AbstractCascaded GMW sequences have two-level autocorrelation functions, which have important applications in communications and cryptology. In this paper, we consider the cascaded GMW sequences corresponding to a fixed finite chain of finite fields, and determine whether the different cascaded GMW sequences are cyclically inequivalent. By introducing the so-called restricted integer systems (RISs), it is proved that all the cascaded GMW sequences can be determined by means of the RISs, and the sequences determined by different RISs are different. Moreover, different cascaded GMW sequences are cyclically inequivalent

Dai, Zong Duo

Golomb, Solomon W.

Gong, Guang

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Discrete Mathematics 219 (2000) 279{285
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Cyclic inequivalence of cascaded GMW-sequences
Solomon W. Golomba ; , Guang Gongb, Zong Duo Daic
aCommunications Sciences Institute, University of Southern California, Los Angeles,
CA 90089-2565, USA
bDepartment of Combinatorics & Optimization, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
cState Key Lab. of Information Security, Department of Mathematics, Graduate School,
University of Science and Technology of China, Beijing 100039-08, China
Received 17 May 1999; revised 28 October 1999; accepted 8 November 1999
Abstract
Cascaded GMW sequences have two-level autocorrelation functions, which have important
applications in communications and cryptology. In this paper, we consider the cascaded GMW
sequences corresponding to a xed nite chain of nite elds, and determine whether the dierent
cascaded GMW sequences are cyclically inequivalent. By introducing the so-called restricted
integer systems (RISs), it is proved that all the cascaded GMW sequences can be determined
by means of the RISs, and the sequences determined by dierent RISs are dierent. Moreover,
dierent cascaded GMW sequences are cyclically inequivalent. c© 2000 Elsevier Science B.V.
All rights reserved.
Keywords: GMW-sequences; Trace functions; Cyclic inequivalence
1. Introduction
Pseudo-random sequences have important applications in communications and cryp-
tology. Cascaded GMW sequences, sometimes called generalized GMW sequence, have
two-level autocorrelation functions. GMW sequences and generalized GMW sequences
have been extensively investigated [1{3,6{8]. The autocorrelations and linear spans of
cascaded GMW sequences have been discussed in the literature [6,2].
Given a nite chain of nite elds, it is known that each of the corresponding
cascaded GMW sequences can be determined by a primitive element and an integer
system. For sequence design in CDMA communications, stream ciphers and various
 Corresponding author.
E-mail addresses: milly@mizar.usc.edu (S.W. Golomb), ggong@cacr.math.uwaterloo.ca (G. Gong),
yangdai@mimi.cnc.ac.cn (Z.D. Dai).
0012-365X/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.
PII: S0012 -365X(00)00026 -1
280 S.W. Golomb et al. / Discrete Mathematics 219 (2000) 279{285
communications applications [9,4], guaranteeing that the sequences generated by a spe-
cic method and predetermined parameters are cyclically distinct (or cyclically in-
equivalent) is an important problem. Hence, to determine whether the cascaded GMW
sequences generated by dierent integer systems are cyclically inequivalent is very im-
portant in the implementation of cascaded GMW sequences. In the published literature,
Ref. [7] stated a result for a given eld chain of characteristic 2 without proof. In this
paper, we completely solve this problem for any characteristic and any given eld chain.
In other words, in this paper we study the problem of how to determine all dierent
cascaded GMW sequences and the problem of whether the dierent cascaded GMW
sequences are cyclically inequivalent. By introducing the so-called restricted integer
systems (RISs), we prove that all the cascaded GMW sequences can be determined
by means of restricted integer systems (RISs), and that the sequences determined by
dierent restricted integer systems (RISs) are dierent. Moreover, dierent cascaded
GMW sequences are cyclically inequivalent.
2. c-GMW sequences and c-RISs
In this paper, we always assume that p is a prime integer, and ci (06i6r) and ni
(16i6r) are positive integers satisfying
ci = ci−1ni; ni > 1; 0<i6r : (1)
The corresponding eld chain
F0F1   Fr; Fi =GF(pci) (2)
will be written simply as c = (c0; c1; : : : ; cr). We call e = (e0; e1; : : : ; er) a c-integer
system if
gcd(ei; pci − 1) = 1; 8i : (3)
Any c-integer system e=(e0; e1; : : : ; er) can determine recursively a functionf(e0 ; e1 ; :::; er)(x)
from Fr onto F0 by
f(er)(x) = x
er ; f(ei ; ei+1 ; :::; er)(x) = Tr
Fi+1
Fi (f(ei+1 ; :::; er)(x))
ei : (4)
The function f(e0 ;e1 ;:::;er)(x) will be called a c-GMW function, which together with a
primitive element  in Fr determines a cascaded GMW sequence. In other words, a
c-GMW sequence is dened by
A(e0; e1; : : : ; er; ) = faig10 ; ai = f(e0 ; e1 ; :::; er)(i) : (5)
We note that if  = k is another primitive element, then
A(e0; e1; : : : ; er; ) = A(e0; e1; : : : ; ker; ) :
Hence, in order to determine all the c-sequences, it is enough to use only one primitive
element. In the sequel, we x a primitive element , and write simply
A(e0; e1; : : : ; er) = A(e0; e1; : : : ; er; ) : (6)
S.W. Golomb et al. / Discrete Mathematics 219 (2000) 279{285 281
We note also that
A(e0; pb1e1; : : : ; pbi ei; : : : ; pbr er) = A(pb1+b2++br e0; e1; : : : ; ei; : : : ; er): (7)
Therefore, in order to reduce this kind of redundancy in determining c-GMW sequences,
we introduce a concept called c-restricted integer systems (RIS). There is a c-integer
system e = (e0; e1; : : : ; er) with the properties that 06e0<pc0 − 1 and ei is a p-coset
leader modulo pci − 1 for each i : 16i6r. Here, by saying an integer e is a p-coset
leader modulo pc − 1 we mean
e =minft j 06t <pc − 1; t  pie (modpc − 1); 06i< cg : (8)
Based on Eq. (7), it is easy to see the following theorem.
Theorem 1. Each c-GMW sequence can be determined by a c-RIS.
3. RISs and cyclic inequivalence
Two periodic sequences A = faig10 and B = fbig10 are called cyclically equivalent
if there exists an integer k such that
ai = bi+k ; 8i>0 (9)
and in this case we write A= Lk(B), or simply A  B.
The main result of this paper is
Theorem 2. Dierent c-RISs determine dierent c-GMW sequences; moreover; all the
dierent c-GMW sequences are cyclically inequivalent.
Note that
A(e) = LkA(d)
if and only if
f(e)(i) = f(d)(k+i); 8i>0 ;
where the latter is equivalent to
f(e)(x) = f(d)(kx) :
Hence, to prove Theorem 2, it is enough to prove the following theorem:
Theorem 3. Let e = (e1; e2; : : : ; er) and d = (d1; d2; : : : ; dr) be two c-RISs; and
0 6=  2 Fr . If
f(e)(x) = f(d)(x); 8x 2 Fr
then
e = d ; = 1:
282 S.W. Golomb et al. / Discrete Mathematics 219 (2000) 279{285
We will prove Theorem 3 in the next section. As an easy consequence of Theorem 2,
we get
Corollary 1. The total number of dierent c-GMW sequences is
’(pc0 − 1)
Y
16i6r
’(pci − 1)
ci
:
4. Proof of Theorem 3
It is well known that a function f(x) from a nite eld GF(q) to itself can be
uniquely represented by a polynomial of the form
f(x) =
X
06i<q
fixi; fi 2 GF(q) :
The number of nonzero coecients in the above expression will be called the weight
of f, written W (f). For an integer e, if e=
P
06i eip
i; 06ei <p, the p-weight of e,
denoted by Wp(e), is dened as
Wp(e) =
X
06i
ei : (10)
We list the following lemma whose proofs for characteristic p = 2 and p> 2 can
be found in [8,5], respectively.
Lemma 1. Let q = pc; n>1; and e =
P
i>0 eip
i; 06ei <p; and (t; qn) = 1; and
0 6=  2 GF(pn). Then
W ((TrGF(q
n)
GF(q) (x
t))e) =
Y
06i<c

n− 1 + ei
n− 1

:
In particular; when n> 1; then W ((TrGF(q
n)
GF(q) (x))
e) = n if and only if Wp(e) = 1.
Lemma 2. Let (e0pu; e1) and (d0pv; d1) be two (c0; c1)-integer systems; where
06u<c0; 06v<c0; let e0 and d0 be p-coset leaders modulo pc0 − 1; and let 
and  be elements in F1. Then
f(d0pu; d1)(x) = f(e0pv; e1)(x)
if and only if
= = 0 ;
or there exists i; 06i<n1; such that
e0 = d0; = ; d1pu  e1pv+c0i (modpc1 − 1) :
S.W. Golomb et al. / Discrete Mathematics 219 (2000) 279{285 283
Proof. It is enough to prove the ‘only if’. We write
e  e0d−10 (modpc0 − 1); 06e<pc0 − 1 :
From the assumption it is easy to see that
f(1;d1pu)(x) = f(e;e1pv)(x) : (11)
Note that Wp(1) = 1. From the assumption, it is clear that both  and  are zeros or
nonzeros. Hence if both  6= 0 and  6= 0 , applying Lemma 1, it follows that
n1 =W (f(1;d1pu)(x)) =W (f(e;e1pv)(x)) ;
and then Wp(e) = 1. So there exists j such that pj  e  e0d−10 (modpc0 − 1). Since
both e0 and d0 are p-coset leaders modulo pc0−1, then j=0 and e=1, hence e0 =d0.
Observing the polynomial expressions on the two sides of Eq. (11), we see that there
exists a monomial (x)d1p
u
on the left-hand side, and that all the monomials on the
right-hand side are of the form (x)e1p
v+c0j ; 06j<n1. So there exists i; 06i<n1,
such that
d1pu  e1pv+c0i (modpc1 − 1) ;
 d1p
u
= e1p
v+c0 i
which leads to = .
From Lemma 2 it is easy to obtain the following corollaries.
Corollary 2. Let K be an extension eld of F1; and both g(x) and h(x) be non-zero
functions from K to F1. Let both (d0pu; d1) and (e0pv; e1) be (c0; c1)-integer systems;
both d0 and e0 be p-coset leaders modulo pc0 − 1; and 06u<c0; 06v<c0. If
f(d0pu; d1)(g(x)y) = f(e0pv; e1)(h(x)y); 8y 2 F1; x 2 K;
then there exists i; 06i<n1; such that
d0 = e0; g(x) = h(x); d1pu = e1pv+c0i :
Corollary 3. Let both (d0; d1) and (e0; e1) be (c0; c1)-RISs; and let  2 F1. If
f(d0 ; d1)(x) = f(e0 ; e1)(x); 8x 2 F1 ;
then
(d0; d1) = (e0; e1); = 1 :
Proof of Theorem 3. We will proceed with induction on r. Theorem 3 is true for r=1
by Corollary 3. Now assume that r > 1. Set
e  e1e2 : : : er (modpc1 − 1) ;
d  d1d2 : : : dr (modpc1 − 1) ;
284 S.W. Golomb et al. / Discrete Mathematics 219 (2000) 279{285
and set
g(x) = f(e1 ; e2 ; :::; er)(x)
e−1 ;
h(x) = f(d1 ; d2 ; :::; dr)(x)
d−1 :
It is not dicult to see that
f(e0 ; e1 ; e2 ; :::; er)(yx) =f(e0 ; e)(yf(e1 ; e2 ; :::; er)(x)
e−1 )
=f(e0 ; e)(yg(x)); 8y 2 F1; x 2 Fr:
Similarly,
f(d0 ; d1 ; d2 ; :::; dr)(yx) = f(d0 ; d)(yh(x)); 8y 2 F1; x 2 Fr :
So we have
f(e0 ; e)(yg(x)) =f(e0 ; e1 ; e2 ; :::; er)(yx) = f(d0 ; d1 ; d2 ; :::; dr)(yx)
=f(d0 ; d)(yh(x)); 8y 2 F1; x 2 Fr :
Now we write
d0  a0pu (modpc0 − 1); 06u<c0 ;
and
e0  b0pv (modpc0 − 1); 06v<c0 ;
where both a0 and b0 are p-coset leaders modulo pc0 − 1. Notice that both f(x) and
g(x) are nonzero functions. From Corollary 2, there exists i1; 06i1<n1 such that
a0 = b0; g(x) = h(x); dpu  epv+c0i1 (modpc1 − 1) :
Then we have
f(e1pv+c0 i1 ; e2 :::er)(x) = g(x)
epv+c0 i1 = h(x)dp
u
=f(d1d2 ::: dr)(x)
pu = f(d1pu;d2 ::: dr)(x) :
By the inductive assumption we have  = 1, (e2; : : : ; er) = (d2; : : : ; dr) and
d1pu  e1pv+c0i1 (modpc1 − 1). Since both d1 and e1 are p-coset leaders modulo
pc1 − 1, we get e1 = d1 and u  v + c0i1 (mod c1), so that u = v. Consequently,
d0 = a0pu = b0pv = e0.
Acknowledgements
The work of the third author was supported by the Chinese Natural Science Foun-
dation (69473019).
S.W. Golomb et al. / Discrete Mathematics 219 (2000) 279{285 285
References
[1] M. Antweiler, L. B}omer, Complex sequences over GF(p) with a two-level autocorrelation function and
a large linear span, IEEE Trans. Inform. Theory IT-38 (1) (1992) 120{130.
[2] G. Gong, On q-ary cascaded GMW sequences, IEEE Trans. Inform. Theory 42 (1) (1996) 263{267.
[3] G. Gong, A new class of q-ary PN sequences, IEEE Trans. Inform. Theory (1997).
[4] S.W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, CA, 1967; Aegean Park Press,
Laguna Hills, CA, 1982.
[5] T. Herlestam, On functions of linear shift register sequences, Advances in Cryptology-EuroCrypt’85,
Lecture Notes in Computer Science, Vol. 219, Springer, Berlin, 1985, pp. 119{129.
[6] A. Klapper, A.H. Chan, M. Goresky, Cascaded GMW sequences, IEEE Trans. Inform. Theory IT-39 (1)
(1993) 177{183.
[7] J.S. No, Generalization of GMW sequences and No sequences, IEEE Trans. Inform. Theory 42 (1)
(1996) 260{262.
[8] R.A. Scholtz, L.R. Welch, GMW sequences, IEEE Trans. Inform. Theory IT-30 (3) (1984) 548{553.
[9] M.K. Simon, J.K. Omura, R.A. Scholtz, B.K. Levitt, Spread Spectrum Communications Handbook,
McGraw-Hill, New York, Revised version, 1994, Computer Science Press, Rockville, MD, 1985.


Cyclic inequivalence of cascaded GMW-sequences 

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Cyclic inequivalence of cascaded GMW-sequences

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