IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.11 NOVEMBER 2010
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LETTER
Special Section on Signal Design and its Application in Communications
Autocorrelation of New Generalized Cyclotomic Sequences of Period pn∗ Seok-Yong JIN†a) , Student Member, Young-Joon KIM†† , Member, and Hong-Yeop SONG† , Nonmember
SUMMARY In this paper, we calculate autocorrelation of new generalized cyclotomic sequences of period pn for any n > 0, where p is an odd prime number. key words: generalized cyclotomic sequence, autocorrelation, linear complexity
good as that of Legendre sequences when n > 1. In Sect. 2, we review new generalized cyclotomic sequences of period pn and present their autocorrelation values. In Sect. 3, we prove our main result.
1.
2.
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. There have been lots of studies about cyclotomy with respect to p or p2 or pq where p and q are distinct odd primes [1]–[3]. In 1998, Ding and Helleseth [4] 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. The linear complexity (LC) of new generalized cyclotomic sequence of order 2 with respect to p2 [2] and p3 [6] is known. Finally, the LC of those sequences of period pn is calculated by by Yan, Li and Xiao [7] and by Kim and Song [8] independently. While the linear complexity is an important measure of pseudo-random sequences for cryptographic application, autocorrelation is another important measure for their application to communication systems for various purposes [5]. In this paper, we compute autocorrelation of new generalized cyclotomic sequences of order 2 with respect to pn for arbitrary positive integer n. Legendre sequences (n = 1) [5], prime square sequences (n = 2) [2], and prime cube sequences (n = 3) [6]–[8] are some important subclasses of these sequences. For simplicity, we will call these sequences as new generalized cyclotomic sequences of period pn . It turned out that their autocorrelation property is not as Manuscript received January 14, 2010. Manuscript revised May 20, 2010. † The authors are with the School of Electrical and Electronics Engineering, Yonsei University, Seoul, 120-749, Korea. †† The author is currently with Samsung Electronics, Co., Ltd., Suwon, Korea. ∗ This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20090083888). a) E-mail:
[email protected] DOI: 10.1587/transfun.E93.A.2345
Generalized Cyclotomic Sequences of Period pn and Its Autocorrlation
Given a prime p ≥ 2, let g be a primitive root of p2 . Then it follows that g is also a primitive root of pk , k ≥ 1. By definition, the order of g modulo pk is pk − pk−1 for 1 ≤ k ≤ k ) n. Let D(p = g2 (mod pk ) be the cyclic group generated 0 k
k
) ) by g2 modulo pk , and let D(p = gD(p (mod pk ) be the 1 0 k
k
k
) ) ) coset of D(p by g. It then follows that D(p ∪ D(p is the 0 0 1 × × multiplicative group Z pk . In fact, Z pk = Z pk ∪ pZ pk−1 , where pZ pk−1 = {0, p, 2p, · · · , (pk−1 − 1)p}. It can be identified that ⎞ ⎛ n ⎞ ⎛ n ⎟ ⎜⎜⎜ n−k (pk ) ⎟⎟⎟ ⎜⎜⎜ n−k (pk ) ⎟ ⎟ ⎜ ⎜ n Z p = ⎜⎝ p D ⎟⎠ ∪ ⎜⎝ p D ⎟⎟⎟⎠ ∪ {0}. 0
k=1
1
k=1
Define C0 and C1 as C0 = C1 =
n k=1 n
k
) pn−k D(p 0 , k
) pn−k D(p ∪ {0}. 1
(1) (2)
k=1
In [4], Ding and Helleseth defined the new generalized pn −1 cyclotomic sequences s = {s(i)}i=0 of period pn as shown below: ⎧ ⎪ ⎪ ⎨0, i ∈ C0 s(i) = ⎪ ⎪ ⎩1, i ∈ C1 . The periodic autocorrelation function C s (τ) of a binary sequence {s(n)} of period N is defined by C s (τ) =
N−1 (−1) s(n+τ)+s(n) , n=0
where n + τ takes modulo N. Theorem. Let p be an odd prime and n be a positive integer. Then the autocorrelation function C s (τ) of new generalized cyclotomic sequence s of period pn is given as follows:
c 2010 The Institute of Electronics, Information and Communication Engineers Copyright
IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.11 NOVEMBER 2010
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1. If p ≡ 1 (mod 4), then C s (τ) is given as ⎧ n ⎪ τ = 0 (mod pn ) p , ⎪ ⎪ ⎪ n u ⎨ C s (τ) = ⎪ p − pu − pu−1 − 2, τ ∈ pn−u D0(p ) ⎪ ⎪ ⎪ ⎩ pn − pu − pu−1 + 2, τ ∈ pn−u D(pu ) 1 for u = 1, . . . , n. 2. If p ≡ 3 (mod 4), then C s (τ) is given as ⎧ ⎪ ⎪ τ = 0 (mod pn ) ⎨ pn , u u C s (τ) = ⎪ ) ⎪ ⎩ pn − pu − pu−1 , τ ∈ pn−u D0(p ) ∪ pn−u D(p 1 for u = 1, . . . , n. 3.
Proof of Theorem
To compute the autocorrelation of new generalized cyclotomic sequences of period pn , we use the generalized cyclotomic numbers of order 2 with respect to pk for k ≥ 1, defined in [10], k ) (pk ) (i, j) pk = D(p + 1 ∩ D i j , where i, j = 0, 1 and k = 1, . . . , n. It is known [4] that: 1. If p = 1 (mod 4), then (0, 0) pk = (0, 1)
pk
pk−1 (p − 5) , 4
= (1, 0)
pk
= (1, 1)
• If l = k,
⎧ ⎪ ⎪ (0, 1) pk , ⎪ ⎪ ⎪ ⎨ Δk,k (τ) = ⎪ (1, 0) pk , ⎪ ⎪ ⎪ ⎪ ⎩0,
pk
1. p = 1 (mod 4) ⎧ l l−1 (pk ) p −p ⎪ ⎪ ⎨ 2 , τ ∈ pn−k D0 , Δl,k (τ) = ⎪ ⎪ ⎩0, othewise. 2. p = 3 (mod 4) ⎧ l l−1 (pk ) p −p ⎪ ⎪ ⎨ 2 , τ ∈ pn−k D1 , Δl,k (τ) = ⎪ ⎪ ⎩0, othewise. • If l > k, we have ⎧ k k−1 p −p ⎪ ⎪ ⎨ 2 , Δl,k (τ) = ⎪ ⎪ ⎩0,
Now we are ready to prove our main result. First we define the difference function d s (i, j; τ) as d (i, j; τ) = C ∩ (C + τ) i
k=1
pk−1 (p − 3) . 4
k=1
n n−k (pk ) n−k (pk ) + p D1 ∩ (p D0 + τ) k=1
+
n n−1
l
k
) |pn−l D1(p ) ∩ (pn−k D(p + τ)| 0
l=1 k=l+1 l−1 n n−l (pl ) n−k (pk ) + p D1 ∩ (p D0 + τ)
Then we have the following two lemmas, which play an important role to prove our main result.
l=2 k=1
n n n−k (pk ) Δk,k (τ) = {0} ∩ (p D0 + τ) + k=1
Lemma 1 Δ∗,k (τ) is given as:
p = 3 (mod 4) :
j
n n−k (pk ) = {0} ∩(p D0 + τ)
Now let us define Δ∗,k (τ) and Δl,k (τ) as (pk ) n−k Δ∗,k (τ) := {0} ∩ p D0 + τ and l k ) . + τ Δl,k (τ) := pn−l D1(p ) ∩ pn−k D(p 0
p = 1 (mod 4) :
l
τ ∈ pn−l D1(p ) othewise.
n k ) +τ) d s (1, 0; τ) = |C1 ∩ (C0 +τ)| = C1 ∩ (pn−k D(p 0
pk−1 (p + 1) , 4
⎧ ⎪ ⎪ ⎨1, Δ∗,k (τ) = ⎪ ⎪ ⎩0, ⎧ ⎪ ⎪ ⎨1, Δ∗,k (τ) = ⎪ ⎪ ⎩0,
k
) τ ∈ pn−k D(p 1 othewise.
for i, j = 0, 1. Since C s (τ) = pn − 4d s (1, 0; τ), we need to calculate d s (1, 0; τ):
pk (p − 1) . = 4
(0, 0) pk = (1, 0) pk = (1, 1) pk =
k
) τ ∈ pn−k D(p 0
• If l < k, then Δl,k (τ) is equal to
s
2. If p = 3 (mod 4), then (0, 1) pk =
Lemma 2 Δl,k (τ) is as follows:
k
(p ) τ ∈ pn−k D0 , otherwise. k
) τ ∈ pn−k D(p 1 , othewise.
+
n n−1 l=1 k=l+1
k=1
l,k (τ) +
l−1 n
Δl,k (τ).
l=2 k=1
We take the case of p = 1 (mod 4), first. If τ ∈ u pn−u D0(p ) , then by Lemma 1 and Lemma 2, we have u−1 2+ pu + pu−1 , d s (1, 0; τ) = 1+(0, 1) pu + Δl,u (τ)+0 = 4 l=1
LETTER
2347
k ) n−u ∩ p Λ = pn−k D(p = |∅| = 0, 1
u
for u = 1, . . . , n. If τ ∈ pn−u D1(p ) , we have d s (1, 0; τ) =
−2 + pu + pu−1 , 4
k
) + b (mod pu ) is a subset of Z ×pu . where Λ := pu−k D(p 0 (2) If u < k, it implies pn−u > pn−k , so
for u = 1, . . . , n.
Now for the case of p = 3 (mod 4), we have in a similar way that d s (1, 0; τ) =
Δk,k (τ) k ) n−k (pk ) n−u n ∩ p D + p b (mod p ) = pn−k D(p 1 0 (pk ) (pk ) (mod pn ) = pn−k D1 ∩ pn−k · D0 + pk−u b k ) (pk ) n−k ∩ p · D = pn−k D(p 1 0 = |∅| = 0.
pu + pu−1 , 4
if τ ∈ pn−u Z ×pu for u = 1, . . . , n. Since C s (τ) = pn − 4d s (1, 0; τ), it completes the proof. Now it remains to prove Lemma 1 and 2. To do that, we need the following two propositions: Proposition 1 For arbitrary integer b, and for any k = k k ) ) 1, . . . , n, we have bp + D(p = D(p (mod pk ), where i = i i 0, 1.
B. Let l < k. k
Δl,k (τ) (pl ) (pk ) n−l n−k n−k n (mod p ) = p D1 ∩ p D0 + p a l k ) n (mod p ) = pn−l D1(p ) − pn−k a ∩ pn−k D(p 0 l k (p ) (p ) (mod pk ) . = pk−l D1 − a ∩ D0
k
) Proposition 2 −1 (mod pk ) ∈ D(p if and only if p = 1 0 (mod 4), for k = 1, . . . , n. proof: It is well known that −1 (mod p) ∈ D(p) 0 if and only if p = 1 (mod 4). Using Proposition 1, we can show that (p2 ) 2 −1 (mod p) ∈ D(p) and −1 0 implies −1 (mod p ) ∈ D0 3
(mod p3 ) ∈ Dg(p ) , and so on. The converse is obvious.
k
(p ) (p ) 1. For τ ∈ pn−k D0 ∪ pn−k D1 , we put τ = pn−k a for × some a ∈ Z pk . Then,
k
k
) ) if and only if −a ∈ D(p (1) p = 1 (mod 4): a ∈ D(p i i , k
Proof of Lemma 1 If p = 1 (mod 4), τ ∈ p
n−k
if and only if −τ ∈ p
n−k
k ) D(p i ,
k ) D(p i
by Proposition 2. Like-
wise, if p = 3 (mod 4), then τ ∈ p
n−k
k
k ) D(p i
if and only if
) −τ ∈ pn−k D(p i+1 (mod 2) . It completes the proof.
Proof of Lemma 2 A. Let l = k. k
k
) ) n−k 1. If τ ∈ pn−k D(p ∪ pn−k D(p a for 0 1 , we can put τ = p × some a ∈ Z pk . Then,
) by Proposition 2. If a ∈ D(p i , for any element x of l
k
) pk−l D1p − a (mod pk ), x becomes an element of D(p i , l
k
) (2) p = 3 (mod 4): if a ∈ D(p i , any element l
x of pk−l D1p − a (mod pk ) becomes an element of k
Δk,k (τ) k ) n−k (pk ) n−k = pn−k D(p ∩ p D + p a 1 0 k ) (pk ) k ∩ D + a (mod p ) = D(p 1 0 ⎧ k n−k (p ) ⎪ ⎪ ⎨ (0, 1) pk , if τ ∈ p D0 =⎪ ⎪ ⎩ (1, 0) k , if τ ∈ pn−k D(pk ) . p
u D0(p )
(mod pn )
1
u D1(p )
2. For τ ∈ p ∪p such that u k, put τ = pn−u b for some b ∈ Z ×pu . Then, (1) If u > k, it implies pn−u < pn−k , so n−u
n−u
Δk,k (τ) k ) n−k (pk ) n−u n = pn−k D(p ∩ p D + p b (mod p ) 1 0 k (p ) = pn−k D1 ∩ pn−u · Λ (mod pn )
k
) by Proposition 1. Hence, pk−l D1p −a ⊂ D(p i . It follows that ⎧ l (pk ) pl −pl−1 k−l (p ) ⎪ ⎪ ⎨ p D1 − a = 2 , a ∈ D0 Δl,k (τ) = ⎪ k ⎪ ) ⎩0, a ∈ D(p 1 .
l
k
) (p ) k−l p D(p i+1 (mod 2) . Hence, p D1 − a ⊂ Di+1 (mod 2) . It follows that ⎧ k ) ⎪ ⎪ a ∈ D(p ⎨0, 0 Δl,k (τ) = ⎪ k l l−1 ⎪ ⎩ p −p , a ∈ D(p ) . 2
u
1
u
2. For τ ∈ pn−u D0(p ) ∪ pn−u D1(p ) such that u k, put τ = pn−u b for some b ∈ Z ×pu . Then, k
=
k
=
) + τ (1) If u > k, note that pn−k D(p 0 k n−u u−k (p ) n−u × p D0 + b ⊂ p Z pu . Hence, p
Δl,k (τ) = |∅| = 0. ) (2) If u < k, note that pn−k D(p + τ 0 k k (p ) n−k k−u n−k (p ) D0 + p b = p D0 . Hence, p
l k ) Δl,k (τ) = pn−l D1(p ) ∩ pn−k D(p 0 = |∅| = 0.
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C. Let l > k. The case of l > k can be done in a similar way to the case of l < k. References [1] J.-H. Kim and H.-Y. Song, “Trace representation of Legendre sequences,” Designs, Codes and Cryptography, vol.24, no.3, pp.343– 348, 2001. [2] T. Yan, R. Sun, and G. Xiao, “Autocorrelation and linear complexity of the new generalized cyclotomic sequences,” IEICE Trans. Fundamentals, vol.E90-A, no.4, pp.857–864, April 2007. [3] E. Bai, X. Liu, and G. Xiao, “Linear complexity of new generalized cyclotomic sequences of order two of length pq,” IEEE Trans. Inf. Theory, vol.51, no.5, pp.1849–1853, 2005. [4] C. Ding and T. Helleseth, “New generalized cyclotomy and its applications,” Finit Fields and Their Applications, vol.4, pp.140–166,
1998. [5] S.W. Golomb and G. Gong, Sequence Design for Good Correlation, Cambridge University Press, 2005. [6] Y.-J. Kim, S.-Y. Jin, and H.-Y. Song, “Linear complexity and autocorrelation of prime cube sequences,” Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, ed. S. Boztas and H.F.F. Lu, LNCS 4851, pp.188–197, Springer, 2007. [7] T. Yan, S. Li, and G. Xiao, “On the linear complexity of generalized cyclotomic sequences with the period pm ,” Applied Mathematics Letters, vol.21, pp.187–193, 2008. [8] Y.-J. Kim and H.-Y. Song, “Linear complexity of prime n-square sequences,” Proc. ISIT 2008, pp.2405–2408, Toronto, July 2008. [9] Y.-J. Kim, Linear Complexity and Correlation of Some Cyclotomic Sequences, PhD Thesis, Yonsei University, 2009. [10] T. Cusick, C. Ding, and A. Renvall, Stream Ciphers and Number Theory, North-Holland, 1998.
