A kind of quaternary sequences of period 2p mq n and their linear complexity

Sequences with high linear complexity have wide applications in cryptography. In this paper, a new class of quaternary sequences over $\mathbb{F}_4$ with period $2p^mq^n$ is constructed using generalized cyclotomic classes. Results show that the linear complexity of these sequences attains the maximum.


Introduction
Stream ciphers divide the plain text into characters and encipher each character with a time-varying function. It is known that the stream cipher plays a dominant role in cryptographic practice and remains a crucial role in military and commercial secrecy systems. e security of stream ciphers now depends on the "randomness" of the key stream [1]. For the system to be secure, the key stream must have a series of properties: balance, long period, low correlation, and so on.
A necessary requirement for unpredictability is a large linear complexity of the key stream, which is defined to be the length of the shortest linear-feedback shift register able to produce the key stream. Let F l denote a finite field with l elements, where l is a prime power. A sequence S � s i is periodic if there exists a positive integer T such that s j+T � s j for all j ≥ 0. Let S � s i be a periodic sequence over F l . e linear complexity of S, denoted by LC(S), is the least integer L of a linear recurrence relation over F l satisfied by S: −c 0 s i+L � c 1 s i+L−1 + · · · + c L s i , for i ≥ 0, where c 0 ≠ 0 and c 0 , c 1 , . . . , c L−1 , c L ∈ F l . By B-M algorithm [2], if LC(S) ≥ (N/2) (N is the least period of S), then S is considered to be good from the viewpoint of linear complexity. Periodic sequences have been intensively studied in the past few years since they are widely used in CDMA (codedivision multiple access), global position systems, and stream ciphers. As special cases, cyclotomic and generalized cyclotomic sequences of different periods and orders have attracted many researchers to deeply explore due to their good pseudorandom cryptographic properties [3][4][5]. In particular, the linear complexity of Legendre sequences and cyclotomic sequences of order r was studied in [6,7], respectively. Generalized cyclotomy, as a natural generalization of cyclotomy, was presented by Whiteman [8] and Ding and Helleseth [9]. It should be noted that Whiteman's generalized cyclotomy is not in accordance with the classic cyclotomy. Ding-Helleseth cyclotomy includes the classic cyclotomy as a special case. Whereafter, the linear complexity of generalized cyclotomic sequences has been determined [10][11][12][13][14][15].
Quaternary sequences are also important from the point of many practical applications; please refer to [16]. Owing to the nice algebraic structure, quaternary sequences also have received a lot of attention. For instance, a kind of almost quaternary cyclotomic sequences was defined in [17] and was proved to have an ideal autocorrelation property [17]. A new class of quaternary sequences of length pq, constructed by the inverse Gray mapping, was studied in [18]. A family of quaternary sequences of period 2p over F 4 was presented and showed to possess high linear complexity [19].
Motivated by the idea in [20,21], we constructed a new class of quaternary sequences over F 4 with period 2p m q n by using the generalized cyclotomic classes in this paper. From the definition of S in (11), we can easily see that the newly proposed sequences have longer period contrast to those in [21]. e linear complexity of these sequences is computed, and the results show that the proposed sequences have high linear complexity.
is paper is organized as follows. In Section 2, the periodic sequence S with period 2p m q n is given. Section 3 determines the linear complexity of the constructed sequence. Finally, we give some remarks on this paper.

Preliminaries
For a positive integer a ≥ 2, use Z a to denote the ring Z a � 0, 1, 2, . . . , a − 1 { } with integer addition modulo a and integer multiplication modulo a. Usually, we use Z * a to denote all invertible elements of Z a , i.e., all elements b in Z a satisfying gcd(a, b) � 1. Obviously, the group Z * a has cardinality ϕ(a), where ϕ(·) denotes the Euler function.
For a subset A ⊂ Z a and an element b ∈ Z a , define where addition and multiplication refer to those in Z a . Let p and q be two distinct odd primes. Let m and n denote two positive integers. Suppose that g 1 is a primitive element of Z * p 2 . en, g 1 is a primitive root of Z * p m for m ≥ 1 [22]. Without loss of generality, assume g 1 is an odd integer. It is known that g 1 is also a primitive root of Z * 2p m [22]. Obviously, g 1 is a common primitive root of Z p i and Z 2p i for all 1 ≤ i ≤ m. By the same argument, there exists an integer g 2 such that g 2 is a common primitive root of Z q j and Z 2q j for any 1 ≤ j ≤ n.
Lemma 1 (see [23]). Let m 1 , . . . , m t be positive integers. For a set of integers a 1 , . . . , a t , the system of congruences has solutions if and only if If (4) is satisfied, the solution is unique modulo l cm(m 1 , . . . , m t ).
Let g be the unique solution of the following congruence equations: Lemma 1 guaranteed the existence and uniqueness of the common primitive root g of p i , 2p i , p j , and 2p j . Similarly, there exists a unique integer y satisfying the following system of congruences: Assume that e i,j � gcd( en, d i,j is the least positive integer that satisfies g d i,j ≡ 1 mod p i q j ([9], Lemma 2), i.e., ord p i q j (g) � d i,j . In the sequel, let i and j be two integers with 1 ≤ i ≤ m and 1 ≤ j ≤ n. e generalized cyclotomic classes with respect to p i q j , similar to Ding-Helleseth's generalized cyclotomic classes ( [9]), are defined as follows: for h � 0, 1. With the above preparations, we get a partition of Z 2p m q n as follows: (10) Let F 4 � 0, 1, α, α 2 be the finite field with 4 elements, where α satisfies α 2 � α + 1. A class of quaternary sequence can be given by allocating each elements of F 4 to each generalized cyclotomic class with respect to 2p m q n . To ensure the constructed sequence has high linear complexity, we should technologically do with it.

Complexity
Let a, b, c, d { } be a set of four tuples over F 4 , and the elements in these tuples are pairwise distinct. A quaternary generalized cyclotomic sequence S � s i of period 2p m q n is defined as It is easily seen that the sequence S � s i is balanced.

Linear Complexity of the Constructed Sequences
In generating running keys, the linear feedback shift register (LFSR) is one of the most useful devices. Also, it is shown that every periodic sequence can be generated by using LFSR. For researchers, what they most concern is the shortest length of LFSR that could produce a given sequence S, which is referred to the linear complexity of S. Let S � s i be a periodic sequence over the finite field F l of period N. We first recall the definition of linear complexity of periodic sequences that is given in Section 1. e linear complexity of S over F l , denoted by LC(S), is the smallest positive integer L satisfying the following linear recurrence relation: where c 0 ≠ 0 and c 0 , associated with the linear recurrence relation (12) is called the characteristic polynomial of S. A characteristic polynomial with the smallest degree is called a minimal polynomial of S [2]. For the periodic sequence S, let , which is called the generating polynomial of S. e following lemma gives a method to compute the linear complexity of S by using the generating polynomial S(x).
Lemma 2 (see [24]). Let S be a sequence over F l of period N. en, the minimal polynomial m(x) of S is and the linear complexity LC(S) of S is given by where S(x) is the generating polynomial of S.
Lemma 3 (Lemma 2, [10] and Lemma 1, [20]). Let notations be defined as above. en, for 1 ≤ i ≤ m and h � 0, 1, we have where δ x,y � 0, if x + py is odd, For the generalized cyclotomic classes D corresponding to p i q j and 2p i q j , we have the following lemma.

Complexity 3
By the method analogous to that used above, we can get the second conclusion of this lemma.
Let d � ord p m q n (8). Assume that β is a primitive p m q n th root of unity in F 4 d . It can be easily checked that By Lemma 2, in order to determine the linear complexity of S, we need to determine gcd(x 2p m q n − 1, S(x)) � gcd((x p m q n − 1) 2 , S(x)) over F 4 [x]. By (22), we should check whether β i , 0 ≤ i ≤ p m q n − 1, is a root of S(x). If it is a root of S(x), we need to verify whether it is a multiple root of S(x).
Recall that H Similarly, we have Combining (24) We first compute where k � p a q b l and gcd(pq, l) � 1. e computation is divided into the following cases.
Case 1: i ≤ a and j ≤ b. With simple derivation, we have Case 2: i � a + 1 and j � b + 1. en, 4 Complexity where ζ pq � β p m−1 q n−1 is a pqth primitive root of unity. Case 3: i > a + 1 or j > b + 1. Let η � β p m+a−i q n+b−j , then η pq ≠ 1. It follows from Lemma 4 that (33) , then ζ q is a qth primitive root of unity. Hence, we obtain Case 5: i � a + 1 and j ≤ b. By Lemma 4, we get where ζ p � β p m−1 q n+b−j .
From the above discussions, we have proved the first part of the following lemma.

Lemma 8.
For k � p a q b l with gcd(l, pq) � 1, 0 ≤ a ≤ m − 1, and 0 ≤ b ≤ n − 1, we have where ζ pq � β p m−1 q n−1 is a pqth primitive root of unity and β is a p m q n th primitive root of unity. Proof.
e proof of the second conclusion of this lemma is similar to the first part and we omit it. □ Lemma 9. For k � p a q b l with gcd(pq, l) � 1, we obtain where ζ p � β p m−1 q n+b and ζ q � β p m+a q n−1 .
Proof. Because (38)-(41) can be proved in a similar way, here we only prove (38). By notations and (25), we get where k � p a q b l with gcd(pq, l) � 1.
If i ≤ a, for each t ∈ p m+a− i q n+b lD (p i ) 0 , it can be easily seen that β t � 1. us, If i � a + 1, then β p m−1 q n+b is a pth primitive root of unity and where erefore, where η � β p m+a−i q n+b l and η p ≠ 1.
In the following, we will determine the terms with a, b, c, d as coefficients in (27), respectively.
First, we compute the terms with a as coefficient.
It follows from Lemmas 8 and 9 that It can be easily checked that