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Linear complexity of quaternary sequences of length pq with low autocorrelation
Journal of Computational and Applied Mathematics 259 (2014) 555–560
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Linear complexity of quaternary sequences of length pq with low autocorrelation V. Edemskiy ∗ , A. Ivanov Department of Applied Mathematics and Informatics, Novgorod State University, Veliky Novgorod, Russia
article
info
Article history: Received 1 February 2013 Received in revised form 23 July 2013 MSC: 11B50 94A55 94A60
abstract We derive the linear complexity of quaternary sequences of length pq with low autocorrelation over the finite field of four elements and over the finite ring of order 4. Also we examine the linear complexity of the other sequences of length pq. © 2013 Elsevier B.V. All rights reserved.
Keywords: Quaternary sequences Linear complexity Cyclotomic sequences
1. Introduction The linear complexity L of a sequence is an important parameter in its evaluation as a key stream cipher for cryptographic applications. The autocorrelation is another measure of pseudorandom sequences significant for practical applications. Ideally, good sequences combine the autocorrelation properties of a random sequence with high linear complexity. Cyclotomic and generalized cyclotomic sequences are important pseudorandom sequences in stream ciphers due to their good pseudorandom cryptographic properties and large linear complexity [1]. Cyclotomic constructions have been repeatedly explored, mostly binary cases. In particular, the linear complexity of sequences of length pq was explored in [2–7] (see also references therein). Quaternary sequences are also interesting in view of the many practical applications; see for example [8,9]. Z. Yang and P. Ke constructed a new family of quaternary sequences of length pq over Zpq by using inverse Gray mapping and generalized binary cyclotomic sequences over Zpq [10]. These new constructions show low autocorrelation, but up to the present they have not been examined from the linear complexity point of view. On the one hand, it is possible to derive the linear complexity of quaternary sequences over the finite ring Z4 . An alternative approach is to consider sequences defined over F4 (the finite field of four elements) with respect to the quaternary sequences by using the Gray mapping. Generally, these two ways lead to different values for the linear complexity because arithmetics of F4 and Z4 differ; see for example [11]. We generalize the method proposed in [5] to explore the linear complexity of quaternary sequences from [10], and also of the other sequences of length pq for both alternatives. First, we recall the definition of sequences from [10]. Let p and q be two different odd primes. According to the Chinese Remainder Theorem, Zpq ∼ = Zp × Zq relatively to isomorphism f (t ) = (t1 , t2 ), where t1 = t mod p, t2 = t mod q. Here and hereafter a mod p denotes the least nonnegative integer that is congruent to a modulo p. Define P = {p, 2p, . . . , (q − 1)p}, Q = {q, 2q, . . . , (p − 1)q}.
∗
Corresponding author. Tel.: +7 8162629972; fax: +7 8162624110. E-mail address: [email protected] (V. Edemskiy).
0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.08.003
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V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560
Let ϕ[a, b] be the inverse Gray map defined by ϕ[0, 0] = 0, ϕ[0, 1] = 1, ϕ[1, 1] = 2, ϕ[1, 0] = 3. By
x y
we
denote the Legendre symbol [12]. Z. Yang and P. Ke examined the autocorrelation function of new quaternary sequences of length pq over Zpq defined as S (t ) = ϕ[S1 (t ), S2 (t )], where
S1 (t ) =
1, 0,
for t ∈ P , for t ∈ {0} ∪ Q ,
1−
(1)
t1 p
t2 q
,
2
∗
for t ∈ ZN
and
S2 (t ) =
0, 1,
for t ∈ P , for t ∈ {0} ∪ Q ,
1−
(2)
t1 p
t2 q
2
,
for t ∈ ZN . ∗
To begin with, S ( t ). anotherdefinitionof the sequence we give Let H0 =
a
a ∈ Zp
p
= 1 , H1 = a ∈ Zp pa = −1 and H2 = {0}. Then H0 , H1 are cyclotomic classes of order 2
modulo p [1]. Similarly, we define G0 , G1 , G2 for q. 2 By definition, put Fk,l = f −1 (Hk × Gl ), k, l = 0, 1, 2. Then, we have a partition Zpq = k,l=0 Fk,l . Immediately from the definitions of S (t ) and the inverse Gray map ϕ[a, b] we have
0, 1, S (t ) = 2, 3,
for t for t for t for t
mod mod mod mod
pq pq pq pq
∈ F0,0 ∪ F1,1 , ∈ {0} ∪ F0,2 ∪ F1,2 , ∈ F0,1 ∪ F1,0 , ∈ F2,0 ∪ F2,1 .
(3)
We can apply the proposed method not only to S (t ), but also to any other quaternary sequences of length pq based on Fk,l . As an example let us examine the linear complexity of a quaternary sequence V (t ) defined as
0, 1, V (t ) = 2, 3,
for t for t for t for t
mod mod mod mod
pq pq pq pq
∈ F0,0 ∪ F0,2 , ∈ F0,1 ∪ F2,0 , ∈ {0} ∪ F1,0 ∪ F1,2 , ∈ F1,1 ∪ F2,1 .
(4)
V (t ) is significantly more well-balanced than S (t ). 2. The linear complexity of quaternary sequences over the finite field of order 4 Let F4 = {0, 1, µ, µ + 1} be the finite field of four elements. If we view F4 as a vector space over F2 with basis µ, 1, then we define a quaternary sequence U (t ) by Gray map as [13]
0, 1, U (t ) = µ + 1 , µ,
for t for t for t for t
mod mod mod mod
pq pq pq pq
∈ F0,0 ∪ F1,1 , ∈ {0} ∪ F0,2 ∪ F1,2 , ∈ F0,1 ∪ F1,0 , ∈ F2,0 ∪ F2,1 .
(5)
The minimal polynomial and the linear complexity of U (t ) are given by the following equations [1]: m(x) = (xpq − 1)/ gcd(xpq − 1, MU (x)) ,
(6)
L (U (t )) = pq − deg gcd(xpq − 1, MU (x)) ,
(7)
pq−1
t where MU (x) is the polynomial of U (t ). Thus, MU (x) = t =0 u(t )x . Let α be a primitive pqth root of unity in the extension of the field F4 . Then by Blahut’s theorem for the linear complexity L of the sequence U (t ) we have
L = pq − |{v|MU (α v ) = 0, v = 0, 1, . . . , pq − 1}| . v
(8)
Let us derive the expression for MU (α ) using the procedure proposed in [5]. Suppose β = α aq and γ = α bp , where a, b are integers satisfying aq + bp = 1. Then β and γ are primitive pth and qth roots of unity in the extension of the field F4 , respectively.
V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560 j Introduce the subsidiary polynomials R2 (x) = j∈H0 x and T2 (x) = It was shown in [5] that if the sequence Y (t ) is defined as
Y (t ) =
1,
if t mod pq ∈
0,
otherwise,
Fk,l ∪
(k,l)∈I
Fj,2 ∪
j∈K
j∈G0
557
xj .
F2,j ,
(9)
j∈M
then for v = 0, 1, . . . , pq − 1 we have MY (α v ) =
R2 β vθ
k
T2 γ vη
l
+
(k,l)∈I
T2 γ vη
j
+
R2 β vθ
j
.
(10)
j∈M
j∈K
Here, θ and η are primitive roots modulo p and q, respectively; I , K , M are subsets of indices. Using (5), (9) and (10) we obtain the following expression for the values of the sequence polynomial: MU (α v ) = (µ + 1) R2 (β v ) T2 (γ ηv ) + R2 β θv T2 (γ v ) +
1
j R2 β θ v + µ
j =0
1
T2 (γ η v ) + 1. j
(11)
j =0
By definition, R2 (1) = (p − 1)/2 and T2 (1) = (q − 1)/2, and hence from (11) we have MU (1) = 1. Properties of R2 (x) and T2 (x) were studied in [14]. In particular, it was shown that with an appropriate choice of β and γ we can assume that
R2 (β v ) =
1, 0,
if v mod p ∈ H0 , if v mod p ∈ H1 .
or T2 (γ v ) =
1, 0,
if v mod q ∈ G0 , if v mod q ∈ G1
(12)
if p ≡ ±1 mod 8 or q ≡ ±1 mod 8, and R2 (β ) = T2 (γ v ) =
µ, µ + 1,
if v mod p ∈ H0 , if v mod p ∈ H1 .
µ, µ + 1,
if v mod q ∈ G0 , if v mod q ∈ G1
v
or
(13)
if p ≡ ±3(mod 8) or q ≡ ±3(mod 8). Without loss of generality, we can choose α such that the relations (12) and (13) are true for R2 (x) and T2 (x). Lemma 1. Let U (t ) be defined by (5), and let MU (x) be the polynomial of U (t ). Then (i) MU (α v ) =
(ii) MU (α v ) =
µ + 1,
if p ≡ 1(mod 4), if p ≡ −1(mod 4),
for v ∈ P,
µ + 1,
if q ≡ −1(mod 4), if q ≡ 1(mod 4),
for v ∈ Q .
0, 0,
Proof. Let v ∈ P; then R2 (β v ) = (p − 1)/2. If p ≡ 1 (mod 4), then R2 (β v ) = 0 and by (11) we have MU (α v ) = v µ (T2 (γ v ) + T2 (γ )) + 1 = µ + 1. In the case where p ≡ −1 (mod 4) we see that R2 (β ) = 1 in the field F4 and MU (α v ) = (µ + 1) T2 (γ v ) + T2 (γ vη ) + µ T2 (γ v ) + T2 (γ vη ) + 1 = 0. Part (ii) is proved in the same way. If v ̸∈ P ∪ Q ∪ {0}, then by (11)–(13) we have MU (α v ) = (µ + 1) R2 (β v ) T2 (γ ηv ) + R2 β θv T2 (γ v ) + µ.
(14)
Now, if v ∈ Fk,l , k, l = 0, 1, then v mod p ∈ Hk and v mod q ∈ Gl , so the form of (14) becomes MU (α v ) =
(µ + 1) R2 (β)T2 (γ η ) + R2 (β θ )T2 (γ ) + µ, (µ + 1) R2 (β θ )T2 (γ η ) + R2 (β)T2 (γ ) + µ,
if v ∈ F0,0 ∪ F1,1 , if v ∈ F1,0 ∪ F0,1 .
(15)
Theorem 2. Let the sequence U (t ) be defined by (5); let p ≡ 1(mod 4) and pq ≡ 1(mod 8). Then L = pq − p + 1 and m(x) = (xpq − 1) /Q (x), where Q (x) = i∈Q (x − α i ). Proof. We consider two cases. (i) Suppose p ≡ 1(mod 8) and q ≡ 1 ( mod 8 ). Then the values of R2 (β v ) and T2 (γ v ) are given by (12). Substituting (12) 1,
for R2 (β v ) and T2 (γ v ) in (15), we get MU (α v ) = µ,
if v ∈ F0,0 ∪ F1,1 , if v ∈ F1,0 ∪ F0,1 .
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V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560 Table 1 Linear complexity of quaternary sequences for p ≡ 1 (mod 4).
1 2 3 4
L
m(x)
pq
pq pq − p + 1 (pq − p + q + 1) /2 (pq + p + q − 1) /2
xpq − 1 (xpq − 1) /Q (x) (xpq − 1) / (Q (x)F (x)) (xpq − 1) /F (x)
pq pq pq pq
≡ −1 (mod 8) ≡ 1 (mod 8) ≡ −3 (mod 8) ≡ 3 (mod 8)
Table 2 Linear complexity of quaternary sequences p ≡ −1 (mod 4).
1 2 3 4
L
m(x)
pq
pq − p − q + 2 pq − q + 1 (pq + p − q + 1) /2 (pq − p − q + 3) /2
(x − 1) / (Q (x)P (x)) (xpq − 1) /P (x) (xpq − 1) / (P (x)F (x)) (xpq − 1) / (Q (x) P (x) F (x))
pq pq pq pq
pq
≡ −1 (mod 8) ≡ 1 (mod 8) ≡ −3 (mod 8) ≡ 3 (mod 8)
(ii) Let p ≡ −3(mod 8) andq ≡ −3(mod 8). Here R2 (β v ) and T2 (γ v ) are given by (13). Similarly, after substitution in 1,
if v ∈ F
,
(15), we again have MU (α v ) = µ, if v ∈ F0,0 ∪ F1,1 . 1,0 0,1 Therefore, by Lemma 1, in both cases we have MU (α v ) = 0 if and only if v ∈ Q . We see that the statement of Theorem 2 follows from (6) and (7). ∪F
In a similar way we can derive the linear complexity and the minimal polynomial of U (t ) for all possible values of p and q (see the results in the Tables 1 and 2). Here Q (x) = i∈Q (x − α i ), P (x) = i∈P (x − α i ) and F (x) = i∈F0,0 ∪F1,1 (x − α i ). Now we consider V (t ). By the Gray map we obtain
0, for t mod pq ∈ F0,0 ∪ F0,2 , 1, for t mod pq ∈ F0,1 ∪ F2,0 , W (t ) = µ + 1, for t mod pq ∈ {0} ∪ F1,0 ∪ F1,2 , µ, for t mod pq ∈ F1,1 ∪ F2,1 . Define G(x) = i∈F0,1 ∪F1,1 (x − α i ), R(x) = i∈H0 (x − β i ) and T (x) = i∈G0 (x − γ i ).
(16)
Let
δ=
1, 0,
if p + q ≡ 0(mod 2), if p + q ≡ 1(mod 2).
Theorem 3. Let the sequence W (t ) be defined by (16). Then: (1) L = (p + 1)q/2 − δ and m(x) = (xpq − 1) / (x − 1)δ G(x)R(x) if q ≡ 1(mod 8) and p ≡ ±1(mod 8) or q ≡ 7(mod 8) and p ≡ ±3(mod 8). (2) L = (pq + p + q − 1)/2 − δ and m(x) = (xpq − 1) / (x − 1)δ Q (x) if q ≡ 1(mod 8) and p ≡ ±3(mod 8) or q ≡ 7(mod 8) and p ≡ ±1(mod 8). (3) L = pq − (p + q − 2)/2 − δ and m(x) = (xpq − 1) / (x − 1)δ R(x)T (x) if q ≡ 3(mod 8) and p ≡ ±3(mod 8) or q ≡ 5(mod 8) and p ≡ ±1(mod 8). (4) L = pq − (q − 1)/2 − δ and m(x) = (xpq − 1) / (x − 1)δ T (x) if q ≡ 3(mod 8) and p ≡ ±1(mod 8) or q ≡ 5(mod 8) and p ≡ ±3(mod 8).
Proof. Under the conditions of Theorem 3, by (11) and (16) we have
µT (γ v ), 2 (µ + 1)T2 (γ v ) + 1, MV (α v ) = v µ ((µ + 1)R2 (β ) + (q − 1)/2) , (µ + 1) ((p − 1)/2 + (q − 1)/2 + 1) ,
if v if v if v if v
∈ Z∗pq , ∈ P, ∈ Q, = 0.
We see that the statement of Theorem 3 follows from (12), (13), and (17).
(17)
3. The linear complexity of quaternary sequences over the ring Z4 The polynomial C (x) = 1 + c1 x + c2 x2 + · · · + cm xm ∈ Z4 [x] is called m an associated connection polynomial of periodic sequence S (t ) over Z4 if coefficients c1 , c2 , . . . , cm satisfy S (t ) = − i=1 ci S (t − i), ∀t ≥ m. The linear complexity L of the periodic sequence S (t ) over Z4 is equal to min{deg C (x) : then C (x) is an associated connection polynomial of S (t )}.
V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560
559
By [15], C (x) is an associated connection polynomial of S (t ) if and only if MS (x)C (x) ≡ 0 (mod (xpq − 1)), where MS (x) is the polynomial of S (t ). Let r be the order of 2 modulo pq, and let R = GF (22r , 22 ) be a Galois ring of characteristic 4. For the knowledge on the Galois ring, please see [16,17]. The group of invertible elements R∗ = R \ 2R of the ring R contains the cyclic subgroup of order 2r − 1 [17]. Hence, there exists an element ζ of order pq in R∗ . Let S (t ) = ϕ[S1 (t ), S2 (t )] be the quaternary sequence where S1 (t ) and S2 (t ) are the binary sequences defined in (1) and (2), respectively. Let us examine the values of the polynomial of the sequence S (t ) in the ring R. A possible path is to introduce auxiliary polynomials in the ring R and subsequently employ the procedure from the previous section. But there is an easier way. Note that in R we have
ζ q + ζ 2q + · · · + ζ (p−1)q = 3 and ζ p + ζ 2p + · · · + ζ (q−1)p = 3.
(18)
By definition, the polynomial of the sequence S (t ) is MS (x) = 1 +
xt + 2
t ∈Q
xt + 3
t ∈F0,1 ∪F1,0
xt .
(19)
t ∈P
Hence, MS (1) = 1 + (p − 1) + 4(p − 1)(q − 1)/4 + 3(q − 1), and thus MS (1) ≡ 1 (mod 2). Lemma 4. Let MS (x) be the polynomial of S (t ). (i) MS (ζ v ) = 0 if v ∈ Q . (ii) MS (ζ v ) = 2 if v ∈ P. Proof. (i) If v ∈ Q , then v F0,1 ∪ F1,0 = Q . Since the order of F0,1 ∪ F1,0 is equal to (p − 1)(q − 1)/2, it follows that
ζ vj = (q − 1)/2
ζ vj = 3(q − 1)/2.
j∈Q
j∈F0,1 ∪F1,0
v
By (18) and (19) we have MS (ζ ) = 1 + 3 + 3(q − 1) + 3(q − 1) = 0 in the ring R. (ii) Let v ∈ P; then similarly we obtain MS (ζ v ) = 1 + (p − 1) + 3(p − 1) + 9 = 2. ∗ Lemma 5. Let MS (x) be the polynomial of the sequence S (t ). If v ∈ Zpq , then MS (ζ v ) ∈ R∗ . ∗ Proof. To prove Lemma 5 it is enough to show that if v ∈ Zpq then 2MS (ζ v ) ̸= 0. From (19) we obtain 2M S (ζ v ) =
ζ vjq + 2
ζ vjp . Now we can apply (18), which concludes the proof of Lemma 5. pq−1 pq−1 From the expansion xpq − 1 = i=0 x − ζ i it follows that pq = i=1 1 − ζ i . Consequently, the elements ζ i − 1 and ζ i − ζ j , i ̸≡ j (mod pq) belong to R∗ . So, if ζ i , ζ j , i ̸≡ j (mod pq) are the roots of P (x), then P (x) is divisible by the product (x − ζ i )(x − ζ j ). However, generally P (x) is not necessarily divisible by the product (x − g )(x − h) in the ring R when g , h are the roots of P (x). For instance, P (x) = 2(x − 1) is not divisible by (x − 1)(x − 3). 2+2
p−1 j=1
q−1 j =1
Theorem 6. Let S (t ) = ϕ[S1 (t ), S2 (t )] be the quaternary sequence where S1 (t ) and S2 (t ) are the binary sequences defined in (1) and (2), respectively. Then the linear complexity of S (t ) is equal to L = pq − p + 1. Proof. Choose C (x) = (xpq − 1) /Q (x). Then all the roots of xpq − 1 are the roots of MS (x)C (x). Hence, C (x) is an associated connection polynomial of S (t ). This implies that L ≤ pq − p + 1. If L ̸= pq − p + 1, then there exists another associated connection polynomial C1 (x) of the sequence S (t ) with degree less than pq − p + 1. Hence, C1 (ζ v )MS (ζ v ) = 0 for v = 0, 1, . . . , pq − 1. Let v ∈ Z∗pq ∪ {0}; then C1 (ζ v ) = 0 by Lemma 5. If v ∈ P, then C1 (ζ v ) ∈ 2R by Lemma 4. We obtain that 2C1 (ζ v ) = 0 for v ∈ Z∗pq ∪ P ∪ {0} and 2C1 (x) ̸= 0 by definition. Thus, 2C1 (x) is divisible by
i∈Z∗ pq ∪P ∪{0}
x − ζi ,
which contradicts the fact that the degree of 2C1 (x) is less than pq − p + 1. Therefore, L = pq − p + 1. This completes the proof. For the linear complexity of V (t ) we have a more complicated dependence which is shown in Theorem 7. Theorem 7. Let V (t ) be the quaternary sequence defined by (4). Then the linear complexity of V (t ) is equal to
L=
Here ∆ =
(p + 1)q/2 − ∆, (pq + p + q − 1)/2 − ∆, pq − (p − 1)/2 − ∆, pq − ∆, 1, 0,
if p ≡ ±1(mod 8) and q ≡ 1(mod 16), if p ≡ ±1(mod 8) and q ≡ −1(mod 16) or p ≡ ±3(mod 8) and q ≡ ±1(mod 16), if p ≡ ±1(mod 8) and q ≡ 5(mod 8) or q ≡ 9(mod 16), otherwise.
if (p − 1)(q − 3) ≡ 4(mod 8), otherwise.
Theorem 7 is proved by the method from Section 2.
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4. Conclusion It was shown in this paper that some of the quaternary sequences of length pq have high linear complexities over the finite ring Z4 . Also we have determined the values of p and q for which these sequences have high linear complexity over the finite field of four elements. For the case of the finite field we have found the minimal polynomial for the sequences concerned. Acknowledgments The authors acknowledge the patient referees for their valuable and constructive comments which helped to improve this work. References [1] T.W. Cusick, C. Ding, A. Renvall, Stream Ciphers and Number Theory, Elsevier, Amsterdam, 1998. [2] 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. [3] E. Bai, X. Fu, G. Xiao, On the linear complexity of generalized cyclotomic sequences of order four over Zpq, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E88-A (1) (2005) 392–395. [4] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order 2, Finite Fields Appl. 3 (2) (1997) 159–174. [5] V. Edemskiy, O. Antonova, About computation of the linear complexity of generalized cyclotomic sequences with period pq, in: Proc. of 2011 International Workshop on Signal Design and its Applications in Communications, IWSDA’011, China, 2011, pp. 9–12. [6] W. Meidl, Remarks on a cyclotomic sequence, Des. Codes Cryptogr. 51 (1) (2009) 33–43. [7] T. Yan, L. Hong, G. Xiao, The linear complexity of new generalized cyclotomic binary sequences of order four, Inform. Sci. 178 (3) (2008) 807–815. [8] S.M. Krone, D.V. Sarwate, Quadriphase sequences for spread spectrum multiple-access communication, IEEE Trans. Inform. Theory IT-30 (3) (1984) 520–529. [9] H.D. Luke, H.D. Schotten, H. Hadinejad-Mahram, Binary and quadriphase sequences with optimal autocorrelation properties: a survey, IEEE Trans. Inform. Theory IT-49 (2003) 3271–3282. [10] Z. Yang, P. Ke, Construction of quaternary sequences of length pq with low autocorrelation, Cryptogr. Commun. 3 (2) (2011) 55–64. [11] D.H. Green, Linear complexity of modulo-m power residue sequences, IEE Proc., Comput. Digit. Tech. 151 (6) (2004) 385–390. [12] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982. [13] X. Du, Z. Chen, Linear complexity of quaternary sequences generated using generalized cyclotomic classes modulo 2p, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E94-A (5) (2011) 1214–1217. [14] C. Ding, T. Helleseth, W. Shan, On the linear complexity of Legendre sequences, IEEE Trans. Inform. Theory IT-44 (1998) 1276–1278. [15] Z.X. Wan, Algebra and Coding Theory, Science Press, Beijing, 1976. [16] B.R. McDonald, Finite Rings With Identity, Marcel Dekker, New York, 1974. [17] Z.X. Wan, Finite Fields and Galois Rings, World Scientific Publisher, Singapore, 2003.
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Linear complexity of quaternary sequences of length pq with low autocorrelation V. Edemskiy ∗ , A. Ivanov Department of Applied Mathematics and Informatics, Novgorod State University, Veliky Novgorod, Russia
article
info
Article history: Received 1 February 2013 Received in revised form 23 July 2013 MSC: 11B50 94A55 94A60
abstract We derive the linear complexity of quaternary sequences of length pq with low autocorrelation over the finite field of four elements and over the finite ring of order 4. Also we examine the linear complexity of the other sequences of length pq. © 2013 Elsevier B.V. All rights reserved.
Keywords: Quaternary sequences Linear complexity Cyclotomic sequences
1. Introduction The linear complexity L of a sequence is an important parameter in its evaluation as a key stream cipher for cryptographic applications. The autocorrelation is another measure of pseudorandom sequences significant for practical applications. Ideally, good sequences combine the autocorrelation properties of a random sequence with high linear complexity. Cyclotomic and generalized cyclotomic sequences are important pseudorandom sequences in stream ciphers due to their good pseudorandom cryptographic properties and large linear complexity [1]. Cyclotomic constructions have been repeatedly explored, mostly binary cases. In particular, the linear complexity of sequences of length pq was explored in [2–7] (see also references therein). Quaternary sequences are also interesting in view of the many practical applications; see for example [8,9]. Z. Yang and P. Ke constructed a new family of quaternary sequences of length pq over Zpq by using inverse Gray mapping and generalized binary cyclotomic sequences over Zpq [10]. These new constructions show low autocorrelation, but up to the present they have not been examined from the linear complexity point of view. On the one hand, it is possible to derive the linear complexity of quaternary sequences over the finite ring Z4 . An alternative approach is to consider sequences defined over F4 (the finite field of four elements) with respect to the quaternary sequences by using the Gray mapping. Generally, these two ways lead to different values for the linear complexity because arithmetics of F4 and Z4 differ; see for example [11]. We generalize the method proposed in [5] to explore the linear complexity of quaternary sequences from [10], and also of the other sequences of length pq for both alternatives. First, we recall the definition of sequences from [10]. Let p and q be two different odd primes. According to the Chinese Remainder Theorem, Zpq ∼ = Zp × Zq relatively to isomorphism f (t ) = (t1 , t2 ), where t1 = t mod p, t2 = t mod q. Here and hereafter a mod p denotes the least nonnegative integer that is congruent to a modulo p. Define P = {p, 2p, . . . , (q − 1)p}, Q = {q, 2q, . . . , (p − 1)q}.
∗
Corresponding author. Tel.: +7 8162629972; fax: +7 8162624110. E-mail address: [email protected] (V. Edemskiy).
0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.08.003
556
V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560
Let ϕ[a, b] be the inverse Gray map defined by ϕ[0, 0] = 0, ϕ[0, 1] = 1, ϕ[1, 1] = 2, ϕ[1, 0] = 3. By
x y
we
denote the Legendre symbol [12]. Z. Yang and P. Ke examined the autocorrelation function of new quaternary sequences of length pq over Zpq defined as S (t ) = ϕ[S1 (t ), S2 (t )], where
S1 (t ) =
1, 0,
for t ∈ P , for t ∈ {0} ∪ Q ,
1−
(1)
t1 p
t2 q
,
2
∗
for t ∈ ZN
and
S2 (t ) =
0, 1,
for t ∈ P , for t ∈ {0} ∪ Q ,
1−
(2)
t1 p
t2 q
2
,
for t ∈ ZN . ∗
To begin with, S ( t ). anotherdefinitionof the sequence we give Let H0 =
a
a ∈ Zp
p
= 1 , H1 = a ∈ Zp pa = −1 and H2 = {0}. Then H0 , H1 are cyclotomic classes of order 2
modulo p [1]. Similarly, we define G0 , G1 , G2 for q. 2 By definition, put Fk,l = f −1 (Hk × Gl ), k, l = 0, 1, 2. Then, we have a partition Zpq = k,l=0 Fk,l . Immediately from the definitions of S (t ) and the inverse Gray map ϕ[a, b] we have
0, 1, S (t ) = 2, 3,
for t for t for t for t
mod mod mod mod
pq pq pq pq
∈ F0,0 ∪ F1,1 , ∈ {0} ∪ F0,2 ∪ F1,2 , ∈ F0,1 ∪ F1,0 , ∈ F2,0 ∪ F2,1 .
(3)
We can apply the proposed method not only to S (t ), but also to any other quaternary sequences of length pq based on Fk,l . As an example let us examine the linear complexity of a quaternary sequence V (t ) defined as
0, 1, V (t ) = 2, 3,
for t for t for t for t
mod mod mod mod
pq pq pq pq
∈ F0,0 ∪ F0,2 , ∈ F0,1 ∪ F2,0 , ∈ {0} ∪ F1,0 ∪ F1,2 , ∈ F1,1 ∪ F2,1 .
(4)
V (t ) is significantly more well-balanced than S (t ). 2. The linear complexity of quaternary sequences over the finite field of order 4 Let F4 = {0, 1, µ, µ + 1} be the finite field of four elements. If we view F4 as a vector space over F2 with basis µ, 1, then we define a quaternary sequence U (t ) by Gray map as [13]
0, 1, U (t ) = µ + 1 , µ,
for t for t for t for t
mod mod mod mod
pq pq pq pq
∈ F0,0 ∪ F1,1 , ∈ {0} ∪ F0,2 ∪ F1,2 , ∈ F0,1 ∪ F1,0 , ∈ F2,0 ∪ F2,1 .
(5)
The minimal polynomial and the linear complexity of U (t ) are given by the following equations [1]: m(x) = (xpq − 1)/ gcd(xpq − 1, MU (x)) ,
(6)
L (U (t )) = pq − deg gcd(xpq − 1, MU (x)) ,
(7)
pq−1
t where MU (x) is the polynomial of U (t ). Thus, MU (x) = t =0 u(t )x . Let α be a primitive pqth root of unity in the extension of the field F4 . Then by Blahut’s theorem for the linear complexity L of the sequence U (t ) we have
L = pq − |{v|MU (α v ) = 0, v = 0, 1, . . . , pq − 1}| . v
(8)
Let us derive the expression for MU (α ) using the procedure proposed in [5]. Suppose β = α aq and γ = α bp , where a, b are integers satisfying aq + bp = 1. Then β and γ are primitive pth and qth roots of unity in the extension of the field F4 , respectively.
V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560 j Introduce the subsidiary polynomials R2 (x) = j∈H0 x and T2 (x) = It was shown in [5] that if the sequence Y (t ) is defined as
Y (t ) =
1,
if t mod pq ∈
0,
otherwise,
Fk,l ∪
(k,l)∈I
Fj,2 ∪
j∈K
j∈G0
557
xj .
F2,j ,
(9)
j∈M
then for v = 0, 1, . . . , pq − 1 we have MY (α v ) =
R2 β vθ
k
T2 γ vη
l
+
(k,l)∈I
T2 γ vη
j
+
R2 β vθ
j
.
(10)
j∈M
j∈K
Here, θ and η are primitive roots modulo p and q, respectively; I , K , M are subsets of indices. Using (5), (9) and (10) we obtain the following expression for the values of the sequence polynomial: MU (α v ) = (µ + 1) R2 (β v ) T2 (γ ηv ) + R2 β θv T2 (γ v ) +
1
j R2 β θ v + µ
j =0
1
T2 (γ η v ) + 1. j
(11)
j =0
By definition, R2 (1) = (p − 1)/2 and T2 (1) = (q − 1)/2, and hence from (11) we have MU (1) = 1. Properties of R2 (x) and T2 (x) were studied in [14]. In particular, it was shown that with an appropriate choice of β and γ we can assume that
R2 (β v ) =
1, 0,
if v mod p ∈ H0 , if v mod p ∈ H1 .
or T2 (γ v ) =
1, 0,
if v mod q ∈ G0 , if v mod q ∈ G1
(12)
if p ≡ ±1 mod 8 or q ≡ ±1 mod 8, and R2 (β ) = T2 (γ v ) =
µ, µ + 1,
if v mod p ∈ H0 , if v mod p ∈ H1 .
µ, µ + 1,
if v mod q ∈ G0 , if v mod q ∈ G1
v
or
(13)
if p ≡ ±3(mod 8) or q ≡ ±3(mod 8). Without loss of generality, we can choose α such that the relations (12) and (13) are true for R2 (x) and T2 (x). Lemma 1. Let U (t ) be defined by (5), and let MU (x) be the polynomial of U (t ). Then (i) MU (α v ) =
(ii) MU (α v ) =
µ + 1,
if p ≡ 1(mod 4), if p ≡ −1(mod 4),
for v ∈ P,
µ + 1,
if q ≡ −1(mod 4), if q ≡ 1(mod 4),
for v ∈ Q .
0, 0,
Proof. Let v ∈ P; then R2 (β v ) = (p − 1)/2. If p ≡ 1 (mod 4), then R2 (β v ) = 0 and by (11) we have MU (α v ) = v µ (T2 (γ v ) + T2 (γ )) + 1 = µ + 1. In the case where p ≡ −1 (mod 4) we see that R2 (β ) = 1 in the field F4 and MU (α v ) = (µ + 1) T2 (γ v ) + T2 (γ vη ) + µ T2 (γ v ) + T2 (γ vη ) + 1 = 0. Part (ii) is proved in the same way. If v ̸∈ P ∪ Q ∪ {0}, then by (11)–(13) we have MU (α v ) = (µ + 1) R2 (β v ) T2 (γ ηv ) + R2 β θv T2 (γ v ) + µ.
(14)
Now, if v ∈ Fk,l , k, l = 0, 1, then v mod p ∈ Hk and v mod q ∈ Gl , so the form of (14) becomes MU (α v ) =
(µ + 1) R2 (β)T2 (γ η ) + R2 (β θ )T2 (γ ) + µ, (µ + 1) R2 (β θ )T2 (γ η ) + R2 (β)T2 (γ ) + µ,
if v ∈ F0,0 ∪ F1,1 , if v ∈ F1,0 ∪ F0,1 .
(15)
Theorem 2. Let the sequence U (t ) be defined by (5); let p ≡ 1(mod 4) and pq ≡ 1(mod 8). Then L = pq − p + 1 and m(x) = (xpq − 1) /Q (x), where Q (x) = i∈Q (x − α i ). Proof. We consider two cases. (i) Suppose p ≡ 1(mod 8) and q ≡ 1 ( mod 8 ). Then the values of R2 (β v ) and T2 (γ v ) are given by (12). Substituting (12) 1,
for R2 (β v ) and T2 (γ v ) in (15), we get MU (α v ) = µ,
if v ∈ F0,0 ∪ F1,1 , if v ∈ F1,0 ∪ F0,1 .
558
V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560 Table 1 Linear complexity of quaternary sequences for p ≡ 1 (mod 4).
1 2 3 4
L
m(x)
pq
pq pq − p + 1 (pq − p + q + 1) /2 (pq + p + q − 1) /2
xpq − 1 (xpq − 1) /Q (x) (xpq − 1) / (Q (x)F (x)) (xpq − 1) /F (x)
pq pq pq pq
≡ −1 (mod 8) ≡ 1 (mod 8) ≡ −3 (mod 8) ≡ 3 (mod 8)
Table 2 Linear complexity of quaternary sequences p ≡ −1 (mod 4).
1 2 3 4
L
m(x)
pq
pq − p − q + 2 pq − q + 1 (pq + p − q + 1) /2 (pq − p − q + 3) /2
(x − 1) / (Q (x)P (x)) (xpq − 1) /P (x) (xpq − 1) / (P (x)F (x)) (xpq − 1) / (Q (x) P (x) F (x))
pq pq pq pq
pq
≡ −1 (mod 8) ≡ 1 (mod 8) ≡ −3 (mod 8) ≡ 3 (mod 8)
(ii) Let p ≡ −3(mod 8) andq ≡ −3(mod 8). Here R2 (β v ) and T2 (γ v ) are given by (13). Similarly, after substitution in 1,
if v ∈ F
,
(15), we again have MU (α v ) = µ, if v ∈ F0,0 ∪ F1,1 . 1,0 0,1 Therefore, by Lemma 1, in both cases we have MU (α v ) = 0 if and only if v ∈ Q . We see that the statement of Theorem 2 follows from (6) and (7). ∪F
In a similar way we can derive the linear complexity and the minimal polynomial of U (t ) for all possible values of p and q (see the results in the Tables 1 and 2). Here Q (x) = i∈Q (x − α i ), P (x) = i∈P (x − α i ) and F (x) = i∈F0,0 ∪F1,1 (x − α i ). Now we consider V (t ). By the Gray map we obtain
0, for t mod pq ∈ F0,0 ∪ F0,2 , 1, for t mod pq ∈ F0,1 ∪ F2,0 , W (t ) = µ + 1, for t mod pq ∈ {0} ∪ F1,0 ∪ F1,2 , µ, for t mod pq ∈ F1,1 ∪ F2,1 . Define G(x) = i∈F0,1 ∪F1,1 (x − α i ), R(x) = i∈H0 (x − β i ) and T (x) = i∈G0 (x − γ i ).
(16)
Let
δ=
1, 0,
if p + q ≡ 0(mod 2), if p + q ≡ 1(mod 2).
Theorem 3. Let the sequence W (t ) be defined by (16). Then: (1) L = (p + 1)q/2 − δ and m(x) = (xpq − 1) / (x − 1)δ G(x)R(x) if q ≡ 1(mod 8) and p ≡ ±1(mod 8) or q ≡ 7(mod 8) and p ≡ ±3(mod 8). (2) L = (pq + p + q − 1)/2 − δ and m(x) = (xpq − 1) / (x − 1)δ Q (x) if q ≡ 1(mod 8) and p ≡ ±3(mod 8) or q ≡ 7(mod 8) and p ≡ ±1(mod 8). (3) L = pq − (p + q − 2)/2 − δ and m(x) = (xpq − 1) / (x − 1)δ R(x)T (x) if q ≡ 3(mod 8) and p ≡ ±3(mod 8) or q ≡ 5(mod 8) and p ≡ ±1(mod 8). (4) L = pq − (q − 1)/2 − δ and m(x) = (xpq − 1) / (x − 1)δ T (x) if q ≡ 3(mod 8) and p ≡ ±1(mod 8) or q ≡ 5(mod 8) and p ≡ ±3(mod 8).
Proof. Under the conditions of Theorem 3, by (11) and (16) we have
µT (γ v ), 2 (µ + 1)T2 (γ v ) + 1, MV (α v ) = v µ ((µ + 1)R2 (β ) + (q − 1)/2) , (µ + 1) ((p − 1)/2 + (q − 1)/2 + 1) ,
if v if v if v if v
∈ Z∗pq , ∈ P, ∈ Q, = 0.
We see that the statement of Theorem 3 follows from (12), (13), and (17).
(17)
3. The linear complexity of quaternary sequences over the ring Z4 The polynomial C (x) = 1 + c1 x + c2 x2 + · · · + cm xm ∈ Z4 [x] is called m an associated connection polynomial of periodic sequence S (t ) over Z4 if coefficients c1 , c2 , . . . , cm satisfy S (t ) = − i=1 ci S (t − i), ∀t ≥ m. The linear complexity L of the periodic sequence S (t ) over Z4 is equal to min{deg C (x) : then C (x) is an associated connection polynomial of S (t )}.
V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560
559
By [15], C (x) is an associated connection polynomial of S (t ) if and only if MS (x)C (x) ≡ 0 (mod (xpq − 1)), where MS (x) is the polynomial of S (t ). Let r be the order of 2 modulo pq, and let R = GF (22r , 22 ) be a Galois ring of characteristic 4. For the knowledge on the Galois ring, please see [16,17]. The group of invertible elements R∗ = R \ 2R of the ring R contains the cyclic subgroup of order 2r − 1 [17]. Hence, there exists an element ζ of order pq in R∗ . Let S (t ) = ϕ[S1 (t ), S2 (t )] be the quaternary sequence where S1 (t ) and S2 (t ) are the binary sequences defined in (1) and (2), respectively. Let us examine the values of the polynomial of the sequence S (t ) in the ring R. A possible path is to introduce auxiliary polynomials in the ring R and subsequently employ the procedure from the previous section. But there is an easier way. Note that in R we have
ζ q + ζ 2q + · · · + ζ (p−1)q = 3 and ζ p + ζ 2p + · · · + ζ (q−1)p = 3.
(18)
By definition, the polynomial of the sequence S (t ) is MS (x) = 1 +
xt + 2
t ∈Q
xt + 3
t ∈F0,1 ∪F1,0
xt .
(19)
t ∈P
Hence, MS (1) = 1 + (p − 1) + 4(p − 1)(q − 1)/4 + 3(q − 1), and thus MS (1) ≡ 1 (mod 2). Lemma 4. Let MS (x) be the polynomial of S (t ). (i) MS (ζ v ) = 0 if v ∈ Q . (ii) MS (ζ v ) = 2 if v ∈ P. Proof. (i) If v ∈ Q , then v F0,1 ∪ F1,0 = Q . Since the order of F0,1 ∪ F1,0 is equal to (p − 1)(q − 1)/2, it follows that
ζ vj = (q − 1)/2
ζ vj = 3(q − 1)/2.
j∈Q
j∈F0,1 ∪F1,0
v
By (18) and (19) we have MS (ζ ) = 1 + 3 + 3(q − 1) + 3(q − 1) = 0 in the ring R. (ii) Let v ∈ P; then similarly we obtain MS (ζ v ) = 1 + (p − 1) + 3(p − 1) + 9 = 2. ∗ Lemma 5. Let MS (x) be the polynomial of the sequence S (t ). If v ∈ Zpq , then MS (ζ v ) ∈ R∗ . ∗ Proof. To prove Lemma 5 it is enough to show that if v ∈ Zpq then 2MS (ζ v ) ̸= 0. From (19) we obtain 2M S (ζ v ) =
ζ vjq + 2
ζ vjp . Now we can apply (18), which concludes the proof of Lemma 5. pq−1 pq−1 From the expansion xpq − 1 = i=0 x − ζ i it follows that pq = i=1 1 − ζ i . Consequently, the elements ζ i − 1 and ζ i − ζ j , i ̸≡ j (mod pq) belong to R∗ . So, if ζ i , ζ j , i ̸≡ j (mod pq) are the roots of P (x), then P (x) is divisible by the product (x − ζ i )(x − ζ j ). However, generally P (x) is not necessarily divisible by the product (x − g )(x − h) in the ring R when g , h are the roots of P (x). For instance, P (x) = 2(x − 1) is not divisible by (x − 1)(x − 3). 2+2
p−1 j=1
q−1 j =1
Theorem 6. Let S (t ) = ϕ[S1 (t ), S2 (t )] be the quaternary sequence where S1 (t ) and S2 (t ) are the binary sequences defined in (1) and (2), respectively. Then the linear complexity of S (t ) is equal to L = pq − p + 1. Proof. Choose C (x) = (xpq − 1) /Q (x). Then all the roots of xpq − 1 are the roots of MS (x)C (x). Hence, C (x) is an associated connection polynomial of S (t ). This implies that L ≤ pq − p + 1. If L ̸= pq − p + 1, then there exists another associated connection polynomial C1 (x) of the sequence S (t ) with degree less than pq − p + 1. Hence, C1 (ζ v )MS (ζ v ) = 0 for v = 0, 1, . . . , pq − 1. Let v ∈ Z∗pq ∪ {0}; then C1 (ζ v ) = 0 by Lemma 5. If v ∈ P, then C1 (ζ v ) ∈ 2R by Lemma 4. We obtain that 2C1 (ζ v ) = 0 for v ∈ Z∗pq ∪ P ∪ {0} and 2C1 (x) ̸= 0 by definition. Thus, 2C1 (x) is divisible by
i∈Z∗ pq ∪P ∪{0}
x − ζi ,
which contradicts the fact that the degree of 2C1 (x) is less than pq − p + 1. Therefore, L = pq − p + 1. This completes the proof. For the linear complexity of V (t ) we have a more complicated dependence which is shown in Theorem 7. Theorem 7. Let V (t ) be the quaternary sequence defined by (4). Then the linear complexity of V (t ) is equal to
L=
Here ∆ =
(p + 1)q/2 − ∆, (pq + p + q − 1)/2 − ∆, pq − (p − 1)/2 − ∆, pq − ∆, 1, 0,
if p ≡ ±1(mod 8) and q ≡ 1(mod 16), if p ≡ ±1(mod 8) and q ≡ −1(mod 16) or p ≡ ±3(mod 8) and q ≡ ±1(mod 16), if p ≡ ±1(mod 8) and q ≡ 5(mod 8) or q ≡ 9(mod 16), otherwise.
if (p − 1)(q − 3) ≡ 4(mod 8), otherwise.
Theorem 7 is proved by the method from Section 2.
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V. Edemskiy, A. Ivanov / Journal of Computational and Applied Mathematics 259 (2014) 555–560
4. Conclusion It was shown in this paper that some of the quaternary sequences of length pq have high linear complexities over the finite ring Z4 . Also we have determined the values of p and q for which these sequences have high linear complexity over the finite field of four elements. For the case of the finite field we have found the minimal polynomial for the sequences concerned. Acknowledgments The authors acknowledge the patient referees for their valuable and constructive comments which helped to improve this work. References [1] T.W. Cusick, C. Ding, A. Renvall, Stream Ciphers and Number Theory, Elsevier, Amsterdam, 1998. [2] 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. [3] E. Bai, X. Fu, G. Xiao, On the linear complexity of generalized cyclotomic sequences of order four over Zpq, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E88-A (1) (2005) 392–395. [4] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order 2, Finite Fields Appl. 3 (2) (1997) 159–174. [5] V. Edemskiy, O. Antonova, About computation of the linear complexity of generalized cyclotomic sequences with period pq, in: Proc. of 2011 International Workshop on Signal Design and its Applications in Communications, IWSDA’011, China, 2011, pp. 9–12. [6] W. Meidl, Remarks on a cyclotomic sequence, Des. Codes Cryptogr. 51 (1) (2009) 33–43. [7] T. Yan, L. Hong, G. Xiao, The linear complexity of new generalized cyclotomic binary sequences of order four, Inform. Sci. 178 (3) (2008) 807–815. [8] S.M. Krone, D.V. Sarwate, Quadriphase sequences for spread spectrum multiple-access communication, IEEE Trans. Inform. Theory IT-30 (3) (1984) 520–529. [9] H.D. Luke, H.D. Schotten, H. Hadinejad-Mahram, Binary and quadriphase sequences with optimal autocorrelation properties: a survey, IEEE Trans. Inform. Theory IT-49 (2003) 3271–3282. [10] Z. Yang, P. Ke, Construction of quaternary sequences of length pq with low autocorrelation, Cryptogr. Commun. 3 (2) (2011) 55–64. [11] D.H. Green, Linear complexity of modulo-m power residue sequences, IEE Proc., Comput. Digit. Tech. 151 (6) (2004) 385–390. [12] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982. [13] X. Du, Z. Chen, Linear complexity of quaternary sequences generated using generalized cyclotomic classes modulo 2p, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E94-A (5) (2011) 1214–1217. [14] C. Ding, T. Helleseth, W. Shan, On the linear complexity of Legendre sequences, IEEE Trans. Inform. Theory IT-44 (1998) 1276–1278. [15] Z.X. Wan, Algebra and Coding Theory, Science Press, Beijing, 1976. [16] B.R. McDonald, Finite Rings With Identity, Marcel Dekker, New York, 1974. [17] Z.X. Wan, Finite Fields and Galois Rings, World Scientific Publisher, Singapore, 2003.