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Trace representation of some generalized cyclotomic sequences of length pq
Available online at www.sciencedirect.com
Information Sciences 178 (2008) 3307–3316 www.elsevier.com/locate/ins
Trace representation of some generalized cyclotomic sequences of length pq Xiaoni Du a,b,*, Tongjiang Yan c, Guozhen Xiao b a
College of Mathematic and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China b National Key Laboratory of ISN, Xidian University, Xi’an, Shaanxi 710071, PR China c Mathematics and Computer Science, China University of Petroleum, Dongying, Shandong 257061, PR China Received 21 April 2007; received in revised form 29 November 2007; accepted 29 November 2007
Abstract This paper contributes to trace representation of some generalized cyclotomic sequences of length pqðp; q primeÞ; which are defined by Ding and Helleseth. From the relations between these sequences and the Legendre sequence, we firstly confirm the defining pairs of these sequences of arbitrary order. Then, we obtain their trace representation, from which we give their linear complexity using Key’s method. It can be seen that Bai et al.’s conclusion is a special case of our result when the order is two. Finally, an example is given to illustrate the validity of our result. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Binary sequence; Finite field; Trace function; Generalized cyclotomic sequence; Linear complexity; Defining pair
1. Introduction Pseudo-random sequences have broad applications in stream cipher, channel coding and spread spectrum communication [13]. The linear complexity, which is the minimal degree of a linear feedback shift register (LFSR) for generating the sequence [8,9], is a valuable measure for unpredictability [2]. Having a large linear complexity implies the difficulty in the analysis of the sequence. The trace [12] representation of sequences is useful for implementing the generator of sequences and analyzing their properties. Thus, it is of great interest to represent the sequence by using the trace function. In 1962, a generalized cyclotomy with respect to pq was introduced by Whiteman [15], by which a generalized cyclotomic sequence of order two with several sound randomness properties was defined by Ding and Helleseth [6]. In 1998, a new generalized cyclotomy with respect to pe11 . . . pet t was proposed [7], which includes classical cyclotomy [3,14] as a special case. Based on this new thoery, Ding and Helleseth defined some new generalized cyclotomic sequences of order two with the length pe11 . . . pet t ; among which Bai et al. [1] considered * Corresponding author. Address: College of Mathematic and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China. E-mail addresses: [email protected], [email protected] (X. Du).
0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.11.023
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the sequence of order two of length pq and determined its linear complexity. In this paper, based on the relations between the generalized cyclotomy sequences and the Legendre sequence, we firstly confirm a defining pair of the sequence of arbitrary order, then we give the trace representations of such sequences. Finally, inspired by Key’s method [11], we obtain their linear complexity. The rest of this paper is organized as follows. Section 2 introduces the generalized cyclotomic sequence and presents some notation. Section 3 constructs a defining pair of the sequence. Section 4 contributes to its trace representation and linear complexity, and then lists an example to illustrate the validity of the result. We draw our conclusions in Section 5. 2. Preliminaries and the generalized cyclotomic sequence Let p and qðp < qÞ be two odd primes with gcdðp 1; q 1Þ ¼ 2k, and GFðpÞ the finite field with p elements. Define N ¼ pq, e ¼ ðp 1Þðq 1Þ=2k. g is a common primitive root of both p and q. The order of g modulo N is e. Let y be an integer satisfying y g mod p; y 1 mod q. Thus, we can get the multiplicative group of the residue ring Z N [15] ZN ¼ fgs y i : s ¼ 0; 1; . . . ; e 1; i ¼ 0; 1 . . . ; 2k 1g: Ding and Helleseth’s generalized cyclotomic classes of order 2k with respect to p and q [6,7] are defined by e 2k Di ¼ g2ktþi ; g2ktþi y; . . . ; g2ktþi y 2k1 : t ¼ 0; 1; . . . ; ; 2k S2k1 where i ¼ 0; 1; . . . ; 2k 1. Then Z N ¼ i¼0 Di , and Di \ Dj ¼ ;; i 6¼ j. If we define p 2k 1 q 2k 1 ðpÞ ðqÞ Di ¼ g2ksþi : s ¼ 0; 1; . . . ; ; Di ¼ g2ksþi : s ¼ 0; 1; . . . ; ; 2k 2k ðpÞ
ðpÞ
ðqÞ
D2i ;
B1 ¼
ðqÞ
then gDi ¼ Diþ1 ; gDi ¼ Diþ1 ; i ¼ 0; 1; . . . ; 2k 1. Assume B0 ¼
k[ 1 i¼0
R ¼ f0g;
k[ 1
D2iþ1 ;
ðpÞ
B0 ¼
i¼0
C0 ¼ R [
ðqÞ pB0
k[ 1
ðpÞ
D2i ;
ðpÞ
B1 ¼
i¼0
[
ðpÞ qB0
[ B0 ;
k[ 1
ðpÞ
D2iþ1 ;
i¼0
C1 ¼
ðqÞ pB1
[
ðpÞ qB1
ðqÞ
B0 ¼
k[ 1
ðqÞ
D2i ;
i¼0
ðqÞ
B1 ¼
k[ 1
ðqÞ
D2iþ1 ;
i¼0
[ B1 :
Then Z N ¼ C 0 [ C 1 , C 0 \ C 1 ¼ ;. Definition 1. The new generalized cyclotomic sequence fsðtÞg of length pq is defined by 0 if tðmod N Þ 2 C 0 sðtÞ ¼ for all t P 0: 1 if tðmod N Þ 2 C 1 Clearly, the sequence is equivalent to 8 0 if t 0ðmod pqÞ; > > > > > ðt=pÞ > if t 0ðmod pÞ and t 6¼ 0ðmod qÞ; > if t 0ðmod qÞ and t ¼ 6 0ðmod pÞ; r ðt=qÞ > p > > > > > :r t if gcdðt; pqÞ ¼ 1; q
where rð1Þ ¼ 0 and rð1Þ ¼ 1, and pt is the Legendre symbol [10] of the integer t mod p, taking the value 1 or 1 according to whether t is a quadratic residue mod p or not. In fact, when k ¼ 1, fsðtÞg is just the sequence defined by Bai et al. in [1]. Let ep ; eq be integers mod pq satisfying 1ðmod pÞ; 1ðmod qÞ; ep eq 0ðmod qÞ; 0ðmod pÞ:
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Note that ep ; eq are unique mod pq due to the Chinese Remainder Theorem. ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ Let Z p ¼ Z p n f0g; A0 ¼ fx2 : x 2 Z p g; A1 ¼ Z p n A0 . The generating polynomials A0 ðxÞ and A1 ðxÞ 2 ðpÞ ðpÞ p GFð2Þ½x=ðx 1Þ of the character sequences A0 and A1 , respectively, are defined as X X ðpÞ ðpÞ xt ðmod xp 1Þ; A1 ðxÞ ¼ xt ðmod xp 1Þ: ð1Þ A0 ðxÞ ¼ ðpÞ
ðpÞ
t2A0
t2A1
Let AðpÞ ðxÞ ¼
p1 ðpÞ ðpÞ þ a0 A0 ðxÞ þ a1 A1 ðxÞðmod xp 1Þ 2
ð2Þ
where ðai ÞðiP0Þ ¼ aiðmod 2Þ
and
ða0 ; a1 Þ ¼
ð0; 1Þ
if p 1ðmod 8Þ;
2
if p 3ðmod 8Þ:
ðx ; xÞ
Assume x 2 GFð4Þ n GFð2Þ to be chosen primitive 3th root of unity. Dai et al. [4] proved that one can always find a pth root a 6¼ 1, such that 8 1 if p 1ðmod 8Þ; > > > <0 if p 1ðmod 8Þ; ðpÞ A0 ðaÞ ¼ 2 > x if p 3ðmod 8Þ; > > : x if p 3ðmod 8Þ: It is known [4] that if a primitive pth root a of unity does not satisfy the above condition, then au must satisfy ðpÞ it, where u is some generator of Z p . This choice of a leads to A1 ðaÞ ¼ 0; 1; x; x2 for p 1; 1; 3; 3ðmod 8Þ, respectively. 3. Defining pair of the generalized cyclotomic sequence By [4], forP a binary sequence fsðtÞg of period N, there exists a primitive Nth root c of unity and a polynomial gðxÞ ¼ 06i
q1 ðqÞ ðqÞ þ b0 A0 ðxÞ þ b1 A1 ðxÞðmod xq 1Þ; 2
where the value of ðb0 ; b1 Þ is the same as that of ða0 ; a1 Þ. There exists a primitive qth root b of unity such that ðqÞ A0 ðbÞ ¼ 1; 0; x2 ; x for q 1; 1; 3; 3ðmod 8Þ, respectively [4]. To determine the defining pair of the sequence fsðtÞg, we need the following Lemmas 1–5. Now, we define the Legendre sequence bp ðtÞ with a slight modification, i.e. ( ðpÞ 1 if t 2 A1 ; bp ðtÞ ¼ 0 otherwise; then we have Lemma 1 [4]. ðAðpÞ ðxÞ; aÞ is a defining pair of the sequence fbp ðtÞg. Similarly, ðAðqÞ ðxÞ; bÞ is a defining pair of the Legendre sequence fbq ðtÞg. The d-sequence of period q, denoted by fdq ðtÞg ¼ fdq ðtÞ : t P 0g, is defined as 1 if t 0ðmod qÞ; dq ðtÞ ¼ 0 otherwise:
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P Define Dq ðxÞ ¼ 06t
¼
bp ðtÞ 0
if t 0ðmod qÞ; otherwise:
It is clear that bpðqÞ ðtÞ ¼ bp ðtÞdq ðtÞ, for all t. Lemma 2. sðtÞ ¼ bq ðtÞ þ bðqÞ p ðtÞ. Proof. The proof of Lemma 2 is straightforward; see Table 1.
h
Lemma 2 indicates that the sequence fsðtÞg has two component sequences fbq ðtÞg and fbðqÞ p ðtÞg. For these two component sequences, we have Lemma 3 [5]. The defining pairs of the two component sequences of fsðtÞg in Lemma 2 are given in Table 2. Proof. Note that ðabÞep ¼ a; ðabÞeq ¼ b, then e t
t
e t
AðpÞ ððabÞ p Þ ¼ AðpÞ ððaÞ Þ ¼ bp ðtÞ; e t
e t
t
AðqÞ ððabÞ q Þ ¼ AðqÞ ððbÞ Þ ¼ bq ðtÞ; 8t; t
AðpÞ ððabÞ p ÞDq ððabÞ q Þ ¼ AðpÞ ððaÞ ÞDq ðbt Þ ¼ bp ðtÞdq ðtÞ ¼ bpðqÞ ðtÞ; 8t: Thus, we complete the proof.
h
Lemma 4 [5]. If f ðxÞ gðxÞðmod xp 1Þ, then f ðxep Þ gðxep Þðmod xpq 1Þ. Lemma 5 [5]. With the symbols defined before, we have X 16i
X
X
xep i ¼
ðpÞ
Ai ðxep Þðmod xpq 1Þ;
i¼0;1
x
ep iþeq j
16i
¼
X
ðpÞ
ðqÞ
Ai ðxep ÞAj ðxeq Þðmod xpq 1Þ:
i¼0;1 j¼0;1
Table 1 Proof of Lemma 2 gcdðt; pqÞ ¼ 1 r qt
s(t)
pqjt
bq ðtÞ
0
bðqÞ p ðtÞ
0
0
r ðt=qÞ p
0
0
r ðt=pÞ q
r ðt=qÞ p
r qt
Sum
pjt, q-t r ðt=pÞ q
qjt, p-t 0
Table 2 Defining pairs of bq ðtÞ and bðqÞ p ðtÞ bq ðtÞ bp ðtÞ bðqÞ p ðtÞ
ðAðqÞ ðxeq Þ; abÞ ðAðpÞ ðxep Þ; abÞ ðAðpÞ ðxep ÞDq ðxeq Þ; abÞ
X. Du et al. / Information Sciences 178 (2008) 3307–3316
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Theorem 1. A defining pair of the sequence fsðtÞg is ðsðxÞ; abÞ, where X X X qþp p 1 ðqÞ eq ðpÞ ðpÞ ðqÞ þ1þ bj þ ai Ai ðxep Þ þ ai Ai ðxep ÞAj ðxeq Þðmod xpq 1Þ: sðxÞ ¼ Aj ðx Þ þ 2 2 j¼0;1 i¼0;1 i¼0;1 j¼0;1
Proof. Lemmas 2 and 3 imply that fsðtÞg has a defining pair ðsðxÞ; abÞ, where sðxÞ ¼ AðqÞ ðxeq Þ þ AðpÞ ðxep ÞDq ðxeq Þðmod xpq 1Þ: By Lemma 5, we get
! ! ! X ðqÞ q1 X p1 X ðqÞ eq ðpÞ ep sðxÞ ¼ þ þ bj Aj ðx Þ þ ai Ai ðx Þ 1þ Aj ðxeq Þ 2 2 j¼0;1 i¼0;1 j¼0;1 X X X qþp p 1 ðqÞ eq ðpÞ ðpÞ ðqÞ þ1þ ¼ bj þ ai Ai ðxep Þ þ ai Ai ðxep ÞAj ðxeq Þðmod xpq 1Þ: Aj ðx Þ þ 2 2 j¼0;1 i¼0;1 i¼0;1 j¼0;1
From Theorem 1, we find that the defining pair of the sequence fsðtÞg is independent of the order 2k.
h
4. Trace representation of the generalized cyclotomic sequence In the remaining part of this paper, let m ¼ ordp 2; n ¼ ordq 2; cp ¼ p1 ; cq ¼ q1 ; d ¼ gcdðm; nÞ; M ¼ mn . u m n d P n=k1 2ki n and v are any given generators of Z p and Z q , respectively. Let Trk ðxÞ ¼ i¼0 x be the trace of x from finite field GFð2n Þ to GFð2k Þ, where kjn. We refer the readers to [12] for detailed properties of the trace function. This section is entirely devoted to the trace representation of fsðtÞg. Our main result is provided in Theorem 2. Lemmas 6–8 are needed to prove this theorem. Lemma 6 [5]. A complete set S of representatives of conjugacy classes of the ðp 1Þðq 1Þ primitive pqth roots of unity over GFð2Þ is given as i
j
S ¼ fau bv : 0 6 i < cp ; 0 6 j < cq dg: ðpÞ
ðpÞ
Lemma 7 [7]. Let a 2 Aj , then aAi ¼ AðiþjÞðmod 2Þ , where i; j ¼ 0; 1. Lemma 8. With the same notations, (1) For p 1ðmod 8Þ, X 2iþj ðpÞ Aj ðxÞ ¼ Trm1 ðxu Þðmod xp 1Þ;
j ¼ 0; 1:
cp
06i< 2
(2) For p 3ðmod 8Þ, X i ðpÞ Trm2 ðxu Þ þ Aj ðxÞ ¼ 06i
X
i
Trm2 ðx2u Þðmod xp 1Þ;
j ¼ 0; 1:
06i
p1 , cp is even. Thus, hucp i ¼ h2i is a Proof. (1) For p 1ðmod 8Þ, 2p ¼ 1, i.e., 2 2 1ðmod pÞ, then mp1 2 proper subgroup of Z p . So we have 0 1 0 1 [ [ B [ C[B [ C ui hucp i ¼ ui h2i ¼ @ ui h2iA @ ui h2iA: Z p ¼ 06i
06i
ðpÞ
Lemma 7 implies that A0 ¼
S
06i
06i
i
ðpÞ
h2i; A1 ¼
S
06i
06i
i
h2i.
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Therefore, ðpÞ A0 ðxÞ
¼
X
cp
t
x ¼
m1 2 1 X X
ðpÞ
i¼0
t2A0 ðpÞ
A1 ðxÞ ¼
X
2i 2k
xu
k¼0
m1 XX
ðpÞ
i¼0
t2A1
2i
Trm1 ðxu Þðmod xp 1Þ;
c 06i< 2p
cp 2 1
xt ¼
X
¼
2iþ1 2k
xu
X
¼
2iþ1
Trm1 ðxu
Þðmod xp 1Þ:
cp 06i< 2
k¼0
Then the result follows. 4 , thus m is even. (ii) (2) For p 3ðmod 8Þ, 2p ¼ 1, weShave (i) mjp 1, but m- p1 p ¼ 1, and h4i is a 2 proper subgroup of Z p . Since h2i ¼ h4i 2h4i, then we obtain ! ! [ [ [ [ i i i Zp ¼ u h2i ¼ u h4i 2u h4i 06i
06i
0
1
B [ ¼@
06i
06i
0
C[B [ ui h4iA @
06i
1
0 [ C B [ 2ui h4iA @
1
06i
0 [ C B [ 2ui h4iA @
06i
1 C ui h4iA:
Lemma 7 implies that 0
ðpÞ
A0
B [ ¼@
1
06i
0 [ C B [ ui h4iA @
1
06i
0
C 2ui h4iA;
ðpÞ
A1
B [ ¼@
1
06i
0 [ C B [ 2ui h4iA @
06i
Then, we get
1 C ui h4iA:
0 1 m1 2 X X X X X i 2k i 2k C i i B ðpÞ xu 2 þ x2u 2 A ¼ Trm2 ðxu Þ þ Trm2 ðx2u Þðmod xp 1Þ A0 ðxÞ ¼ @ k¼0
and ðpÞ
A1 ðxÞ ¼
06i
X
06i
X
i
Trm2 ðxu Þ þ
06i
06i
06i
i
Trm2 ðx2u Þðmod xp 1Þ:
06i
Therefore, we have X i ðpÞ Trm2 ðxu Þ þ Aj ðxÞ ¼ 06i
X
i
Trm2 ðx2u Þðmod xp 1Þ;
j ¼ 0; 1:
06i
Theorem 2. The generalized cyclotomic sequence has a trace representation as follows (1) For q 1ðmod 8Þ, p þ q X X 2iþ1 ui t vj t þ1 þ sðtÞ ¼ Trm1 ðau t Þ þ TrM 1 ða b Þ 2 cp 06i
8 P 2i > Trn1 ðbv t Þ > < cq 06i< 2 þ P 2iþ1 > Trn1 ðbv t Þ > : cq 06i< 2
06j
if p 1ðmod 8Þ; if p 1ðmod 8Þ:
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0 1 p þ q X X i i C B þ 1 þ x2 @ sðtÞ ¼ Trm2 ðau t Þ þ Trm2 ða2u t ÞA 2 06i
i1 mod 2
0
1 X X X i i C B ui t vj t þ x@ Trm2 ðau t Þ þ Trm2 ða2u t ÞA þ x2 TrM 1 ða b Þ 06i
X
þx
06i
j
i
ut vt TrM 1 ða b Þ þ
06i
06i
8 P 2iþ1 > Trn1 ðbv t Þ > > c q > < 06i< 2
if p 3ðmod 8Þ;
> P 2i > > Trn1 ðbv t Þ > : cq
if p 3ðmod 8Þ:
06i< 2
(2) For q 3ðmod 8Þ, p þ q X X 2iþ1 ui t vj t sðtÞ ¼ þ1 þ Trm1 ðau t Þ þ TrM 1 ða b Þ 2 cp 06i
þ
06j
1 8 0 > > P i i C > B P > > Trn2 ðbv t Þ þ Trn2 ðb2v t ÞA x@ > > > 06i > i0 mod 2 i1 mod 2 > > > > > 0 1 > > > > > > P i i C B P > > Trn2 ðbv t Þ þ Trn2 ðb2v t ÞAif p 1ðmod 8Þ; > þx2 @ > > 06i > < i1 mod 2 i0 mod 2 0 1 > > > > > P i i C > B P > > x2 @ Trn2 ðbv t Þ þ Trn2 ðb2v t ÞA > > 06i > > i0 mod 2 i1 mod 2 > > > > > 0 1 > > > > > P > i i C B P > > Trn2 ðbv t Þ þ Trn2 ðb2v t ÞAif p 1ðmod 8Þ: > þx@ > : 06i
i0 mod 2
0 1 p þ q X X i i C B sðtÞ ¼ þ 1 þ x2 @ Trm2 ðau t Þ þ Trm2 ða2u t ÞA 2 06i
i1 mod 2
0
1 X X i i C B þ x@ Trm2 ðau t Þ þ Trm2 ða2u t ÞA 06i
þ x2
X 06i
06i
i
j
ut vt TrM 1 ða b Þ þ x
X 06i
i
j
ut vt TrM 1 ða b Þ
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1 8 0 > > > P i i C B P > > Trn2 ðbv t Þ þ Trn2 ðb2v t ÞA x2 @ > > > 06i > > i0 mod 2 i1 mod 2 > > > 0 1 > > > > > P > i i C B P > > þx@ Trn2 ðbv t Þ þ Trn2 ðb2v t ÞAif p 3ðmod 8Þ; > > > 06i < i1 mod 2 i0 mod 2 þ 0 1 > > > > P P i i C > B > > x Trn2 ðbv t Þ þ Trn2 ðb2v t ÞA > > @ 06i 06i > i0 mod 2 i1 mod 2 > > > > 0 1 > > > > > P i i C > B P > > þx2 @ Trn2 ðbv t Þ þ Trn2 ðb2v t ÞAif p 3ðmod 8Þ: > > : 06i
i0 mod 2
where ðaÞ ¼ 1; 0 for a 1; 0ðmod 2Þ, respectively. Proof. For p 1ðmod 8Þ, Lemmas 4 and 8 lead to X 2iþj ðpÞ Aj ðxep Þ ¼ Trm1 ðxep u Þðmod xpq 1Þ; j ¼ 0; 1: cp
06i< 2
t
Substituting x ¼ ðabÞ into the above gives X 2iþj ðpÞ ðpÞ Trm1 ðau t Þ; Aj ðxep Þjx¼ðabÞt ¼ Aj ðat Þ ¼
j ¼ 0; 1:
c
06i< 2p
For p 3ðmod 8Þ, Lemmas 4 and 8 imply that X X i i ðpÞ Trm2 ðau t Þ þ Trm2 ða2u t Þ; Aj ðxep Þjx¼ðabÞt ¼ 06i
j ¼ 0; 1:
06i
Similarly, we have, if q 1ðmod 8Þ, X 2iþj ðqÞ ðqÞ Aj ðxeq Þjx¼ðabÞt ¼ Aj ðbt Þ ¼ Trn1 ðbv t Þ;
j ¼ 0; 1;
c
06i< 2q
and if q 3ðmod 8Þ,
X
ðqÞ
Aj ðxeq Þjx¼ðabÞt ¼
X
i
Trn2 ðbv t Þ þ
06i
i
Trn2 ðb2v t Þ;
j ¼ 0; 1:
06i
From Lemma 5, we have X X X X X iþ2t jþ2s ðpÞ ðqÞ ai Ai ðxep ÞAj ðxeq Þ ¼ ai ðxep t Þ ðxeq s Þ ¼ ai xep u 1 þeq v 1 i¼0;1 j¼0;1
i¼0;1 j¼0;1
¼
ðpÞ
ðqÞ
t2Ai
X
i;j¼0;1 06t1
s2Aj
06s1
j
ai xep u þeq v ðmod xpq 1Þ :¼ nðxÞ:
06i
nððabÞ Þ ¼
X 06i
ai ðabÞ
tðep ui þeq vj Þ
¼
X 06i
i
j
t
ai ðau bv Þ ¼
X 06i
i
j
ai au t bv t ¼
X 06i
i
j
ut vt ai TrM 1 ða b Þ:
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Therefore, Theorem 1 implies that X X X pþq p 1 ðqÞ t ðpÞ ui t vj t þ1þ ÞAj ðb Þ þ sðtÞ ¼ aj Aj ðat Þ þ ðbj þ ai TrM 1 ða b Þ: 2 2 j¼0;1 j¼0;1 06i
Thus, with the definition of ða0 ; a1 Þ; ðb0 ; b1 Þ 8 > ð0; 1Þ > > < ð1; 0Þ p1 p1 ; b1 þ b0 þ ¼ > 2 2 ðx; x2 Þ > > : 2 ðx ; xÞ
and the following fact, the theorem can be proven if q 1ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ; if q 1ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ; if q 3ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ; if q 3ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ:
Key [11] has shown that if a linear feedback shift register can be represented in terms of the roots of its minimal characteristic polynomial, then the number of the roots required to represent the generator is equal to the length of the shortest linear feedback shift register that produces the sequences. Based on this idea, from Theorem 2, we have the following corollary. Corollary 1. The linear complexity LðsÞ of fsðtÞg is given by (1) If q 1ðmod 8Þ, then p þ q LðsÞ ¼ þ1 þ 2 (2) If q 3ðmod 8Þ, then p þ q þ1 þ LðsÞ ¼ 2
( pq1
if p 1ðmod 8Þ;
2
(
pq
q1 2
ðpþ1Þq 2
1 if p 1ðmod 8Þ;
pq 1
1 if p 3ðmod 8Þ:
if p 3ðmod 8Þ:
Remark. This is just Bai et al.’s conclusion when k ¼ 1 [1]. Corollary 1 determined the linear complexity of fsðtÞg for arbitrary order, i.e. 2k. Thus, it can be seen that their conclusion is only a special case of our result. Theorem 2 indicates that the trace representation of the sequence is independent of the order 2k. Thus, we have solved the problem of determining the trace representation and the linear complexity of Ding and Helleseth’s generalized cyclotomic classes with period pq of arbitrary order. Example. Without the loss of generality, we consider the case ðp; qÞ ¼ ð5; 7Þ which gives a binary sequence fsðtÞg of period 35. It is clear that p 3ðmod 8Þ and q 1ðmod 8Þ, and that ð5Þ
ð5Þ
A0 ¼ f1; 4g; A1 ¼ f2; 3g; n ¼ 3; c7 ¼ 2; e7 ¼ 15;
m ¼ 4; d ¼ 1;
c5 ¼ 1; e5 ¼ 21; M ¼ 12
ð7Þ
A0 ¼ f1; 2; 4g;
ð7Þ
A1 ¼ f3; 5; 6g;
and that ð5Þ
ð5Þ
ð7Þ
ð7Þ
A0 ðxÞ ¼ x þ x4 ; A1 ðxÞ ¼ x2 þ x3 ; A0 ðxÞ ¼ x þ x2 þ x4 ; A1 ðxÞ ¼ x3 þ x5 þ x6 : We may take u ¼ 2 and v ¼ 3, since 2 and 3 are generators of Z 5 and Z 7 , respectively. It is known that there ð5Þ exists a 5th primitive root a of unity such that A0 ðaÞ ¼ x, where x is a 3th primitive root of unity, and a 7th ð7Þ primitive root of unity b such that A0 ðbÞ ¼ 0. With these choices of a; x and b, based on Theorems 1 and 2, we get the following fact: Fact. Given the notations above, we have the generalized cyclotomic sequence fsðtÞg with the defining pair ðsðxÞ; abÞ, where X ð5Þ X ð5Þ ð7Þ ð5Þ ð5Þ ð7Þ ð7Þ sðxÞ ¼ 1 þ A1 ðx15 Þ þ x2 A0 ðx21 Þ þ xA1 ðx21 Þ þ x2 A0 ðx21 ÞAi ðx15 Þ þ x A1 ðx21 ÞAi ðx15 Þ: i¼0;1
i¼0;1
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X. Du et al. / Information Sciences 178 (2008) 3307–3316
Therefore, the trace representation for fsðtÞg is t 3 t 12 sðtÞ ¼ 1 þ Tr31 ðb3t Þ þ x2 Tr42 ðat Þ þ xTr42 ða2t Þ þ x2 Tr12 1 ððabÞ Þ þ xTr1 ððab Þ Þ
8t:
Thus, the linear complexity of the sequence is LðsÞ ¼ 1 þ 3 þ 2 þ 2 þ 12 þ 12 ¼ 32: 5. Conclusions In this correspondence, we determined the trace representation and linear complexity of the generalized cyclotomic sequence of length pq with arbitrary order. Results have shown that both of them are independent of this order. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant Nos. 60473028 and 60603010, the tackle key problem of Science Technique Committee of Gansu Province under Grant No. 2GS064-A52-035-03 and the Funds of the Education Department of Gansu Province (No. 0701-16). The authors wish to thank the anonymous referees for their detailed and very helpful comments and suggestions that improved this article. References [1] E. Bai, X. Liu, G. Xiao, Linear complexity of new generalized cyclotomic sequences of order two of length pq, IEEE Trans. Inform. Theory 51 (5) (2005) 1849–1853. [2] L. Blum, M. Blum, M. Shub, A simple unpredictable pseudo-random number generator, SIAM J. Comput. 15 (1986) 364–383. [3] Z. Chen, X. Du, G. Xiao, Sequences related to Legendre/Jacobi sequences, Inform. Sci. 177 (21) (2007) 4820–4831. [4] Z. Dai, G. Gong, H. Song, Trace representation of eth residue sequences of period p, preprint, 2002. [5] Z. Dai, G. Gong, H. Song, Trace representation of binary Jacobi sequences, http://www.cacr.math.uwaterloo.ca/, Technical Reports CORR 2002-32,2002. [6] C. Ding, Linear complexity of generalized cyclotomic binary sequence of order 2, Finite Field Appl. 3 (2) (1997) 159–174. [7] C. Ding, T. Helleseth, New generalized cyclotomy and its application, Finite Field Appl. 4 (2) (1998) 140–166. [8] S.W. Golomb, Shift Register Sequences, Holden-Day, CA, San Francisco, 1967, Revised edition: Aegean Park, CA, Laguna Hills, 1982. [9] S.W. Golomb, Theory of transformation groups of polynomials over GF(2) with applications to linear shift register sequences, Inform. Sci. 1 (1) (1968) 87–109. [10] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, 1982. [11] E.L. Key, An analysis of the structure and complexity of nonlinear binary sequence generators, IEEE Trans. Inform. Theory 22 (6) (1976) 732–736. [12] R. Lidl, H. Niederreiter, Finite Fields, in: Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, MA, 1983. [13] F.F. Shueh, W.S.E. Chen, Optimum OVSF code reassignment in Wideband CDMA forward radio link, Inform. Sci. 174 (2005) 81– 101. [14] T. Storer, Cyclotomy and Difference Set, Markham, Chicago, 1967. [15] A.L. Whiteman, A family of difference sets, Illinois J. Math. 6 (1962) 107–121.
Information Sciences 178 (2008) 3307–3316 www.elsevier.com/locate/ins
Trace representation of some generalized cyclotomic sequences of length pq Xiaoni Du a,b,*, Tongjiang Yan c, Guozhen Xiao b a
College of Mathematic and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China b National Key Laboratory of ISN, Xidian University, Xi’an, Shaanxi 710071, PR China c Mathematics and Computer Science, China University of Petroleum, Dongying, Shandong 257061, PR China Received 21 April 2007; received in revised form 29 November 2007; accepted 29 November 2007
Abstract This paper contributes to trace representation of some generalized cyclotomic sequences of length pqðp; q primeÞ; which are defined by Ding and Helleseth. From the relations between these sequences and the Legendre sequence, we firstly confirm the defining pairs of these sequences of arbitrary order. Then, we obtain their trace representation, from which we give their linear complexity using Key’s method. It can be seen that Bai et al.’s conclusion is a special case of our result when the order is two. Finally, an example is given to illustrate the validity of our result. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Binary sequence; Finite field; Trace function; Generalized cyclotomic sequence; Linear complexity; Defining pair
1. Introduction Pseudo-random sequences have broad applications in stream cipher, channel coding and spread spectrum communication [13]. The linear complexity, which is the minimal degree of a linear feedback shift register (LFSR) for generating the sequence [8,9], is a valuable measure for unpredictability [2]. Having a large linear complexity implies the difficulty in the analysis of the sequence. The trace [12] representation of sequences is useful for implementing the generator of sequences and analyzing their properties. Thus, it is of great interest to represent the sequence by using the trace function. In 1962, a generalized cyclotomy with respect to pq was introduced by Whiteman [15], by which a generalized cyclotomic sequence of order two with several sound randomness properties was defined by Ding and Helleseth [6]. In 1998, a new generalized cyclotomy with respect to pe11 . . . pet t was proposed [7], which includes classical cyclotomy [3,14] as a special case. Based on this new thoery, Ding and Helleseth defined some new generalized cyclotomic sequences of order two with the length pe11 . . . pet t ; among which Bai et al. [1] considered * Corresponding author. Address: College of Mathematic and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China. E-mail addresses: [email protected], [email protected] (X. Du).
0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.11.023
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X. Du et al. / Information Sciences 178 (2008) 3307–3316
the sequence of order two of length pq and determined its linear complexity. In this paper, based on the relations between the generalized cyclotomy sequences and the Legendre sequence, we firstly confirm a defining pair of the sequence of arbitrary order, then we give the trace representations of such sequences. Finally, inspired by Key’s method [11], we obtain their linear complexity. The rest of this paper is organized as follows. Section 2 introduces the generalized cyclotomic sequence and presents some notation. Section 3 constructs a defining pair of the sequence. Section 4 contributes to its trace representation and linear complexity, and then lists an example to illustrate the validity of the result. We draw our conclusions in Section 5. 2. Preliminaries and the generalized cyclotomic sequence Let p and qðp < qÞ be two odd primes with gcdðp 1; q 1Þ ¼ 2k, and GFðpÞ the finite field with p elements. Define N ¼ pq, e ¼ ðp 1Þðq 1Þ=2k. g is a common primitive root of both p and q. The order of g modulo N is e. Let y be an integer satisfying y g mod p; y 1 mod q. Thus, we can get the multiplicative group of the residue ring Z N [15] ZN ¼ fgs y i : s ¼ 0; 1; . . . ; e 1; i ¼ 0; 1 . . . ; 2k 1g: Ding and Helleseth’s generalized cyclotomic classes of order 2k with respect to p and q [6,7] are defined by e 2k Di ¼ g2ktþi ; g2ktþi y; . . . ; g2ktþi y 2k1 : t ¼ 0; 1; . . . ; ; 2k S2k1 where i ¼ 0; 1; . . . ; 2k 1. Then Z N ¼ i¼0 Di , and Di \ Dj ¼ ;; i 6¼ j. If we define p 2k 1 q 2k 1 ðpÞ ðqÞ Di ¼ g2ksþi : s ¼ 0; 1; . . . ; ; Di ¼ g2ksþi : s ¼ 0; 1; . . . ; ; 2k 2k ðpÞ
ðpÞ
ðqÞ
D2i ;
B1 ¼
ðqÞ
then gDi ¼ Diþ1 ; gDi ¼ Diþ1 ; i ¼ 0; 1; . . . ; 2k 1. Assume B0 ¼
k[ 1 i¼0
R ¼ f0g;
k[ 1
D2iþ1 ;
ðpÞ
B0 ¼
i¼0
C0 ¼ R [
ðqÞ pB0
k[ 1
ðpÞ
D2i ;
ðpÞ
B1 ¼
i¼0
[
ðpÞ qB0
[ B0 ;
k[ 1
ðpÞ
D2iþ1 ;
i¼0
C1 ¼
ðqÞ pB1
[
ðpÞ qB1
ðqÞ
B0 ¼
k[ 1
ðqÞ
D2i ;
i¼0
ðqÞ
B1 ¼
k[ 1
ðqÞ
D2iþ1 ;
i¼0
[ B1 :
Then Z N ¼ C 0 [ C 1 , C 0 \ C 1 ¼ ;. Definition 1. The new generalized cyclotomic sequence fsðtÞg of length pq is defined by 0 if tðmod N Þ 2 C 0 sðtÞ ¼ for all t P 0: 1 if tðmod N Þ 2 C 1 Clearly, the sequence is equivalent to 8 0 if t 0ðmod pqÞ; > > > > > ðt=pÞ > if t 0ðmod pÞ and t 6¼ 0ðmod qÞ; >
where rð1Þ ¼ 0 and rð1Þ ¼ 1, and pt is the Legendre symbol [10] of the integer t mod p, taking the value 1 or 1 according to whether t is a quadratic residue mod p or not. In fact, when k ¼ 1, fsðtÞg is just the sequence defined by Bai et al. in [1]. Let ep ; eq be integers mod pq satisfying 1ðmod pÞ; 1ðmod qÞ; ep eq 0ðmod qÞ; 0ðmod pÞ:
X. Du et al. / Information Sciences 178 (2008) 3307–3316
3309
Note that ep ; eq are unique mod pq due to the Chinese Remainder Theorem. ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ Let Z p ¼ Z p n f0g; A0 ¼ fx2 : x 2 Z p g; A1 ¼ Z p n A0 . The generating polynomials A0 ðxÞ and A1 ðxÞ 2 ðpÞ ðpÞ p GFð2Þ½x=ðx 1Þ of the character sequences A0 and A1 , respectively, are defined as X X ðpÞ ðpÞ xt ðmod xp 1Þ; A1 ðxÞ ¼ xt ðmod xp 1Þ: ð1Þ A0 ðxÞ ¼ ðpÞ
ðpÞ
t2A0
t2A1
Let AðpÞ ðxÞ ¼
p1 ðpÞ ðpÞ þ a0 A0 ðxÞ þ a1 A1 ðxÞðmod xp 1Þ 2
ð2Þ
where ðai ÞðiP0Þ ¼ aiðmod 2Þ
and
ða0 ; a1 Þ ¼
ð0; 1Þ
if p 1ðmod 8Þ;
2
if p 3ðmod 8Þ:
ðx ; xÞ
Assume x 2 GFð4Þ n GFð2Þ to be chosen primitive 3th root of unity. Dai et al. [4] proved that one can always find a pth root a 6¼ 1, such that 8 1 if p 1ðmod 8Þ; > > > <0 if p 1ðmod 8Þ; ðpÞ A0 ðaÞ ¼ 2 > x if p 3ðmod 8Þ; > > : x if p 3ðmod 8Þ: It is known [4] that if a primitive pth root a of unity does not satisfy the above condition, then au must satisfy ðpÞ it, where u is some generator of Z p . This choice of a leads to A1 ðaÞ ¼ 0; 1; x; x2 for p 1; 1; 3; 3ðmod 8Þ, respectively. 3. Defining pair of the generalized cyclotomic sequence By [4], forP a binary sequence fsðtÞg of period N, there exists a primitive Nth root c of unity and a polynomial gðxÞ ¼ 06i
q1 ðqÞ ðqÞ þ b0 A0 ðxÞ þ b1 A1 ðxÞðmod xq 1Þ; 2
where the value of ðb0 ; b1 Þ is the same as that of ða0 ; a1 Þ. There exists a primitive qth root b of unity such that ðqÞ A0 ðbÞ ¼ 1; 0; x2 ; x for q 1; 1; 3; 3ðmod 8Þ, respectively [4]. To determine the defining pair of the sequence fsðtÞg, we need the following Lemmas 1–5. Now, we define the Legendre sequence bp ðtÞ with a slight modification, i.e. ( ðpÞ 1 if t 2 A1 ; bp ðtÞ ¼ 0 otherwise; then we have Lemma 1 [4]. ðAðpÞ ðxÞ; aÞ is a defining pair of the sequence fbp ðtÞg. Similarly, ðAðqÞ ðxÞ; bÞ is a defining pair of the Legendre sequence fbq ðtÞg. The d-sequence of period q, denoted by fdq ðtÞg ¼ fdq ðtÞ : t P 0g, is defined as 1 if t 0ðmod qÞ; dq ðtÞ ¼ 0 otherwise:
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P Define Dq ðxÞ ¼ 06t
¼
bp ðtÞ 0
if t 0ðmod qÞ; otherwise:
It is clear that bpðqÞ ðtÞ ¼ bp ðtÞdq ðtÞ, for all t. Lemma 2. sðtÞ ¼ bq ðtÞ þ bðqÞ p ðtÞ. Proof. The proof of Lemma 2 is straightforward; see Table 1.
h
Lemma 2 indicates that the sequence fsðtÞg has two component sequences fbq ðtÞg and fbðqÞ p ðtÞg. For these two component sequences, we have Lemma 3 [5]. The defining pairs of the two component sequences of fsðtÞg in Lemma 2 are given in Table 2. Proof. Note that ðabÞep ¼ a; ðabÞeq ¼ b, then e t
t
e t
AðpÞ ððabÞ p Þ ¼ AðpÞ ððaÞ Þ ¼ bp ðtÞ; e t
e t
t
AðqÞ ððabÞ q Þ ¼ AðqÞ ððbÞ Þ ¼ bq ðtÞ; 8t; t
AðpÞ ððabÞ p ÞDq ððabÞ q Þ ¼ AðpÞ ððaÞ ÞDq ðbt Þ ¼ bp ðtÞdq ðtÞ ¼ bpðqÞ ðtÞ; 8t: Thus, we complete the proof.
h
Lemma 4 [5]. If f ðxÞ gðxÞðmod xp 1Þ, then f ðxep Þ gðxep Þðmod xpq 1Þ. Lemma 5 [5]. With the symbols defined before, we have X 16i
X
X
xep i ¼
ðpÞ
Ai ðxep Þðmod xpq 1Þ;
i¼0;1
x
ep iþeq j
16i
¼
X
ðpÞ
ðqÞ
Ai ðxep ÞAj ðxeq Þðmod xpq 1Þ:
i¼0;1 j¼0;1
Table 1 Proof of Lemma 2 gcdðt; pqÞ ¼ 1 r qt
s(t)
pqjt
bq ðtÞ
0
bðqÞ p ðtÞ
0
0
r ðt=qÞ p
0
0
r ðt=pÞ q
r ðt=qÞ p
r qt
Sum
pjt, q-t r ðt=pÞ q
qjt, p-t 0
Table 2 Defining pairs of bq ðtÞ and bðqÞ p ðtÞ bq ðtÞ bp ðtÞ bðqÞ p ðtÞ
ðAðqÞ ðxeq Þ; abÞ ðAðpÞ ðxep Þ; abÞ ðAðpÞ ðxep ÞDq ðxeq Þ; abÞ
X. Du et al. / Information Sciences 178 (2008) 3307–3316
3311
Theorem 1. A defining pair of the sequence fsðtÞg is ðsðxÞ; abÞ, where X X X qþp p 1 ðqÞ eq ðpÞ ðpÞ ðqÞ þ1þ bj þ ai Ai ðxep Þ þ ai Ai ðxep ÞAj ðxeq Þðmod xpq 1Þ: sðxÞ ¼ Aj ðx Þ þ 2 2 j¼0;1 i¼0;1 i¼0;1 j¼0;1
Proof. Lemmas 2 and 3 imply that fsðtÞg has a defining pair ðsðxÞ; abÞ, where sðxÞ ¼ AðqÞ ðxeq Þ þ AðpÞ ðxep ÞDq ðxeq Þðmod xpq 1Þ: By Lemma 5, we get
! ! ! X ðqÞ q1 X p1 X ðqÞ eq ðpÞ ep sðxÞ ¼ þ þ bj Aj ðx Þ þ ai Ai ðx Þ 1þ Aj ðxeq Þ 2 2 j¼0;1 i¼0;1 j¼0;1 X X X qþp p 1 ðqÞ eq ðpÞ ðpÞ ðqÞ þ1þ ¼ bj þ ai Ai ðxep Þ þ ai Ai ðxep ÞAj ðxeq Þðmod xpq 1Þ: Aj ðx Þ þ 2 2 j¼0;1 i¼0;1 i¼0;1 j¼0;1
From Theorem 1, we find that the defining pair of the sequence fsðtÞg is independent of the order 2k.
h
4. Trace representation of the generalized cyclotomic sequence In the remaining part of this paper, let m ¼ ordp 2; n ¼ ordq 2; cp ¼ p1 ; cq ¼ q1 ; d ¼ gcdðm; nÞ; M ¼ mn . u m n d P n=k1 2ki n and v are any given generators of Z p and Z q , respectively. Let Trk ðxÞ ¼ i¼0 x be the trace of x from finite field GFð2n Þ to GFð2k Þ, where kjn. We refer the readers to [12] for detailed properties of the trace function. This section is entirely devoted to the trace representation of fsðtÞg. Our main result is provided in Theorem 2. Lemmas 6–8 are needed to prove this theorem. Lemma 6 [5]. A complete set S of representatives of conjugacy classes of the ðp 1Þðq 1Þ primitive pqth roots of unity over GFð2Þ is given as i
j
S ¼ fau bv : 0 6 i < cp ; 0 6 j < cq dg: ðpÞ
ðpÞ
Lemma 7 [7]. Let a 2 Aj , then aAi ¼ AðiþjÞðmod 2Þ , where i; j ¼ 0; 1. Lemma 8. With the same notations, (1) For p 1ðmod 8Þ, X 2iþj ðpÞ Aj ðxÞ ¼ Trm1 ðxu Þðmod xp 1Þ;
j ¼ 0; 1:
cp
06i< 2
(2) For p 3ðmod 8Þ, X i ðpÞ Trm2 ðxu Þ þ Aj ðxÞ ¼ 06i
X
i
Trm2 ðx2u Þðmod xp 1Þ;
j ¼ 0; 1:
06i
p1 , cp is even. Thus, hucp i ¼ h2i is a Proof. (1) For p 1ðmod 8Þ, 2p ¼ 1, i.e., 2 2 1ðmod pÞ, then mp1 2 proper subgroup of Z p . So we have 0 1 0 1 [ [ B [ C[B [ C ui hucp i ¼ ui h2i ¼ @ ui h2iA @ ui h2iA: Z p ¼ 06i
06i
ðpÞ
Lemma 7 implies that A0 ¼
S
06i
06i
i
ðpÞ
h2i; A1 ¼
S
06i
06i
i
h2i.
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Therefore, ðpÞ A0 ðxÞ
¼
X
cp
t
x ¼
m1 2 1 X X
ðpÞ
i¼0
t2A0 ðpÞ
A1 ðxÞ ¼
X
2i 2k
xu
k¼0
m1 XX
ðpÞ
i¼0
t2A1
2i
Trm1 ðxu Þðmod xp 1Þ;
c 06i< 2p
cp 2 1
xt ¼
X
¼
2iþ1 2k
xu
X
¼
2iþ1
Trm1 ðxu
Þðmod xp 1Þ:
cp 06i< 2
k¼0
Then the result follows. 4 , thus m is even. (ii) (2) For p 3ðmod 8Þ, 2p ¼ 1, weShave (i) mjp 1, but m- p1 p ¼ 1, and h4i is a 2 proper subgroup of Z p . Since h2i ¼ h4i 2h4i, then we obtain ! ! [ [ [ [ i i i Zp ¼ u h2i ¼ u h4i 2u h4i 06i
06i
0
1
B [ ¼@
06i
06i
0
C[B [ ui h4iA @
06i
1
0 [ C B [ 2ui h4iA @
1
06i
0 [ C B [ 2ui h4iA @
06i
1 C ui h4iA:
Lemma 7 implies that 0
ðpÞ
A0
B [ ¼@
1
06i
0 [ C B [ ui h4iA @
1
06i
0
C 2ui h4iA;
ðpÞ
A1
B [ ¼@
1
06i
0 [ C B [ 2ui h4iA @
06i
Then, we get
1 C ui h4iA:
0 1 m1 2 X X X X X i 2k i 2k C i i B ðpÞ xu 2 þ x2u 2 A ¼ Trm2 ðxu Þ þ Trm2 ðx2u Þðmod xp 1Þ A0 ðxÞ ¼ @ k¼0
and ðpÞ
A1 ðxÞ ¼
06i
X
06i
X
i
Trm2 ðxu Þ þ
06i
06i
06i
i
Trm2 ðx2u Þðmod xp 1Þ:
06i
Therefore, we have X i ðpÞ Trm2 ðxu Þ þ Aj ðxÞ ¼ 06i
X
i
Trm2 ðx2u Þðmod xp 1Þ;
j ¼ 0; 1:
06i
Theorem 2. The generalized cyclotomic sequence has a trace representation as follows (1) For q 1ðmod 8Þ, p þ q X X 2iþ1 ui t vj t þ1 þ sðtÞ ¼ Trm1 ðau t Þ þ TrM 1 ða b Þ 2 cp 06i
8 P 2i > Trn1 ðbv t Þ > < cq 06i< 2 þ P 2iþ1 > Trn1 ðbv t Þ > : cq 06i< 2
06j
if p 1ðmod 8Þ; if p 1ðmod 8Þ:
X. Du et al. / Information Sciences 178 (2008) 3307–3316
0 1 p þ q X X i i C B þ 1 þ x2 @ sðtÞ ¼ Trm2 ðau t Þ þ Trm2 ða2u t ÞA 2 06i
i1 mod 2
0
1 X X X i i C B ui t vj t þ x@ Trm2 ðau t Þ þ Trm2 ða2u t ÞA þ x2 TrM 1 ða b Þ 06i
X
þx
06i
j
i
ut vt TrM 1 ða b Þ þ
06i
06i
8 P 2iþ1 > Trn1 ðbv t Þ > > c q > < 06i< 2
if p 3ðmod 8Þ;
> P 2i > > Trn1 ðbv t Þ > : cq
if p 3ðmod 8Þ:
06i< 2
(2) For q 3ðmod 8Þ, p þ q X X 2iþ1 ui t vj t sðtÞ ¼ þ1 þ Trm1 ðau t Þ þ TrM 1 ða b Þ 2 cp 06i
þ
06j
1 8 0 > > P i i C > B P > > Trn2 ðbv t Þ þ Trn2 ðb2v t ÞA x@ > > > 06i
i0 mod 2
0 1 p þ q X X i i C B sðtÞ ¼ þ 1 þ x2 @ Trm2 ðau t Þ þ Trm2 ða2u t ÞA 2 06i
i1 mod 2
0
1 X X i i C B þ x@ Trm2 ðau t Þ þ Trm2 ða2u t ÞA 06i
þ x2
X 06i
06i
i
j
ut vt TrM 1 ða b Þ þ x
X 06i
i
j
ut vt TrM 1 ða b Þ
3313
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X. Du et al. / Information Sciences 178 (2008) 3307–3316
1 8 0 > > > P i i C B P > > Trn2 ðbv t Þ þ Trn2 ðb2v t ÞA x2 @ > > > 06i
i0 mod 2
where ðaÞ ¼ 1; 0 for a 1; 0ðmod 2Þ, respectively. Proof. For p 1ðmod 8Þ, Lemmas 4 and 8 lead to X 2iþj ðpÞ Aj ðxep Þ ¼ Trm1 ðxep u Þðmod xpq 1Þ; j ¼ 0; 1: cp
06i< 2
t
Substituting x ¼ ðabÞ into the above gives X 2iþj ðpÞ ðpÞ Trm1 ðau t Þ; Aj ðxep Þjx¼ðabÞt ¼ Aj ðat Þ ¼
j ¼ 0; 1:
c
06i< 2p
For p 3ðmod 8Þ, Lemmas 4 and 8 imply that X X i i ðpÞ Trm2 ðau t Þ þ Trm2 ða2u t Þ; Aj ðxep Þjx¼ðabÞt ¼ 06i
j ¼ 0; 1:
06i
Similarly, we have, if q 1ðmod 8Þ, X 2iþj ðqÞ ðqÞ Aj ðxeq Þjx¼ðabÞt ¼ Aj ðbt Þ ¼ Trn1 ðbv t Þ;
j ¼ 0; 1;
c
06i< 2q
and if q 3ðmod 8Þ,
X
ðqÞ
Aj ðxeq Þjx¼ðabÞt ¼
X
i
Trn2 ðbv t Þ þ
06i
i
Trn2 ðb2v t Þ;
j ¼ 0; 1:
06i
From Lemma 5, we have X X X X X iþ2t jþ2s ðpÞ ðqÞ ai Ai ðxep ÞAj ðxeq Þ ¼ ai ðxep t Þ ðxeq s Þ ¼ ai xep u 1 þeq v 1 i¼0;1 j¼0;1
i¼0;1 j¼0;1
¼
ðpÞ
ðqÞ
t2Ai
X
i;j¼0;1 06t1
s2Aj
06s1
j
ai xep u þeq v ðmod xpq 1Þ :¼ nðxÞ:
06i
nððabÞ Þ ¼
X 06i
ai ðabÞ
tðep ui þeq vj Þ
¼
X 06i
i
j
t
ai ðau bv Þ ¼
X 06i
i
j
ai au t bv t ¼
X 06i
i
j
ut vt ai TrM 1 ða b Þ:
X. Du et al. / Information Sciences 178 (2008) 3307–3316
3315
Therefore, Theorem 1 implies that X X X pþq p 1 ðqÞ t ðpÞ ui t vj t þ1þ ÞAj ðb Þ þ sðtÞ ¼ aj Aj ðat Þ þ ðbj þ ai TrM 1 ða b Þ: 2 2 j¼0;1 j¼0;1 06i
Thus, with the definition of ða0 ; a1 Þ; ðb0 ; b1 Þ 8 > ð0; 1Þ > > < ð1; 0Þ p1 p1 ; b1 þ b0 þ ¼ > 2 2 ðx; x2 Þ > > : 2 ðx ; xÞ
and the following fact, the theorem can be proven if q 1ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ; if q 1ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ; if q 3ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ; if q 3ðmod 8Þ and ðp 1ðmod 8Þ or p 3ðmod 8ÞÞ:
Key [11] has shown that if a linear feedback shift register can be represented in terms of the roots of its minimal characteristic polynomial, then the number of the roots required to represent the generator is equal to the length of the shortest linear feedback shift register that produces the sequences. Based on this idea, from Theorem 2, we have the following corollary. Corollary 1. The linear complexity LðsÞ of fsðtÞg is given by (1) If q 1ðmod 8Þ, then p þ q LðsÞ ¼ þ1 þ 2 (2) If q 3ðmod 8Þ, then p þ q þ1 þ LðsÞ ¼ 2
( pq1
if p 1ðmod 8Þ;
2
(
pq
q1 2
ðpþ1Þq 2
1 if p 1ðmod 8Þ;
pq 1
1 if p 3ðmod 8Þ:
if p 3ðmod 8Þ:
Remark. This is just Bai et al.’s conclusion when k ¼ 1 [1]. Corollary 1 determined the linear complexity of fsðtÞg for arbitrary order, i.e. 2k. Thus, it can be seen that their conclusion is only a special case of our result. Theorem 2 indicates that the trace representation of the sequence is independent of the order 2k. Thus, we have solved the problem of determining the trace representation and the linear complexity of Ding and Helleseth’s generalized cyclotomic classes with period pq of arbitrary order. Example. Without the loss of generality, we consider the case ðp; qÞ ¼ ð5; 7Þ which gives a binary sequence fsðtÞg of period 35. It is clear that p 3ðmod 8Þ and q 1ðmod 8Þ, and that ð5Þ
ð5Þ
A0 ¼ f1; 4g; A1 ¼ f2; 3g; n ¼ 3; c7 ¼ 2; e7 ¼ 15;
m ¼ 4; d ¼ 1;
c5 ¼ 1; e5 ¼ 21; M ¼ 12
ð7Þ
A0 ¼ f1; 2; 4g;
ð7Þ
A1 ¼ f3; 5; 6g;
and that ð5Þ
ð5Þ
ð7Þ
ð7Þ
A0 ðxÞ ¼ x þ x4 ; A1 ðxÞ ¼ x2 þ x3 ; A0 ðxÞ ¼ x þ x2 þ x4 ; A1 ðxÞ ¼ x3 þ x5 þ x6 : We may take u ¼ 2 and v ¼ 3, since 2 and 3 are generators of Z 5 and Z 7 , respectively. It is known that there ð5Þ exists a 5th primitive root a of unity such that A0 ðaÞ ¼ x, where x is a 3th primitive root of unity, and a 7th ð7Þ primitive root of unity b such that A0 ðbÞ ¼ 0. With these choices of a; x and b, based on Theorems 1 and 2, we get the following fact: Fact. Given the notations above, we have the generalized cyclotomic sequence fsðtÞg with the defining pair ðsðxÞ; abÞ, where X ð5Þ X ð5Þ ð7Þ ð5Þ ð5Þ ð7Þ ð7Þ sðxÞ ¼ 1 þ A1 ðx15 Þ þ x2 A0 ðx21 Þ þ xA1 ðx21 Þ þ x2 A0 ðx21 ÞAi ðx15 Þ þ x A1 ðx21 ÞAi ðx15 Þ: i¼0;1
i¼0;1
3316
X. Du et al. / Information Sciences 178 (2008) 3307–3316
Therefore, the trace representation for fsðtÞg is t 3 t 12 sðtÞ ¼ 1 þ Tr31 ðb3t Þ þ x2 Tr42 ðat Þ þ xTr42 ða2t Þ þ x2 Tr12 1 ððabÞ Þ þ xTr1 ððab Þ Þ
8t:
Thus, the linear complexity of the sequence is LðsÞ ¼ 1 þ 3 þ 2 þ 2 þ 12 þ 12 ¼ 32: 5. Conclusions In this correspondence, we determined the trace representation and linear complexity of the generalized cyclotomic sequence of length pq with arbitrary order. Results have shown that both of them are independent of this order. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant Nos. 60473028 and 60603010, the tackle key problem of Science Technique Committee of Gansu Province under Grant No. 2GS064-A52-035-03 and the Funds of the Education Department of Gansu Province (No. 0701-16). The authors wish to thank the anonymous referees for their detailed and very helpful comments and suggestions that improved this article. References [1] E. Bai, X. Liu, G. Xiao, Linear complexity of new generalized cyclotomic sequences of order two of length pq, IEEE Trans. Inform. Theory 51 (5) (2005) 1849–1853. [2] L. Blum, M. Blum, M. Shub, A simple unpredictable pseudo-random number generator, SIAM J. Comput. 15 (1986) 364–383. [3] Z. Chen, X. Du, G. Xiao, Sequences related to Legendre/Jacobi sequences, Inform. Sci. 177 (21) (2007) 4820–4831. [4] Z. Dai, G. Gong, H. Song, Trace representation of eth residue sequences of period p, preprint, 2002. [5] Z. Dai, G. Gong, H. Song, Trace representation of binary Jacobi sequences, http://www.cacr.math.uwaterloo.ca/, Technical Reports CORR 2002-32,2002. [6] C. Ding, Linear complexity of generalized cyclotomic binary sequence of order 2, Finite Field Appl. 3 (2) (1997) 159–174. [7] C. Ding, T. Helleseth, New generalized cyclotomy and its application, Finite Field Appl. 4 (2) (1998) 140–166. [8] S.W. Golomb, Shift Register Sequences, Holden-Day, CA, San Francisco, 1967, Revised edition: Aegean Park, CA, Laguna Hills, 1982. [9] S.W. Golomb, Theory of transformation groups of polynomials over GF(2) with applications to linear shift register sequences, Inform. Sci. 1 (1) (1968) 87–109. [10] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, 1982. [11] E.L. Key, An analysis of the structure and complexity of nonlinear binary sequence generators, IEEE Trans. Inform. Theory 22 (6) (1976) 732–736. [12] R. Lidl, H. Niederreiter, Finite Fields, in: Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, MA, 1983. [13] F.F. Shueh, W.S.E. Chen, Optimum OVSF code reassignment in Wideband CDMA forward radio link, Inform. Sci. 174 (2005) 81– 101. [14] T. Storer, Cyclotomy and Difference Set, Markham, Chicago, 1967. [15] A.L. Whiteman, A family of difference sets, Illinois J. Math. 6 (1962) 107–121.