homomorphic image of the complex roots of unity. The kernel of this homomorphism is the group (exp(2aim/2k)), consisting of the complex Ist, 2nd, 4th, 8th, 16th, etc. roots of unity. If w # 0 is an element of B, then w is a primitive rth root of unity for some odd r. Then the minimal polynomial for w over GF(2) has degree n, where n is the smallest positive integer such that 2’ = 1 (mod r). Since there are #(I) primitive rth roots of unity, and they occur n at a time in irreducible polynomials of degree 12,there are $@)/a distinct minimal polynomials for the primitive rth roots of unity, over GF(2). Each of these polynomials is a factor of x2” + X, which in fact is the product of all irreducible polynomials, over GF(2), whose degrees divide n. t This research was supported in part by the Army Research Office under Contract No. DA-ARO-D-31-1244930. ~~~~~atjo~ Sciences 1 (1968), 87-109 Copyright 0 1968 by American Etsevier ~blishing

Company, Inc.

SOLOMON W. GOLOMB

88

If a set of complex roots of unity sums to 0, 1, or -1, the corresponding elements of B sum to 0, 1, or 1, respectively, by homomorphic correspondence. However, Euclidean distances between points on the complex unit circle have no discernible meaning when applied to the unit circle representation for B. A finite subfield of B, which is the same as a finite extension of GJ’(2), consists of the (2” - 1)st roots of unity, with 0 adjoined. The multiplication operation for such a set of 2” elements is obvious. In order to describe the addition operation, one must introduce an nth degree irreducible polynomial over GF(Z), and consider the 2” linear combinations of its n roots. 2. A~OMORP~IS~S

AND GALOIS THEORY

Let

be any polynomial of degree n over GF(2). We will show that the operation u, which squares all the roots of f(x), in fact leaves f(x) unaltered. It will be convenient to use the substitution t 2 = x. Then :

=[fjw+Lftt)12=fm=fo, as asserted. We have used the special rules of GF(2) that $1 = -1, and (a + Qz = ~9 + b2. An operator u on the roots off(x) which leaves the coefficients unchanged is called an automorphism. Along with cr, also u*, u3, u4, are all automorphisms. Thus for any polynomial over GF(2), and in particular for any irreducible polynomial, if w is a root, then also w*, w4, w*, c@, etc. are roots. Iff(x) is irreducible of degree n, then w2*-r = 1, and w*”= w, so the distinct roots of f(x) are w, CZJ*, fu4, . . ., w2”‘. Moreover, the distinct automorphisms off(x) are I,a,02 ,..., cf-’ ; since c?’takes o into w2”= wand is thus the identity operator. The principal conclusion embodied in the “Fundamental Theorem of Galois Theory” is that an irreducible polynomial&) of degree n (over a field F) has exactly n distinct automorphisms (with respect to F). Thus, for polynomials over GF(2), we know the group of automorphisms (the so-called Galois group) completely: An operator ?P on the roots of f(x) is an automorphism if and only if it is a power of the root-squaring operator u. The roots of the factors of x2” + x can thus be distinguished as follows.

6.

THE

(26 1)ST

=

63RD

ROOTS

d7,

w45,

w21, d

21,42

0

= 1

042

~18,

~9,

d4,

~30,

d,

9, 18, 36 45, 27, 54

~7,

7, 14, 28, 56,49, 35

~3,

~‘5,

3, 6, 12, 24,48, 33 15, 30, 60, 57, 51, 39

d4

~3’5

w*8,

do,

d2,

~5~,

w5’,

w24,

w38, w13, 0~~6,d2,

w49,

0, w35

w39

0.148, cd3

w41, cd9

+ x5

+ x4

x4

+ x2

x+1

x2+x+1

x3+x+1

x3

x-5 + x3

x6

x6

x6 +

+ x5

+ x5

d9

x6

w46, x6

d3,

w58,

w43,

+ x5

x6

d3,

+ x5

1

+

+

1

1

+ x4

+ x2

+ x3

+ x4

+

+ x3

+ x2

+ x2

+ x +

+ x +

+ x +

+ x2

+ x +

Polynomial

x6

x6+x+1

OF UNITY

cd, ~2, ~4, ~9, ~‘6, uP2 us, do, do, u40, d7, w34 &, do, w37, UJl, w**, oJ44 ~~62,d’, ~59, ~55, cu4’, ~03’

1,2,4,8, 16, 32 5, 10, 20,40, 17, 34 25, 50,37, 11,22,44 62,61, 59, 55,47, 31 58,53,43, 23,46,29 38, 13, 26, 52,41, 19

n =

Set of Roots

WlTH

EXAMPLE

Coset

I

TABLE

+

+

1

1 1

1

1

1

1

2

3 3

6

1

3

7 7

9

21 21

63 63 63 63 63 6 6

63

6 6 6 6 6

Primitivity

6

Degree

90

SOLOMON

W. GOLOMB

Let w be a primitive (2” - I)st root of unity. Then w, w*, w4, . . ., w*“-’ are all roots of the same irreducible polynomial. Taking the set R = { 1,2,4,. . ., 2+*} as a subgroup of the multiplicative group modulo 2” - 1, we may form all “generalized cosets” qR = (q,2q,4q,. . .,2’-‘q}. When (q,2” - 1) = 1 we get a true coset in the usual sense of group theory; and these cosets correspond to other irreducible polynomials of degree n with primitive (2” - 1)st roots of unity as roots. If q has a factor in common with 2” - 1, but qR consists of n distinct elements, we get the roots of an irreducible polynomial of degree n whose roots are not primitive (2” - 1)st roots of unity. If qR has only m -C n distinct elements, these correspond to the roots of an irreducible polynomial of degree m (Table I). 3.

THE UNIMODULAR

GROUP

Let T[f(x)] =f(x + l), and Slf(x)] = x”f(l/x), wheref(x) is a polynomial of degree n over GF(2). Clearly both Tand Sareinvolutions. That is, T* = S* = I, the identity transformation. We can show that the group generated by S and T is abstractly the noncommutative group of order 6. We do this by observing the effects of S and T on the roots off(x). If

then

WW

= fi (x + 1 - 4 = fi b - bi - l>l,

so that over any field, the effect of

T is to translate all the roots by a unit amount. (It is only over characteristic 2 that two such translations bring one back to the starting point.) On the other hand,

w..fWl = x” fj (; - w) = fi (1 - =4. i=l If

is non-zero then

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GROUPS OF POLYNOMIALS

which replaces each root wi off(x) by the root I/q. Thus, over any field, the effect of S on any polynomial which does not have 0 as a root is to replace all roots by their reciprocals. w, it is clear that over any field, the FromT:w-+w+landS:w-+l/ effect of the group generated by S and T is to perform transformations of the type w -+ (ao + ~)~(e~ + d). Moreover, since 5’ and T have inverses, all transformations

in the group they generate are nonsingular, so that a

c

b

Ic

diO* I

b

d with integer > coefhcients, such that ad - bc = 1. Over GF(2), this group has six eIements:

In

general, we get the “unimodular group” of matrices

a

t

corresponding to transforming the root w into:

w,

1 -&’

W+l w



w-i-1,

w

W4-1’

The structure of this group is the noncommutative 4. POLYNOMIAL

1

w+l’

respectively.

group of order 6.

WITH UNIMO~ULAR SYMMETRIES

We are now in a position to characterize those irreducible polynomials which are left invariant by the various operators of the unimodular group. First, we observe that the six members of the unimodular group lie in the conjugate classes :

Every polynomial over GF(2) has I as a symmetry. We next consider S as a possible symmetry. If S[jc(x)] =f(x), then along with w, f(x) has I/w as a root. If f(x) is irreducible of degree n, however, then its only roots are wl, m2, 04, . . ., CT-‘. Hence w-i = wzk for some k, 0 G k G n - I. Moreover, w’ = 1 for some r which divides 2” - 1. Thus o’-~ = w-i = 02’, and r - 1 = 2*(mod 2” - l), so r=2k+1sincebothr-land2KarebetweenOand2”-1.Ifr=2ktI,then r divides 22L- 1, and r cannot divide any smaller number of the form 2” - 1. Thus n = 2k. In summary, S is a symmetry of the irreducible polynomialf(x) I~for~~otio~Sciences 1(1968), 87-109

92

SOLOMON W. GOLOMB

over GJ’(2) if and only iff(x) has degree n = 2k, and its roots are (2“ + 1)st roots of unity. There are such polynomials of every even degree, and of course the S-symmetry is evident as a left-to-right symmetry of the coefficients. In fact, the number of such polynomials of degree 2k is ~$(2~+ 1)/2k.The first few cases are shown in Table II. TABl,E II k

Degree 2k

Period 2k + 1

1 2 3 4

2 4 6 8

3 5 9 17

Polynonlials X2-+X-k

Coefficient sequence

1

111

x4 + x3 + x2 + x -I- 1

x6 +

11111 1001001

x3 + 1

.*+x’+x~+x4+x*+x+1

111010111

x8 + XJ + x4 + x3 + 1

100111001

Iff(x) has the translation operator T as a symmetry, thenf(x + 1) =f(x), and iff(x) is irreducible, then along with o,f(x) has 1 + w as a root, so that 1 + w = wzLfor some k, since all roots off(x) are of the type mZk.Thusf(x) must be a factor of x2’ + x + 1, since the root w off(x) is also a root of x2’ -i-x + 1. The irreducible factors of x2&+ x -t- 1 can be shown to be all irreducible polynomials whose degrees divide 2k but not k, and which satisfy certain other conditions. (There is an incomplete discussion of this question in “On the Factorization of Trinomials over GF(2),” in reference[ l].)However, it is easiest to list the polynomials having Tas a symmetry by taking the list of those with S as a symmetry, and then using the fact that Sand Tare in the same conjugate class of the unimodular group to transform these into polynomials with T as a symmetry. Abstractly, suppose we have Sf, =fi, and T = P-i SP. Then let f2 = P-“f,, and we find: TV;) = (P-’ SP)(P-‘f,)

= P-’ Sfi = P-‘fi

Explicitly, since T = U-l SU, and U = U-t =

=f2_

1 0 1 1 , we can apply the ( )

transformation U, which takes w into O/(UJ+ 1), to the polynomials left fixed by S to obtain those left fixed by T. Similarly, we can apply T to the poly nomials left fixed by S to get those left fixed by U. On the assumption that S and Tare the easiest transformations to apply in practice, we can start with the list for S, apply T to get the list for U, and then apply S to those to get the list for T, as shown in Table III. The rows of this table are the “orbits” with respect to the unimodular group. It is easy to show that only the polynomial x2 + x + 1 is an orbit unto itself. rn~r~tion

Sciences 1 (1968), 87-109

THEORY OF TRANSFORMATION

GROUPS OF POLYNOMIALS

c

% -t 2 _

A++ N %

*

CI

2:

z

+++

Information Sciences 1 (1968), 87-109

93

94

SOLOMON W. GOLOMB

(One may even regard the triad x + 1, x, 1 as an orbit, whose elements, though not all of same degree, transform like the triads in the table, with the fictitious root of “1” acting as the reciprocal of the root of “x.“) Polynomials having the operators V or Was symmetries occur in dyads, and in fact symmetry under V implies symmetry under W, and conversely, since W = V2 and Y= W2. The criterion for V-symmetry is that V(w) = w2’, hence that (w + 1)/o = wzk, and that w2’[+l+ w + 1 = 0. Thus the candidates for V- Wsymmetry are precisely the factors of x2”+’ + x + 1. (The factorization of this polynomial is also discussed in “On the Factorization of Trinomials over GF(2),” in reference [l], and the degrees of the factors are seen to divide 3~) The first few factorizations are as follows: irreducible x3+x+1 x5+x+ 1 =(x2+x+ 1)(xX$x2+ 1) x9 + x + 1 irreducible xi7+x+ 1 =(x2+x+ 1)(x3+x+ 1) x (xi2 + xii + xi0 + x9 + xs + x6 + x4 + x + 1). The first dyad consists of x3 + x + 1 and x3 + x2 + 1, and the next consists of x9 + x + 1 and x9 + x8 + 1. In general, iff(x) is one member of the dyad, then the other member is S[f(x)] = TLf(x)] = Ulf(x)].

5. SHIFT REGISTER SEQUENCES, FIELD CHARACTERS, CLOSURE OF

AND FOURIER ANALYSIS ON THE

GF(2)

It is well known that maximum-length linear shift register sequences are in one-to-one correspondence with “primitive polynomials” over GF(2), i.e. polynomials over GF(2) of degree n whose roots are primitive (2” - 1)st roots of unity. It is also known that the shift register sequence may in fact be regarded as a group character on the additive group of GF(2’7. In fact, the cyclic shifts of the shift register sequence (taking the terms as +I and -l), with a constant value assigned to the O-element of the field, and the “principal character” adjoined, form a complete character table for the group. Since the shift register sequence also bears a very close relationship to the multiplicative structure of GF(2”), it might be appropriate to define jieId characters for finite fields, where the maximum-length shift register sequences are examples for the case of characteristic 2. If we take the “binary unit circle,” with the 2” - 1 roots of x2”-i = 1 marked around its circumference, then any maximum-length linear shift register sequence of degree II can be put in its “natural orientation,” so that it is constant on cyclotomic cosets-i.e., it is invariant under the automorphisms of the Information Sciences 1 (1968), 87-109

THEORY OF TRANSFORMATION

GROUPS

OF POLYNOMIALS

95

field. This invariance property is the appropriate way to define “field characters” in general. Fourier analysis abstractly is the study of group characters and their use as a basis for the representation of numerical functions defined on the group. Classical Fourier Analysis involves the characters of the multiplicative group comprising the complex unit circle. There are many analogies with the “binary unit circle.” Moreover, for communications engineering applications, the computation of the classical spectrum of a shift register sequence is of considerable practicai importance. The computation involved here is the same as first performed by Gauss in the evaluation of “Gaussian Sums,” which invoke the quadratic residue group character. Details of these calculations are explained in “Structural Properties of PN Sequences,” in reference [l]. Frequently there are other binary sequences of period 2” - 1, besides the linear shift register sequences, which have the same two-level correlation property, and balance between t-1 and -1 terms. From the theory of Hadamard difference sets, these examples all have 2 as a “multiplier,” which means they have the same automorphisms as the shift register examples. Hence they too can be obtained by assigning the values i-1 and -1 to entire cyclotomic cosets, rather than to individual terms of the sequence.

6. CROSS-CORRELATION

OF SHIFT REGISTER SEQUENCES

Let A := {a,} be a linear maximum-length shift register sequence of degree II and period p = 2” - 1, written in +l, -1 notation. Let B = {bk} be any other such sequence. If A and B are in natural orientation, then (bk} = (a,J, where (4,P) = 1.

We define the cross-correlation

c,,(T) =

of the sequences A and B to be the function

&?, ukbk++ = f akuqk+qi = 2 a&arrk+ k21 k=l

More precisely, this is the “unnormalized cross-correlation function of A and B.“’ It is quite easy to show that C.&T,) = CAB(7J whenever 71 and Q belong to the same cyclotomic coset. See, for example, “Structural Properties of PN Sequences” [ 11. We may regard a, as x(w~), where x is a mapping into {1-1,-l}, and then b, = x(w”“). If

Zn~r~a~io~ Sciences 1(1968), 87-109

SOLOMON W. GOLOMB

96

then we define

since x is constant on each Q. Moreover, IX(q)1 = r, the “size” of the cyclotomic coset corresponding to q. Letting qs stand for the cyclotomic coset of which oT is a member, we derive the following important result: MAIN THEOREM.

where where

x(w”) = @Z, where x(7,,,) = x(J) 3 ut E q,,, Wrlfn) = &

and

x(w’)*

By a theorem of Gauss, each of the products viqj can be rewritten as a linear combination of the q’s, and X(QQ) is the result of applying X as a linear operator on the expansion of qlqj. By jjr]511 we mean the number of o’s in rlT9i.e., llr/A = IWrl.,)l= X(S) -UrlT). Proof. First, we know that C,,(T) has the same value for all 7 E q. Hence,

C,a(4 ll~,ll= JT

=

JT

Cr.,&)= z: Ifi

ak-tbk

,t

=

tet)~k=l

X(~k-f)X(~Qk)

tz,k;vt

z_

X(~Qk)Xbk-f)

from which the Main Theorem follows. Examples.

1. Take A = +l, +l, t-1, -1, +l, -1, -1, and B = +l, -1, -1, +l, -1, +I, +l, with q = 3. Here the q’s are: 110= (O),

rll = (1,2,4),

Information Sciences 1(1968), 87-109

113 =

(3AO

THEORY OF T~~SFO~TION

97

GROUPS OF POLYNOMIALS

By direct calculation,

CMi@~ = -5, and

CA,(l) = C.&I C,,(3) = C,,(S) = C,,(6) = +3.

= C&?(4) = -1,

The multipfication table for the q’s is:

and since x(?o> = 1, X(Q) = 1, X(Q) = -1, we get the following table for WIi. 4 : -~

x

5%

70

I

Tr

3

Q-3

rlt

5%

3 -3 -3 3 3 3

Using the formula of the Main Theorem, we then get:

CAB(3)= Hx(7;io)X(rlo%) + X(Q) X(rtr *It,>+ x(TIJ X(rlJ *%)I =3[1*(3)+(-l)(-3)+ l-(3)1-3 which agrees with direct calculation. (Note that C,,(3) has the largest value it possibly could from the form of the representation presented by the Main Theorem.)

2. Take A =-I, +I, i-1, -i-l, +I, --I, +I, -I, +I, +I, -1, -I, +I, and ~--1,-1,-1,+1,-1,-1,~1,~1,--1,+1,--1,$-1,+1,$-~,+t, with 4=-l, and yO=(0), (3,6,9, IQ, qs = (5, IO).

~~=(1,2,4,8),

q7=(7,11,13,14),

-1,

-I,

773=

98

SOLOMON

W. GOLOMB

Then x(70) = x(715)= X(Q) = -1, while x(7,) = ~(7~) = +1 ; and we may compute the following tables :

70

170 rll

r)l

‘?I

r/7

rl3

117

‘?3

-I7

711+h + 2%

4710+

75

171

+ 7?3+ r/7

47?0+7?1 + rl3 + r/7

rl7 + 2773

‘hi-h7

r/7 + 2%

rls

‘?l +

277

r/3 + r/7

+ 2’?5 7?7+ 2%

+ 2775

r1+

r/3

+ 2115 4rlo + 37?3

rll + r/7

rll + r/7

270 + 775

+ 2%

+ 2115

115

rl3

171f 173

r/3+77

from which we obtain

-1 +4 -4 $4 -2

70

771 117 773 rl5

+4 -4 0 $8 0 0 -8 0 0 +8

+4 -2 -8 0 0 +8 $8 0 0 -4

Note that the row-sums of the X-table are the sizes of the corresponding 7)‘s. (This phenomenon may also be observed in Example 1.) We may do the computation required for the Main Theorem as follows: Regard the X-table as a 5 x 5 matrix, and normalize the rows by dividing the T*-row by l[q& Let the vector (am), x(r/A X(Q), X(Q), x(vs)) = (-1, -1, 1, 1, -1) be operated on by the normalized X-matrix as follows:

;(j

_i

-i

-;

~~[;~~~~,

indicating that C(qo) = -1, C(q7) = -5, C(Q) = -1, C(vJ = +3, and C(q5) = +7, where C(rlr) means CAB(~)for any T in the set ql. For Example 1, the corresponding matrix transformation is : z(_i

-;

Information Sciences 1 (1968), 87-109

-;j

(4)

= (-ij.

THEORY

7.

OF TRANSFORMATION

THE CYCLOTOMIC

GROUPS

OF POLYNOMIALS

99

ALGEBRA

The cyclotomic cosets for an extension of degree IZover GF(2) form the basis for an algebra over the rational field. That is, by virtue of Gauss’s result on the products of the cyclotomic cosets (or “cyclotomic periods,” as the classical literature calls them), qi*rlj = 1 CijkTh, where the scalars Cijkare k

rationals, and in fact non-negative integers. The algebra we actually obtain has a vector multiplication which is both associative and commutative, and the vector q,, acts as a multiplicative identity. The function X which occurs in the Main Theorem is a linear functional on this “Cyclotomic Algebra.” A linear shift register sequence is merely a kind of function from the basis vectors of this algebra (viz. the cyclotomic cosets themselves) to the set (1 ,-l), defined by the function x of the Main Theorem. The cross-correlation of two such sequences is itself a function on the cyclotomic cosets, and as we have seen in Example 2, the cross-correlation function can be obtained as a certain linear transformation on the x-function, in which the linear functional X is used to specify the linear transformation, by providing the coefficients of the transformation matrix.

8.

THE CYCLOTOMIC

TRANSFORMATION

MATRIX

The number of cyclotomic cosets for the (2” - 1)st roots of unity is given by

where summation is extended over all divisors d of n, where #(d) is Euler’s &function, and e,(d) is the smallest positive exponent e such that 2’~ 1 (mod d). Thus Y’”is the dimension of the cyclotomic coset algebra. As we have seen in the examples in Section 6, the Main Theorem may be interpreted as a matrix of order Y, operating on vectors of dimension Y,, in the cyclotomic algebra. The matrix elements involve X(Q*~_~), which is the same matrix for all the cross-correlations, and the trial vector whose typical element is x(~r). The product vector belongs to the same cyclotomic algebra, and gives the cross-correlation function between the “test vector” and the “reference vector” (or reference sequence) used in the computation of the transformation matrix. In fact, if the test vector is the same as the reference vector, then the product vector is the auto-correlation function of the reference vector. Information Sciences 1(1968), 87-109

100

SOLOMON W. GOLOMB

For the cases n = 3 and n = 4, we illustrate both auto- and cross-correlation, as follows : n=3 Auto

Cross

(_i-; -;)(_i);(;;),(_; -; -;)(-i)-(;). n=4 Auto

Cross

For TI= 5, there are seven cosets: q,,, ql, Q, r/9, Q,, T,~, and vz6, all except r], having six elements, and we may take ~(7~) = ~(7~) = X(Q) = ~(7,~) = +I, x(79) = x(r/27) = x(7lza> = -1. From properties of the cyclotomic transform matrix already mentioned, we can write it in the form:

MS=

lb

g

h

-1c -Id

h i

1 mn mp q

o r

lejnqst --If

k

o

u

Information Sciences 1 (1968), 87-109

i

r

j

k

t

.

THEORY OF TRANSFORMATION

GROUPS OF POLYNOMIALS

101

We further know that each row must sum to unity, and that the effect of this matrix on the column vector which is the same as the first column of the matrix must be to give us the auto-correlation vector, as follows:

These conditions give us twelve linear equations in the twenty-one unknowns a,b,c ,..., u. Actually, there are enough other constraints, based on the way the

matrix must act on the other shift register sequences of period 31, to fully specify all the matrix elements. However, we go through the direct computation of these coefficients in the next section as a further illustration of the Main Theorem.

9. THE CYCLOTOMIC TRANSFORM FOR n =

5

For n = 5, we first list the seven cyclotomic cosets: Cosets yr 770=

/

x(771)

(0)

+l +l +l -1 -1 +1 -1

rl1=11,2,4,8,14)

7j3= (3,6,12,24,17) yg = (9,18,5,10,20 yz7 = (27,23,15,30,29) 77r9=(19,7,14,28,25) 1726=(26,21,11,22,13)

Then we compute 77,*vi by Gauss’s principle for the products of the cyclotomic cosets: Basic Products of Cosets

~nfor~at~o~ Sfiences 1(1968}, 87-109

102

SOLOMON W. GOLOMB

771 ~77~ =

17[4,7,13,25,181

=‘% + = w727

%9+1726+%9+~9=r]l

v9+%9+r/3

+??26+r)26=r/3

~1.~26

-t-r/9+7)19

+2q26

=r1[w4,16,0,301 =1719+r/3

rl1*~19

+r/9+%9+'j26

57)0+7I27

+n+

=d20,8,

l&29,261

= r/9+%

+7)27+q27+v26=%

=?1[2w =727

=%0+%+~3+r/27

+%9

+r/9+2727+q26

12?23,141 +q26+rl3

+q27

+%9=r/3

+$27

+q19+r/26

This enables us to fill in the complete multiplication table (Table IV) for the cosets. Applying the function X to the entries in this table, we get the matrix: X(rlt.rlJ

770

,

773 779 r/27 719

Tz,j

73

r/9

1 5 5 5 5 5 5 5 -15 -5 -5 -5 -5 15 -5 5 -15 5 -5 -5 15

rl0

7)r

71

/

-5 -5 -5 -5 15 15 -5

r127

??19

-5 5 15 -15 -5 5 15 15 -5 -5 -5 5 -5 -5

726

-5 -5 15 -5 -5 -5 15

Next, we normalize the rows, dividing them by the sizes of the corresponding Q’S, to get: -

M=

-

1 5 5 1 1 1 1 1 -3 -1 -1 -1 -1 3 -1 1 -3 1 -1 -1 3

which is the Cyclotomic Transformation Information Sciences 1 (1968), 87-109

-5 -1 -1 -1 3 3 -1

-5 5 3 -3 -1 1 3 3 -1 -1 -1 1 -1 -1

Matrix.

-5-1 3 -1 -1 -1 3

THEORY OF TRANSFORMATION

GROUPS OF POLYNOMIALS

Znformation Sciences 1(1968),

103

87-109

104

SOLOMON W. GOLOMB

We apply M to each of the six vectors corresponding sequences, with the following reasults: _ _

M

1 1 1 -1 = -1 1 -1 _ _

zz

31 -1 -1 -1 -1 -1 -1

to shift register

d_

-9

1 -1 -1 1 -1 1 1 --

‘J L,

-11 3 -5 7 , -9 3 -1 _I

7 7 -1 -1 -9 -1

=

M

7 -1 -1 7 -1 I

--9

7 -1 = -1 7 -1 -9 --

M

-9 -9

=

-9

-1 -1 -9 -1 7 7 I

We may apply M to any sequence of period 31 which is constant on the cosets. For example, if we apply M to the quadratic residue sequence of period 31, we get:

-11 -5 3 -1 7 3 -9 -_ which is merely a rearrangement of the cross-correlation function for the case that a shift register sequence of period 31 is compared with its time-reversal.

10. THE CYCLOTOMIC

TRANSFORM

FOR ?I =

7

We start with the computations needed for the Main Theorem in this case (Tables V-VII). Information Sciences

1(1968), 87-109

THEORY OF T~NS~~ATION TABLE

105

GROUPS OF ~LYNOMIALS

V

THE COSET STRUCTURE FOR n = 7 ; p = 127

0

01

2

4

8

16

32

64

03

6

12

24

48

96

65

09

18

36

72

17

34

68

027

54

108

89

51

102

77

81

35

70

013

26

52

104

116

105

83

39

78

029

58

94

61

122

117

107

87

047

28

56

112

97

67

84

4i

82

37

74

125

123

119

111

95

063

126

121

115

103

79

031

62

124

I09

91

110

93

59

118

73

019

38

76

25

50

100

92

57

114

101

75

023

46

22

44

88

49

98

69

@

10

20

40

80

33

30

60

120

113

99

90

53

106

85

043

66 71 86

05 015 45

055

07 0tl

14 42

I~fur~a~i~n Sciences 1(1968), 87409

106

SOLOMON W. GOLOMB

TABLE VI FORp = 127, THE

BASIC 7]

1.?jJPRODUCTS

r11-~1= $2,3,5,9,17,33,65] = 71 -t 272 + 2~~ + 2~~~ 7)l.r/~=?14,7,13,25,49,97,66l=rll+r)~f211~+77,3+rf~~+?1~ r)l*r)3 = $10,19,37,73,18,35,69] = rf3 -I- qs + qg + 2q13 + 11115 + rlla %‘~4 = $2fJ, 55,109,90,52,103,78] = q5 + 76 + 7s + vj11+ 2q12 -i-7jlB ?~z’rfs= ~182,36,71,14,27,53,1051= 113 -I- 7j4-k 16 -I- Q + qg+ ~7 + qS t71’% = qP17,1~,~,40,79,30,591= 77 f 119 + -911 f qrz + 916 + 91, + rll’r17=~[95,62,123,118,108,88,483=r/2+ q4+27j~o+~~l+7jlz+7jls

QS

‘?t’qs=r)~29,57,113,98,68,8,15J=~,+t/3+~6+~,4+llls+211r, 91’79= ~185,42,83,38,75,22,43J= r]~+ 79 + 7113 + ~4 + 415+ 27j,8 ?Il’WO

=

$126,124,120,112,96,61,1271=

710 + ?), + 112+ 18 + ,)I,, + 11, + 1117

TABLE VII SUMMARY

(9)

OF PAlTERNS

Patterns: Coefficientsequence

(1) (3) (9) (27) (13) (29) (47) (7) (21) (63)

Structure 11.23

15-21 1’

Number of Ievets (2)

(3) (3) (3) (3)

(3) 15.21 0.16

(7) (7) (7) (11)

interpretationof patterns The coefficient patterns displayed in Table VII bear a very close relationship to the types of cross-correlation functions obtained. The patterns for q1 ‘71 and ?I~‘~10 are each unique. This corresponds to the fact that C4(t) is two-level if and only if q E vl, while C*(t) is maximally multi-leveled (eleven levels, in this instance) if and only if 4 E qlo. Next we observe that the coefficient patterns for ql *vs and 7, *r/e are quite similar to one another, and very different from all the others. This apparently corresponds to the fact that if and only if q E(qs or Q or 71;’ or 71;;I), the correlation function C4(t) is three-valued but nof directly transformable into a maximal linear shift register sequence. ~u~rficially, it appears that the products Q -ri2, v1 -q3, and Q *7j4,on the Information Sciences 1 j1968), 87-109

THEORY OF ~NSFO~TION

GROUPS

107

OF POLYNOMIALS

one hand, and the products Q *y7, q +Q, and y1 *r/9,on the other hand, exhibit the same pattern behavior. However the former correspond to correlation functions CJt) which are not only three-level, but also transformable into maximal linear shift register sequences; whereas the latter correspond to cor-

Key: Shading

UIInIn

71 -~iPattern

C,(t)Behavior 2-valued

11.23

.16-2’ (caseA)

3.valued"linear" 3-valued"nonlinear"

1'

7-valued

15u2' (case8)

Il-valued

0.16 l%x 1. Summaryofn==7;p=127.

relation functions C4(f) which are seven-valued ! The conjecture which immediately suggests itself is that the patterns of the two cases are somehow dissimilar-i.e., that there exists a set of permitted transformations T on coefficient sequence patterns which, if known, would interchange the patterns within a class, but would not transform between patterns of the former group and patterns of the latter group. Znfor~~iion Sciences I (1968), 87-109

108

SOLOMON

W. GOLOMB

The relevance of the q1 ‘vi coefficient sequences to the behavior of the cross-correlation functions is evident from the Main Theorem, since the “correlation transform” matrix is composed entirely of entries X(V~.qm), and every Q-T,,, is merely a cyclic permutation on the coefficient sequence of q1 *~m_t+l= ql ‘vi. The only other fact that comes into the computation of X(~r*~,,,) is the evaluation of the function x(~~), and the way the coefficient pattern interacts with the sequence {x(~~)} must be the determining factor in deciding which coefficient patterns are “equivalent” in determining crosscorrelation behavior. We may summarize both the coefficient patterns and the cross-correlation levels observed for n = 7, p = 127, by the picture shown in Fig. 1. 11.

OBSERVATIONS

CONCERNING

THREE-LEVEL

CORRELATION

As before, we define

C,(d = I=15 4 aqttr where p = 2” - 1 is the period of the maximal linear shift register sequence {a,}, of degree n. IfqE~1={1,2,4,8 ,..., r-i}, the correlation function Cq(7) becomes an auto-correlation function, and takes on only two values, C,(O) =p and C,(T) = -1 for 7 $0 (mod p). For q $ rll, the cross-correlation function Cq(7) takes on more than two distinct values. Of special interest is the case where exactly three values occur. Various conditions sufficient for this to occur have been observed : SuJikient conditions for three-valued cross-correlation 1. For degree n, any q = 2k + 1 such that n/[GCD(k,n)]

is odd leads to three-valued correlation, where the three values which occur are: -1, -1 f 2(n+e)/2where e = GC!D(K,n), (This result has been derived independently by various authors, including T. Kasami ([3]; Theorem 5), G. Solomon and R. McEleice [4], and R. Gold [5, 61). 2. If n/(GCD(n,k)) is odd, where n is the degree of the shift register, then q = 22k- 2’ + 1 leads to three-valued correlation, where the three values which occur are : -1, -1 * 2(n+e)i2where e = GCD(n, k). (This result was conjectured by the present author, and was first proved by L. R. Welch [7].) 3. For any q which leads to three-level correlation, all other members of its cyclotomic coset clearly do likewise. So too do the members of the inverse cyclotomic coset, i.e. the coset containing q’ such that qq’ E 1 (mod 2” - 1). Information

Sciences

1(1968),

87-109

THEORY OF TRANSFORMATION GROUPS OF POLYNOMIALS

109

4. It is further conjectured (by 1;. R. Welch) that the specific choice 2(“-1)‘2+ 3 leads to three-level correlation for all odd n. This conjecture has been verified for odd n G 15, and generalized to q = 2(n-‘)/2 i- 2d -t 1 for any d which divides n - 1. The method used in obtaining these results involves properties of trace mappings from larger to smaller extension fields of GF(2). The interested reader is referred to refs. 13-71 for further information.

q =

REFERENCES

I S. W. Golomb, Shift Register Sequences, Holden-Day, 1967. 2 E. S. Selmer, Linear Recurrence Relations Over Finite Fields, Department of Mathematics, Univ. of Bergen, Norway, 1966. 3 Tadao Kasami, Weight Distribution Formula For Some Class of Cyclic Codes, Report R-285, Coordinated Science Lab., Univ. of Illinois, Urbana, Ill., April 1966. 4 G. Solomon and R. McEleice, Weights of cyclic codes, Journal of Combinatorial Theory, Vol. 1, No. 4, December, 1966. 5 R. Gold, Optimum binary sequences for spread-spectrum multiplexing, ZEEE Trans. un Zn~~rmation Theory, October, 1967. 6 R. Gold, Maximal recursive sequences with three-valued recursive cross-correlation function, IEEE Trans. on Znformation Theory, January, 1968. 7 L. R. Welch, Trace Mappings in Finite Fields and Shift Register Cross-Correlation Properties, Electrical Engineering Department Report, Univ. of Southern California, to appear, 1969.

Received April 9, 1968

Information Sciences 1(1968), 87-109