Linear Complexity and Autocorrelation of Prime Cube ... - Springer Link

Linear Complexity and Autocorrelation of Prime Cube Sequences Young-Joon Kim, Seok-Yong Jin, and Hong-Yeop Song Department of Electrical and Electronic Engineering Yonsei University, Seoul, 121-749, Korea {yj.kim, sy.jin, hysong}@yonsei.ac.kr

Abstract. We review a binary sequence based on the generalized cyclotomy of order 2 with respect to p3 , where p is an odd prime. Linear complexities, minimal polynomials and autocorrelation of these sequences are computed.

1

Introduction

Let n ≥ 2 be a positive integer and Zn∗ be the multiplicative group of the integer ring Zn . For a partition {Di |i = 0, 1, · · · , d − 1} of Zn∗ , if there exist elements g1 , · · · , gd of Zn∗ satisfying Di = gi D0 for all i where D0 is a multiplicative subgroup of Zn∗ , the Di are called generalized cyclotomic classes of order d. In 1998, Ding and Helleseth [1] introduced the new generalized cyclotomy with respect to pe11 · · · pet t and defined a balanced binary sequence based on their own generalized cyclotomy, where p1 , · · · , pt are distinct odd primes and e1 , · · · , et are positive integers. Before them, there have been lots of studies about cyclotomy, but they are only about ones with respect to p or p2 or pq where p and q are distinct odd primes [1,4,7,8]. In [1] they also introduced how to construct a balanced binary sequence based on their generalized cyclotomy. Let it call the generalized cyclotomic sequences. Those sequences includes the binary quadratic residue sequences also known as Legendre Sequences because these sequences can be understood as the generalized cyclotomic sequences with respect to p. In 1998, C. Ding [4] presented some cyclotomy sequences with period p2 which are not balanced. They are defined in a slightly different way from the generalized cyclotomic sequences with respect to p2 . In that paper, he calculated the linear complexities with minor errors. Y.-H. Park and others [5] corrected the errors. The linear complexity of the sequence is not so good. In general, the linear complexity of a sequence is considered as good when it is not less than half of the period of the sequence. Recently, in [7], Yan et al. calculated the linear complexity and autocorrelation of generalized cyclotomic sequences of order 2 with respect to p2 . In this paper, we compute the linear complexity and autocorrelation of the generalized cyclotomic sequences with respect to p3 . Hereafter we will call these sequences as prime cube sequences. S. Bozta¸s and H.F. Lu (Eds.): AAECC 2007, LNCS 4851, pp. 188–197, 2007. c Springer-Verlag Berlin Heidelberg 2007 

Linear Complexity and Autocorrelation of Prime Cube Sequences

2

189

Prime Cube Sequences

Let p be an odd prime. Let g be a primitive root of p2 . Then it’s well known that g is also a primitive root of pk for k ≥ 1[2]. The order of g modulo p is p − 1, the order of g modulo p2 is p(p − 1) and the order of g modulo p3 is p2 (p − 1). Define (p) (p) (p) D0 = (g 2 ) (mod p) D1 = gD0 (mod p) (p2 )

D0

3

(p ) D0

Then Zp∗ = D0

(p)

(p2 )

= (g 2 ) (mod p2 ) D1 = (g 2 ) (mod p3 ) (p2 )

∪ D1 ,Zp∗2 = D0 (p)

3

(p ) D1

= gD0

(p2 )

(mod p2 )

=

(p3 ) gD0

(mod p3 )

(p2 )

(p3 )

and Zp∗3 = D0

∪ D1

(p3 )

∪ D1

. For

(pi ) Dj

are called generalized cyclotomic classes of order 2 with i = 0, 1, 2, the j respect to p . Note that (p3 )

Zp3 = D0

(p3 )

∪ D1

(p2 )

∪ pD0

(p2 )

∪ pD1

(p)

(p)

∪ p2 D0 ∪ p2 D1 ∪ {0}.

i p3 (p ) are sets of elements pi Dj (pi ) Dj over Zp3 for i = 0, 1, 2 and

Here and hereafter,

obtained by multiplying

p3 pi

to

j = 0, 1. the elements of In [1], the authors define the binary prime cube sequence {s(n)} as follows[1]:  0, if (i mod p3 ) ∈ C0 s(i) = (1) 1, if (i mod p3 ) ∈ C1 . where C0 =

3



p3 (d) d|p3 ,d>1 d D0

and C1 = {0} ∪



p3 (d) d|p3 ,d>1 d D1 .

Linear Complexity and Minimal Polynomial

Let {s(n)} be a sequence of period L over a field F . The linear complexity of {s(n)} is defined to be least positive integer l such that there are constants c0 = 1, c1 , · · · , cl ∈ F satisfying −s(i) = c1 s(i − 1) + c2 s(i − 2) + · · · + cl s(i − l) for all l ≤ i < L The polynomial c(x) = c0 + c1 x + · · · + cl xl is called a minimal polynomial of {s(n)}. Let {s(n)} be a sequence of period L over a field F , and S(x) = s(0) + s(1)x + · · · + s(L − 1)xL−1 . It is well known that[3] 1. the mimimal polynomial of {s(n)} is given by c(x) = (xL − 1)/ gcd(xL − 1, S(x)) 2. the linear complexity of {s(n)} is given by CL = L − deg(gcd(xL − 1, S(x)))

190

Y.-J. Kim, S.-Y. Jin, and H.-Y. Song

Lemma 1. For a ∈ Zp∗3 and 1 ≤ i ≤ 3  (pi ) aD0

=

(pi )

, if a ∈ D0

D1

, if a ∈ D1

(pi )



(pi )

D0

(pi )

(pi ) aD1

,

=

(pi )

(pi )

D1

, if a ∈ D0

D0

, if a ∈ D1

(pi )

(pi )

.

Proof. It can be proved in the same way as [4]. (p3 )

Lemma 2. Let b be any integer. Then Di for i = 0, 1.

(p3 )

+bp = Di

(p2 )

and Di

(p2 )

+bp = Di

Proof. It can also be proved in the same way as [4]. (p3 )

(p2 )

Lemma 3. −1 (mod p3 ) ∈ D0 if and only if −1 (mod p2 ) ∈ D0 (p) only if −1 (mod p) ∈ D0 if and only if p ≡ 1 (mod 4). (p)

Proof. It is well known that −1 (mod p) ∈ D0 Using Lemma 2, we can show −1 (mod p) ∈ 3

(p )

and −1 (mod p3 ) ∈ D0 (p3 )

Lemma 4. 2 ∈ Di

if and

if and only if p ≡ 1 (mod 4)[2].

(p) D0

(p2 )

implies −1 (mod p2 ) ∈ D0

. The converse is obvious. (p2 )

if and only if 2 ∈ Di

(p)

if and only if 2 ∈ Di

for i = 0, 1.

Proof. It can be proved in the same way as [4]. Let m be the order of 2 modulo p3 and θ a primitive p3 th root of unity in GF (2m ). Define S(x) =



xi = 1 + (



(p3 ) i∈D1

i∈C1



+

(p2 ) i∈pD1

+



)xi ∈ GF (2)[x].

(p) i∈p2 D1

Then S(x) is generating function of the prime cube sequence {s(n)} defined before. To compute S(θ), we use the generalized cyclotomic numbers of order 2 with respect to pi for i ≥ 1 defined by (pk )

(i, j)pk = |(Di

(pk )

+ 1) ∩ Dj

|

i, j = 0, 1, and k = 0, 1, 2.

Lemma 5. [1] If p ≡ 3 (mod 4), then (1,0)pk = (0,0)pk = (1,1)pk =

pk−1(p−3) pk−1(p+1) , and (0,1)pk = . 4 4

If p ≡ 1 (mod 4), then (0,1)pk = (1,0)pk = (1,1)pk =

pk−1(p−1) pk−1(p−5) , and (0,0)pk = . 4 4

(2)

Linear Complexity and Autocorrelation of Prime Cube Sequences

191

Note that 3

2

2

2

2

2

0 = θp − 1 = (θp )p − 1 = (θp − 1)(1+θp +θ2p + · · · +θ(p−1)p ). It follows that 2

2



2

1 + θp + θ2p + · · · + θ(p−1)p = 1 +



θi +

(p) i∈p2 D0

θi = 0.

(3)

(4)

(p) i∈p2 D1

(3) can be rewritten as follows: 3

2

2

0 = θp − 1 = (θp )p − 1 = (θp − 1)(1 + θp + · · · + θ(p It follows that 2

1 + θp + · · · + θ(p

−1)p



=1+

−1)p

).

θi = 0.

(5)

(p) (p) (p2 ) (p2 ) i∈p2 D0 ∪p2 D1 ∪pD0 ∪pD1

From (4) and (5), we obtain



(p2 ) i∈pD0

Since

p3 −1 i=0

θi =



θi .

(6)

(p2 ) i∈pD1

θi = 0, by (5) we obtain   θi = θi . (p3 ) i∈D0

(7)

(p3 ) i∈D1

2

Assume θ1 = θp , θ2 = θp , then θ1 is a primitive p2 th root of unity and θ2 is a primitive pth root of unity in GF (2m ). Define   t1 (θ1 ) = θ1i and t2 (θ2 ) = θ2i . (p2 )

(p)

i∈D1

i∈D1

  Lemma 6. [5] i∈pZp θ1i + i∈D(p2 ) θ1i = 0 if p is an odd prime. 1   i i Lemma 7. (p2 ) θ1 = (p2 ) θ1 = t1 (θ1 ) = 0. i∈D i∈D 0

1

Proof. From (4),(6) and Lemma 6, obvious. (p)

Lemma 8. [6] t2 (θ2 ) ∈ {0, 1} if and only if 2 ∈ D0

if and only if p ≡ ±1 (mod 8)

Lemma 9. Let the symbols be the same as before, ⎧ p+1 (mod 2), if a = 0 ⎪ 2 ⎪ ⎪ (p3 ) ⎪ ⎪ S(θ), if a ∈ D0 ⎪ ⎪ ⎪ (p3 ) ⎪ ⎪ if a ∈ D1 ⎨ S(θ) + 1, (p2 ) S(θa ) = p+1 + t2 (θ2 ), if a ∈ pD0 2 ⎪ ⎪ (p2 ) p−1 ⎪ ⎪ if a ∈ pD1 ⎪ 2 + t2 (θ2 ), ⎪ ⎪ (p) ⎪ if a ∈ p2 D0 ⎪ 1 + t2 (θ2 ), ⎪ ⎩ (p) t2 (θ2 ), if a ∈ p2 D1 .

192

Y.-J. Kim, S.-Y. Jin, and H.-Y. Song p3 +1 2

Proof. For the case a = 0, we have S(θa ) = S(1) = a∈

3

(p ) D0 ,

≡

p+1 2

(mod 2). If

2s

by definition there is an integer s such that a = g . It follows that (p3 )

(p3 )

= {g 2s+2t+1 |t = 0, 1, · · · , p2 (p − 1) − 1} = D1

aD1

(p2 )

apD1

(p2 )

= p{g 2s+2t+1 |t = 0, 1, · · · , p(p − 1) − 1} = pD1

(p2 )

(p)

ap2 D1 = p2 {g 2s+2t+1 |t = 0, 1, · · · , (p − 1) − 1} = p2 D1 Hence S(θa ) = 1 + (



(p3 )

i∈D1



+

(p2 )

(p3 )

+

(p3 )

(p3 )

(p2 )

For a = a1 p, a1 ∈ Zp∗2 = D0



(p2 )

(p3 )

(p3 )

(p2 )

S(θa ) =

(p3 )

, a1 D 1



+ 

)θai

(p) i∈p2 D1

θa1 pi +

(p3 )

= D1



θa1 pi

(p)

i∈p2 D1

θ2i +

(p2 )

i∈a1 D1

If a1 ∈ D0

, we have



θ1i +

. By

2

i∈pD1



=1+

(p2 )

= p2 D0

(p)

(p2 )

i∈D1

(p)

, ap2 D1

i∈p2 D0

(p2 ) i∈pD1

θa1 pi +

(p)

i∈p2 D1

)θi = S(θ) + 1.

(p)



+

(p3 ) i∈D1

=1+

+

)θi = S(θ).

| = p2 |D1 |, θ1p = 1 and θ2p = 1.

∪ D1



S(θa ) = 1 + (



(p3 )

mod p=D1 , |D1

(p2 )

 i∈pD0

(p)

(p2 )

= pD0



+

i∈pD1

, apD1

(p2 )

i∈D0

Note that D1

(p2 )



+

i∈D1

= D0



S(θa ) = 1 + (



(p3 )

(p)

(p3 )

If a ∈ D1 , then aD1 (4), (6) and (7)

)θai = 1 +(

i∈p2 D1

i∈pD1

(p3 )



+

.

p−1 . 2

i∈a1 D1

(p2 )

and a1 D1

(p2 )

= D1

. we have

    p+1 p+1 + +p θ1i + θ2i = θ1i + p θ2i 2 2 3 2 2 (p) (p )

(p )

i∈D1

(p )

i∈D1

i∈D1

p+1 p+1 + t1 (θ1 ) + t2 (θ2 ) = + t2 (θ2 ). = 2 2 (p2 )

If a1 ∈ D1

(p3 )

, a1 D 1

S(θa ) =

(p3 )

= D0

(p2 )

and a1 D1

. we have

  p+1 + θ1i + θ2i 2 3 2 (p )

i∈D0

=

(p2 )

= D0

(p )

i∈D0

p+1 p−1 + t1 (θ1 ) + 1 + t2 (θ2 ) = + t2 (θ2 ). 2 2

i∈D1

Linear Complexity and Autocorrelation of Prime Cube Sequences

193

For a = a2 p2 , a2 ∈ Zp∗ = D0 ∪ D1 , we have (p)

(p)



S(θa ) = 1 + (

(p3 )

i∈pD1



2



(p)



2

θ1a2 p i +

(p2 )

i∈a2 D1

=1+

)θai

i∈p2 D1



θp i +

(p3 )



+

(p2 )

i∈D1

=1+



+

2

θ2a2 p

i

(p)

i∈D1

i∈D1

p2 − p p − 1 + . 2 2

θ2i +

(p3 )

i∈a2 D1 (p3 )

(p)

If a2 ∈ D0 , a2 D1 S(θa ) =

(p3 )

= D1

(p2 )

and a2 D1

(p2 )

= D1

  p2 + 1 p2 + 1 + + p2 θ2i = θ2i = 1 + t2 (θ2 ). 2 2 3 (p) (p )

i∈D1

i∈D1

(p2 )

If a2 ∈ D1

(p3 )

, a2 D 1

(p3 )

= D0

(p2 )

and a2 D1

(p2 )

= D0

(p )

i∈D0

i∈D0

(p3 )



(p)

a∈Di

. we have

  p2 + 1 p2 + 1 θ2i = θ2i = t2 (θ2 ). + + p2 2 2 3 (p)

S(θa ) =

Define di

. we have

(x) =



(p2 )

(p3 )

a∈Di

(x−θa ), di

(x) =



(x − θ2a ), i = 0, 1. Then 3

(p)

(p2 )

(p)

xp − 1 = (x − 1)d0 (x)d1 (x)d0 (p2 )

(p)

Lemma 10. di (x), di

(p3 )

(x), di

(p)

(p2 )

a∈Di

(p2 )

(x)d1

(x−θ1a ) and di (x) =

(p3 )

(x)d0

(p3 )

(x)d1

(x).

(x) ∈ GF (2)[x] if and only if p ≡ ±1 mod 8.

Proof. Almost the same proof in [4] can be applied . If p ≡ ±1 mod 8, from (p) (p2 ) (p3 ) Lemma 4 and 8, 2 ∈ D0 ∩ D0 ∩ D0 . Then for i = 0, 1, 2, we have (pi )

(di

(x))2 =



i

x2 − θ2p a ) =

(pi )

(pi )

2∈

(p) D1



i

(pi )

(x2 − θp a ) = di

(x2 ).

(pi )

a∈2Di

a∈Di

(x) ∈ GF (2)[x], i = 0, 1, 2. If p ≡ ±3 mod 8, from Lemma 4 and 8, (p2 )

∩ D1

(pi )

(di

(p3 )

∩ D1

(x))2 =

. Then for i = 0, 1, 2, we have (pi )

(pi )

i

(pi )

(pi )

(x2 − θp a ) = di+1(mod 2) (x2 ) = di

a∈Di+1(mod 2)

Hence di

i

(x2 − θp a ) =

(pi )

a∈Di

Thus di



(x) ∈ GF (2)[x], i = 0, 1, 2.

(x2 ).

194

Y.-J. Kim, S.-Y. Jin, and H.-Y. Song

Theorem 1. Let p be an odd prime and {s(n)} be a prime cube sequence of period p3 . Then the linear complexity CL of {s(n)} is as follows: ⎧ p3 +1 ⎪ ⎪ 32 , if p ≡ 1 mod 8 ⎨ p − 1, if p ≡ 3 mod 8 CL = p3 , if p ≡ 5 mod 8 ⎪ ⎪ ⎩ p3 −1 , if p ≡ 7 mod 8. 2 Proof. If p ≡ 1 mod 8, from Lemmas 8, t2 (θ2 ) ∈ {0, 1}. Furthermore, since (p) (p2 ) (p3 ) 2 ∈ D0 ∩ D0 ∩ D0 by Lemma 4 and 8, S(θ2 ) = S(θ). Hence, S(θ) ∈ {0, 1}. Applying Lemma 9, we have ⎧ (p3 ) (p2 ) (p) ⎪ (x−1)d1 (x)d0 (x)d0 (x),if (S(θ),t2 (θ2 )) = (0, 0) ⎪ ⎪ ⎪ 3 2 3 ⎨ (p ) (p ) (p) xp − 1 (x−1)d1 (x)d1 (x)d1 (x),if (S(θ),t2 (θ2 )) = (0, 1) = c(x) = 3 2 (p ) (p ) (p) gcd(xp3 − 1, S(x)) ⎪ ⎪(x−1)d0 (x)d0 (x)d0 (x),if (S(θ),t2 (θ2 )) = (1, 0) ⎪ ⎪ 3 2 ⎩ (p ) (p ) (p) (x−1)d0 (x)d1 (x)d1 (x),if (S(θ),t2 (θ2 )) = (1, 1) p +1 It follows that CL = deg (c(x)) = 1 + p −p + p 2−p + p−1 2 2 = 2 . For the cases of p ≡ 3, 5 and 7 mod 8, we can reach easily by similar procedure with the case p ≡ 1 mod 8. 3

4

2

2

3

Autocorrelation

The periodic autocorrelation of a binary sequence {s(n)} of period N is defined L by Cs (τ ) = n=0 (−1)s(n+τ )−s(n) where 0 ≤ τ < L. Define ds (i, j; τ ) = |Ci ∩ (Cj + τ )|, 0 ≤ τ < L, i, j = 0, 1 Theorem 2. Let p be an odd prime. Then the autocorrelation profile of the binary prime cube sequence of period p3 which is defined at (1) is as follows: 1. p ≡ 1 (mod 4)

2. p ≡ 3 (mod 4)

⎧ 3 p , τ ⎪ ⎪ ⎪ 3 ⎪ τ p − p − 3, ⎪ ⎪ ⎪ ⎪ 3 ⎪ p − p + 1, τ ⎪ ⎨ 3 2 Cs (τ ) = p − p − p − 2, τ ⎪ ⎪ ⎪ p3 − p2 − p + 2, τ ⎪ ⎪ ⎪ ⎪ ⎪ −p2 − 2, τ ⎪ ⎪ ⎩ 2 −p + 2, τ ⎧ 3 p , τ ⎪ ⎪ ⎪ ⎨ p3 − p − 1, τ Cs (τ ) = ⎪ p3 − p2 − p, τ ⎪ ⎪ ⎩ 2 −p , τ

= 0 (mod p3 ) (p) ∈ p2 D0 (p) ∈ p2 D1 (p2 )

∈ pD0

(p2 )

∈ pD1

(p3 )

∈ D0 (p3 ) ∈ D1

= 0 (mod p3 ) (p) (p) ∈ p2 D0 ∪ p2 D1 (p2 )

∈ pD0

(p3 )

∈ D0

(p2 )

∪ pD1

(p3 )

∪ D1

.

Linear Complexity and Autocorrelation of Prime Cube Sequences

195

Proof. Since Cs (τ ) = p3 − 4ds (1, 0; τ ), we need to calculate ds (1, 0; τ ). Note that ds (1, 0; τ ) =|C1 ∩ (C0 + τ )| (p2 )

(p)

=|C1 ∩(p2 D0 +τ )| + |C1 ∩(pD0

(p3 )

+τ )| + |C1 ∩(D0

+τ )|

(8)

Denote the first, the second and the third term in (8) as A(τ ), B(τ ) and C(τ ), respectively. To begin with, we are going to compute A(τ ). Note that (p)

(p)

(p)

(p)

A(τ ) = |C1 ∩ (p2 D0 + τ )| = |{0} ∩ (p2 D0 + τ )| + |p2 D1 ∩ (p2 D0 + τ )| (p2 )

+ |pD1

(p3 )

(p)

∩ (p2 D0 + τ )| + |D1

(p)

∩ (p2 D0 + τ )|.

(9)

Denote the first, the second, the third and the fourth term in (9) as A1 (τ ), A2 (τ ), A3 (τ ) and A4 (τ ), respectively. Let us compute A1 (τ ) first. When τ = 0, (p) (p2 A1 (τ ) = |{0} ∩ p2 D0 | = 0. When τ ∈ pDi for i = 0, 1, by Lemma 2, any (p2 )

(p)

element of p2 D0 + τ is an element of pDi when τ ∈

(p3 ) Di

for i = 0, 1, respectively. Similarly, (p3 )

(p)

2

for i = 0, 1, any element of p D0 + τ is an element of Di (p2 ) {0} ∪ pD0

(p2 ) ∪ pD1

(p3 ) ∪ D0

for

(p3 ) ∪ D1 ,

i = 0, 1, respectively. Therefore, when τ ∈ A1 (τ ) = 0. Next thing to do is to compute the value of A1 (τ ) when τ belongs (p) (p) (p) to the set p2 D0 ∪ p2 D1 . From Lemma 1 and 3, if p ≡ 1 mod 4, τ ∈ p2 Di (p) implies −τ ∈ p2 Di for i = 0, 1, respectively. Hence, in this case, A1 (τ ) = 1 if (p) (p) (p) τ ∈ p2 D0 and A1 (τ ) = 0 if τ ∈ p2 D1 . Likewise if p ≡ 1 mod 4, τ ∈ p2 Di (p) for i = 0, 1, respectively. Hence, A1 (τ ) = 0 if implies −τ ∈ p2 D i+1 mod 2 (p) 2 (p) τ ∈ p D0 and A1 (τ ) = 1 if τ ∈ p2 D1 . Summarizing these, we have ⎧ (p2 ) (p2 ) (p3 ) (p3 ) ⎪ ⎪ ⎪ 0, τ ∈ {0} ∪ pD0 ∪ pD1 ∪ D0 ∪ D1 ⎪ (p) ⎪ ⎪ ⎨ 1, τ ∈ p2 D0 and p ≡ 1 mod 4 A1 (τ ) = 0, τ ∈ p2 D0(p) and p ≡ 3 mod 4 (10) ⎪ ⎪ (p) 2 ⎪ ⎪ 0, τ ∈ p D1 and p ≡ 1 mod 4 ⎪ ⎪ ⎩ 1, τ ∈ p2 D(p) and p ≡ 3 mod 4 1 (p2 )

Next let us consider A2 (τ ). Similarly A2 (τ ) = 0 if τ ∈ {0} ∪ pD0

(p2 )

∪ pD1

∪

(p3 ) (p3 ) (p) (p) (p) (p) D0 ∪ D1 . When τ ∈ p2 D0 ∪ p2 D1 , A2 (τ ) = |p2 D1 ∩ (p2 D0 + τ )| = (p) (p) (p) (p) (p) |p2 D1 ∩ (p2 D0 + p2 a)| for some a ∈ D0 ∪ D1 . Therefore A2 (τ ) = |D1 ∩ (p) (p) (p) (D0 + a)| = |a−1 D1 ∩ (a−1 D0 + 1)| and by Lemma 1 and the definition of

the generalized cyclotomic numbers of order 2 with respect to p, we have ⎧ (p2 ) (p2 ) (p3 ) (p3 ) ⎪ τ ∈ {0} ∪pD0 ∪pD1 ∪D0 ∪D1 ⎨0, . A2 (τ ) = (0, 1)p ,τ ∈ p2 D0(p) ⎪ ⎩ 2 (p) (1, 0)p ,τ ∈ p D1 (p3 )

In the case of A3 (τ ), A3 (τ ) = 0 if τ ∈ {0} ∪ D0 as A1 (τ ) and A2 (τ ). If τ ∈

(p2 ) ∪pD0

∪

(p2 ) pD1 ,

(p3 )

∪ D1

with the same reason

then for i = 0, 1, any element

196

Y.-J. Kim, S.-Y. Jin, and H.-Y. Song (p)

of p2 Di (p2 ) pD1 .

+ τ is a multiple of p2 mod p3 so that it can not be an element of (p2 )

Thus, in these cases, A3 (τ ) = 0. In the case of τ ∈ pDi (p)

2

(p )

we have p2 D0 + τ ⊂ pDi (p) D0

(p2 ) pD1 .

+ τ| = if τ ∈ A3 (τ ) = |p Summarizing these calculation, we have 2

p−1 2

 A3 (τ ) =

for i = 0, 1, (p2 )

. Therefore, A3 (τ ) = |∅| = 0 if τ ∈ pD0

Similarly, we can compute A4 (τ ). 

(p2 )

0,

τ ∈ Zp3 \ pD1

p−1 2 ,

τ ∈ pD1

(p2 )

,

and

A4 (τ ) =

(p3 )

0,

τ ∈ Zp3 \ D1

p−1 2 ,

τ ∈ D1

(p3 )

.

Combining the results of A1 (τ ), A2 (τ ), A3 (τ ) and A4 (τ ), we have ⎧ 0, τ ⎪ ⎪ ⎪ p+3 ⎪ ⎪ ⎪ 4 , τ ⎪ p+1 ⎪ ⎪ ⎪ 4 , τ ⎪ ⎪ p−1 ⎪ ⎪ ⎪ 4 ,τ ⎨ p+1  A(τ ) = Ai (τ ) = 4 , τ ⎪ ⎪ 1≤i≤4 ⎪ 0, τ ⎪ ⎪ ⎪ p−1 ⎪ ⎪ ⎪ 2 ,τ ⎪ ⎪ ⎪ ⎪ 0, τ ⎪ ⎪ ⎩ p−1 2 ,τ

=0 (p) ∈ p2 D0 (p) ∈ p2 D0 (p) ∈ p2 D1 (p) ∈ p2 D1

p≡1 p≡3 p≡1 p≡3

and and and and

(p2 )

mod mod mod mod

4 4 4 4

(11)

∈ pD0

(p2 )

∈ pD1

(p3 )

∈ D0 (p3 ) ∈ D1

Next we are going to compute B(τ ) and C(τ ). Note that (p2 )

B(τ ) = |{0} ∩ (pD0 2

(p )

+ |pD1

(p3 )

C(τ ) = |{0} ∩ (D0 2

(p )

+ |pD1

2

(p )

∩ (pD0

3

(p )

+ τ )| + |D1

(p )

(p3 )

(p)

3

3

(p )

+ τ )| + |D1

+ τ )|

2

∩ (pD0

+ τ )| + |p2 D1 ∩ (D0

(p )

∩ (D0

(p2 )

(p)

+ τ )| + |p2 D1 ∩ (pD0

(12)

+ τ )|

3

(p )

∩ (D0

+ τ )|.

+ τ )|.

(13)

Denote the first, the second, the third and the fourth term in (12) as B1 (τ ), B2 (τ ), B3 (τ ) and B4 (τ ), respectively. Likewise denote the first, the second, the third and the fourth term in (13) as C1 (τ ), C2 (τ ), C3 (τ ) and C4 (τ ), respectively. With almost the same way, we can reach the following: p ≡ 1 mod 4 B1 (τ ) B2 (τ ) B3 (τ ) B4 (τ ) B(τ ) (p2 )

τ ∈ pD0

2

(p )

τ ∈ pD1

(p3 ) D1

τ ∈ otherwise

1

p−1 2

(0, 1)p2

0

0

(1, 0)p2

0 0

0 0

0 0

p2 −p 2

p2 +p+2 4 p(p−1) 4 p2 −p 2

0

0

0 0

(14)

Linear Complexity and Autocorrelation of Prime Cube Sequences

197

p ≡ 3 mod 4 B1 (τ ) B2 (τ ) B3 (τ ) B4 (τ ) B(τ ) (p2 )

pD0

(p )

τ ∈ pD1

0

0

(0, 1)p2

1

p−1 2

(1, 0)p2

0 0

0 0

2

3

(p ) D1

τ∈ otherwise

0 0

p2 −p 2

p(p+1) 4 p2 −p+2 4 p2 −p 2

0

0

0 0

(15)

By doing the same procedure repeatedly, we can reach the following: p ≡ 1 mod 4 C1 (τ ) C2 (τ ) C3 (τ ) C4 (τ ) (p3 )

τ ∈ D0

1

p−1 2

p2 −p 2

(0, 1)p3

τ ∈ D1 otherwise

0 0

0 0

0 0

0

(p3 )

C(τ )

p3 +p2 +2 4 3 2 (1, 0)p3 p −p 4

0

(16) p ≡ 3 mod 4 C1 (τ ) C2 (τ ) C3 (τ ) C4 (τ ) (p3 )

τ ∈ D0

3

(p )

τ ∈ D1 otherwise

0

0

1 0

p−1 2

0

C(τ )

p3 +p2 4 p3 −p2 +2 p2 −p (1, 0) 3 p 2 4

0

(0, 1)p3

0

0

0

Combining (11),(14), and (16), we can compute ds (1, 0; τ ). Since Cs (τ ) = p3 − 4ds (1, 0; τ ), it completes the proof.

References 1. Ding, C., Helleseth, T.: New Generalized Cyclotomy and Its Application. Finite Fields and Their Applications 4, 140–166 (1998) 2. Burton, D.M.: Elementary Number Theory, 4th edn. McGraw-Hill, New York (1998) 3. Golomb, S.W.: Shift Register Sequences, Revised edn. Aegean Park Press, Laguna Hills (1982) 4. Ding, C.: Linear Complexity of Some Generalized Cyclotomic Sequences. Int. J. Algebra and Computation 8, 431–442 (1998) 5. Park, Y.H., Hong, D., Chun, E.: On the Linear Complexity of Some Generalized Cyclotomic Sequences. Int. J. Algebra and Computation 14, 431–439 (2004) 6. Cusick, T., Ding, C., Renvall, A.: Stream Ciphers and Number Theory. Elservier Science, Amsterdam (1998) 7. Yan, T., Sun, R., Xiao, G.: Autocorrelation and Linear Complexity of the New Generalized Cyclotomic Sequences. IEICE Trans. Fundamentals E90-A, 857–864 (2007) 8. Bai, E., Liu, X., Xiao, G.: Linear Complexity of New Generalized Cyclotomic Sequences of Order Two of Length pq. IEEE Trans. Inform. Theory 51, 1849–1853 (2005)