Linear Complexity of a New Class of Quaternary Generalized Cyclotomic Sequence with Period 2 p m

. Sequences with high linear complexity property are of importance in applications. In this paper, based on the theory of generalized cyclotomy, new classes of quaternary generalized cyclotomic sequences with order 4 and period 2 p m are constructed. In addition, we determine their linear complexities over ﬁnite ﬁeld F 4 and over Z 4 , respectively.


Introduction
Pseudorandom sequence has a wide range of applications in the spread spectrum communication, radar navigation, code division multiple access, stream cipher, and so on [1]. e linear complexity L( s (u) { }) of a sequence s(u) { } is defined as the smallest order of linear feedback shift register (LFSR) that can generate the whole sequence. According to Berlekamp-Massey algorithm [2], if the linear complexity of the sequence is l, then 2l consecutive terms of the sequence can be used to restore the whole sequence. Hence, a "high" linear complexity L( s (u) { }) should be no less than one half of the length (or minimum period) of the sequence [3]. For cryptographic applications, sequences with high linear complexity are required.
Sequences with high linear complexity can be constructed based on cyclotomic classes. Generally, sequences based on classical cyclotomic classes and generalized cyclotomic classes are called classical cyclotomic sequences and generalized cyclotomic sequences, respectively. ere are lots of works on linear complexity of binary cyclotomic sequences, see [4][5][6][7], for instance. Recently, Xiao et al. presented a new class of cyclotomic binary sequences of period p 2 and determined the linear complexity of the sequences in the case of f � 2 r for a positive integer r and showed that the sequences have large linear complexity if p is a non-Wieferich prime [8]. Moreover, Edemskiy et al. examined the linear complexity of sequences and extended the result to more general even integers f [9]. At the same time, the results were also generalized by Ye and Ke in [10] by considering a new construction with a flexible support set, which includes the original construction as a special case. Very recently, in [11], Ouyang and Xie extended the construction to the case of 2p m and showed that the constructed sequences have high linear complexity when m ≥ 2.
Compared with the binary case, less attention has been paid to quaternary sequences with high linear complexity. In [12], Kim et al. constructed a class of generalized cyclotomic sequences with period 2p over Z 4 and analyzed the autocorrelation properties of these sequences. In [13], Chen and Edemskiy determined the linear complexity of these sequences over Z 4 . In [14], Du and Chen defined a family of quaternary sequences over F 4 of period 2p using generalized cyclotomic classes over the residue class ring modulo 2p. ey determined the linear complexities of the sequences and then showed that the sequences have large linear complexity.
e autocorrelation values of the sequences were further studied in [15]. In [16], Ke et al. generalized the construction of [14] to the case of 2p m . ey proved that the constructed sequences are balanced and also possess high linear complexity. Recently, in [17], Liu et al. further extended the sequence construction of [16,18]. A class of quaternary generalized cyclotomic sequences with order 2 d period 2p m over F 4 are constructed, and their linear complexities are studied. In addition, in [19], Edemskiy and Ivanov constructed another kind of new quaternary generalized cyclotomic sequences with period 2p based on the Chinese Remainder eorem and studied their autocorrelation properties and linear complexities over F 4 as well as Z 4 .
Although the sequence they constructed are both quaternary sequences, it can be seen by comparison that [16,17,19] have their advantages and disadvantages. On the one hand, by definition in [16,17], the elements in a fixed support set of the sequence are all even or odd, while the sequence proposed by Edemskiy and Ivanov in [19] takes value more random. On the other hand, in [19], the sequence has period 2p, while in [17], the sequence has a more general period 2p m . In this paper, combining the advantages of the above two constructions, we will define a new class of quaternary generalized cyclotomic sequences of order 4 with period 2p m over F 4 as well as Z 4 . Furthermore, we determine the linear complexities of the new defined sequences over F 4 and Z 4 , respectively.
is paper is organized as follows. In Section 2, we introduce some necessary preliminary concepts and present a general construction of quaternary generalized cyclotomic sequences with period 2p m over Z 4 . In addition, based on a mapping from Z 4 to F 4 , quaternary sequences over F 4 can be derived directly from the quaternary sequences over Z 4 . In Section 3, we compute the linear complexity of sequences we constructed over F 4 . In Section 4, we compute the linear complexity of sequences that we constructed over Z 4 . In Section 5, we conclude this paper.

Perliminaries
Let p be an odd prime with p ≡ 1(mod 4). Let g be a primitive root of Z * p 2 , where Z * n denotes the set of all invertible elements of Z n . It is well known that g is also a primitive root of Z * p m (m ≥ 1). For j � 1, 2, . . . , m and l � 0, 1, 2, 3, we define where D It is easy to verify that For simplicity, we denote (3) en, we define the sequences s 1 (u) u ≥ 0 and s 2 (u) u≥0 over Z 4 of length 2p m by respectively. e well-known Gray-mapping ϕ: Let F 4 � μF 2 + F 2 be a finite field of order 4, where µ satisfies the relation μ 2 + μ + 1 � 0. Let σ be a map from F 2 × F 2 to F 4 which is defined as follows: where a, b ∈ F 2 . en, we have σ ∘ ϕ: By this way, the sequences over Z 4 defined in (4) and (5) can be modified to be sequences over F 4 as follows, i.e., 2 Complexity Remark 1. Obviously, the support sets of sequences defined in this paper are more random than those in [16,17], where the support sets all consist of even or odd numbers. In addition, the quaternary sequences constructed in [19] can be considered as a special case of the sequences defined above by taking m � 1. In other words, in the case of length 2p, the constructions in [19] and this paper, although they were expressed in different forms, are the same in essence. In detail, the difference between these two constructions is that, in [19], the sequences were constructed by using the Chinese Remainder eorem, while the sequences in this paper are defined directly. erefore, this paper can be regarded as a generalization of [19].

Linear Complexity of the Sequence over F 4
In this section, we discuss the linear complexity over F 4 . Hence, we focus on the quaternary sequences in (9) and (10). Let s ∞ � (s 0 , · · · , s i , · · ·) be a sequence over F 4 , and its linear complexity L(s ∞ ) is the smallest positive integer L for which there exist constants c 1 , c 2 , · · · , c L such that Let s ∞ � s u N u�0 be a periodic sequence of period N over F 4 . And S(x) � N− 1 u�0 s u x u is called the generating polynomial of the sequence. en, the minimal polynomial of s u N u�0 is given by and the linear complexity of s u N u�0 is given by Let d be the order of 4 modulo p m , that is, d is the smallest integer such that 4 d ≡ 1(mod p m ). Let ξ be a primitive element of F 4 d ; then, the order of α � ξ 4 d − 1/p m is p m , and θ � α p m− 1 has order p.
According to the definition of the linear complexity of the sequence, we should compute the number of common roots of the generating polynomial S(x) and polynomials , we need to verify if it is a multiple root of S(x) [20].

Complexity 3
Proof. Since the proof is similar, only the case of R (2) If j � i + 1, then where θ � α p m− 1 .
e proof is thus completed.
Theorem 1. Let p ≡ 1(mod 4) be an odd prime. Let b 1 (u) u≥0 be a generalized cyclotomic quaternary sequence of period 2p m defined in (9). en, the linear complexity of b 1 (u) u≥0 over F 4 is given by Proof. By the definition of the sequence b 1 (u) u≥0 , the generating polynomial S(x) of b 1 (u) u≥0 is given by It is easily seen that if v � 0, then S(1) � μ + 1 ≠ 0. Hence, 1 is not root of S(x). We can write where the last equality follows from the fact that T 2 (θ tg ) � 1 + T 2 (θ t ).
is called the generating polynomial of the sequence.
en, an LFSR with a connection polynomial C(X) generates s u if and only if [22] S(X)C(X) ≡ 0 mod X T − 1 , where C(X) ∈ Z 4 [X] satisfies C(0) � 1. at is, , Let R � GR(4 r , 4) be a Galois ring of characteristic 4 and cardinality 4 r , where r is the order of 2 modulo p [23]. Let R * � R/2R be the group of units of R, and it contains a cyclic subgroup of order 2 r − 1 [23]. en, let β ∈ R be of order p. We can write c � 3β, and then the order of γ is 2p m and Complexity c p m � − 1. By [23], we know that 2R is the maximal ideal of ring R. e natural homomorphism R ⟶ R � R/2R will be denoted by "-" and the image r ∈ R in R � R/2R by r.
Notice that, in the ring R, the number of roots of a polynomial can be greater than its degree. Hence, we need some lemmas.
Lemma 7 (see [13]). Let P(x) ∈ Z 4 [x] be a nonconstant polynomial. If a ∈ R is a root of polynomial P(x), then there exists a polynomial Q 1 (x) ∈ Z 4 [x] such that Moreover, if b ∈ R is also the root of P(x) and a − b ∈ R * , then where Q 1 (x) � (x − b)Q 2 (x).
By the definition of the connection polynomial and linear complexity over Z 4 , we have L( s u ) ≤ 2p m .
Without loss of generality, assume that L( s u ) < 2p m ; then, there exists a connection polynomial C(x), and it satisfies where the degree of C(x) is less than 2p m . By the definition of γ, we have c p m � 1 and c ≠ 0, 1. en, in the ring R, p m − 1 j�1 c j � 1 holds. By (4), we get