Research Article | Open Access
Pinhui Ke, Yan Zhong, Shengyuan Zhang, "Linear Complexity of a New Class of Quaternary Generalized Cyclotomic Sequence with Period ", Complexity, vol. 2020, Article ID 6538970, 7 pages, 2020. https://doi.org/10.1155/2020/6538970
Linear Complexity of a New Class of Quaternary Generalized Cyclotomic Sequence with Period
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 are constructed. In addition, we determine their linear complexities over finite field and over , respectively.
Pseudorandom sequence has a wide range of applications in the spread spectrum communication, radar navigation, code division multiple access, stream cipher, and so on . The linear complexity of a sequence is defined as the smallest order of linear feedback shift register (LFSR) that can generate the whole sequence. According to Berlekamp–Massey algorithm , if the linear complexity of the sequence is l, then consecutive terms of the sequence can be used to restore the whole sequence. Hence, a “high” linear complexity should be no less than one half of the length (or minimum period) of the sequence . 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. There are lots of works on linear complexity of binary cyclotomic sequences, see [4–7], for instance. Recently, Xiao et al. presented a new class of cyclotomic binary sequences of period and determined the linear complexity of the sequences in the case of for a positive integer r and showed that the sequences have large linear complexity if p is a non-Wieferich prime . Moreover, Edemskiy et al. examined the linear complexity of sequences and extended the result to more general even integers f . At the same time, the results were also generalized by Ye and Ke in  by considering a new construction with a flexible support set, which includes the original construction as a special case. Very recently, in , Ouyang and Xie extended the construction to the case of and showed that the constructed sequences have high linear complexity when .
Compared with the binary case, less attention has been paid to quaternary sequences with high linear complexity. In , Kim et al. constructed a class of generalized cyclotomic sequences with period over and analyzed the autocorrelation properties of these sequences. In , Chen and Edemskiy determined the linear complexity of these sequences over . In , Du and Chen defined a family of quaternary sequences over of period using generalized cyclotomic classes over the residue class ring modulo . They determined the linear complexities of the sequences and then showed that the sequences have large linear complexity. The autocorrelation values of the sequences were further studied in . In , Ke et al. generalized the construction of  to the case of . They proved that the constructed sequences are balanced and also possess high linear complexity. Recently, in , Liu et al. further extended the sequence construction of [16, 18]. A class of quaternary generalized cyclotomic sequences with order period over are constructed, and their linear complexities are studied. In addition, in , Edemskiy and Ivanov constructed another kind of new quaternary generalized cyclotomic sequences with period based on the Chinese Remainder Theorem and studied their autocorrelation properties and linear complexities over as well as .
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  takes value more random. On the other hand, in , the sequence has period , while in , the sequence has a more general period . 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 over as well as . Furthermore, we determine the linear complexities of the new defined sequences over and , respectively.
This 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 over . In addition, based on a mapping from to , quaternary sequences over can be derived directly from the quaternary sequences over . In Section 3, we compute the linear complexity of sequences we constructed over . In Section 4, we compute the linear complexity of sequences that we constructed over . In Section 5, we conclude this paper.
Let p be an odd prime with . Let be a primitive root of , where denotes the set of all invertible elements of . It is well known that is also a primitive root of .
For and , we definewhere are called generalized cyclotomic classes of order 4 with respect to and are called generalized cyclotomic classes of order 4 with respect to .
It is easy to verify that
For simplicity, we denote
Definition 1. Let . Then, we define the sequences and over of length byrespectively.
The well-known Gray-mapping is defined asLet be a finite field of order 4, where µ satisfies the relation . Let σ be a map from to which is defined as follows:where .
Then, we have , i.e.,By this way, the sequences over defined in (4) and (5) can be modified to be sequences over as follows, i.e.,Similarly,
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  can be considered as a special case of the sequences defined above by taking . In other words, in the case of length , the constructions in  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 , the sequences were constructed by using the Chinese Remainder Theorem, while the sequences in this paper are defined directly. Therefore, this paper can be regarded as a generalization of .
3. Linear Complexity of the Sequence over
In this section, we discuss the linear complexity over . Hence, we focus on the quaternary sequences in (9) and (10). Let be a sequence over , and its linear complexity is the smallest positive integer L for which there exist constants such that
Let be a periodic sequence of period N over . And is called the generating polynomial of the sequence. Then, the minimal polynomial of is given byand the linear complexity of is given by
Let d be the order of 4 modulo , that is, d is the smallest integer such that . Let ξ be a primitive element of ; then, the order of is , and 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 and polynomials . Notice that , so we may check that if is a root of . Furthermore, if it is a root of , we need to verify if it is a multiple root of .
To this end, we need some auxiliary polynomials:where and .
Lemma 1. (see ). For , , we have
Lemma 2. (see ). For , , we have(i)(ii)where
Lemma 3. Let , , and . For , we have
Proof. Since the proof is similar, only the case of is considered here. By Lemma 1, we have (1)If , note that for any . Obviously, . Hence,(2)If , then where .(3)If , suppose that , , and ; then, . So,The proof is thus completed.
Lemma 4. (see ). For , let θ be a primitive pth root of unity in the extension of the field , and define . Then, we have
Lemma 5. (see ). With the above notation, then if ; otherwise, .
Lemma 6. (see ). With notations defined as above, if , then .
Theorem 1. Let be an odd prime. Let be a generalized cyclotomic quaternary sequence of period defined in (9). Then, the linear complexity of over is given by
Proof. By the definition of the sequence , the generating polynomial of is given byIt is easily seen that if , then . Hence, 1 is not root of . We can writeAccording to Lemma 3, if , we havewhere the last equality follows from the fact that .(1)If , by Lemmas 4 and 5, we obtain . In other words, when .(2)If , by Lemmas 4 and 5 again, we know that (if not, then we can simply replace with ). So,Thus, if , then ; if , then .
To complete the proof, we should verify that if is a multiple root of when .According to Lemma 3, if , we haveAs it was shown in , if , then or , where ζ satisfies the relation and over .(i)When , without loss of generality, suppose that . By Lemma 6, we can write . Obviously, for any . Since and , then we get if . But, this means that , we get a contradiction. Thus, we have for any .(ii)When , without loss of generality, suppose that . By Lemma 6, we can write . Obviously, for any . Similarly, sinceand , we derive that if . But, this means that . By , we get , i.e., . It is also a contradiction. Thus, we have for any .
In conclusion, we know for all . Then, we complete the proof.
Theorem 2. Let be an odd prime. Let be a generalized cyclotomic quaternary sequence of period defined in (10). Then, the linear complexity of over is given by
Proof. The proof of this theorem is similar to that of Theorem 1. Thus, we omit it here.
4. Linear Complexity of the Sequence over
Let be a quaternary sequence of period T over , and its linear complexity is the smallest order L of a linear feedback shift register (LFSR) in , for which there exist constants such that
Also, is called that connection polynomial. is called the generating polynomial of the sequence. Then, an LFSR with a connection polynomial generates if and only if where satisfies . That is,
Let be a Galois ring of characteristic 4 and cardinality , where r is the order of 2 modulo p . Let be the group of units of R, and it contains a cyclic subgroup of order . Then, let be of order p. We can write , and then the order of γ is and . By , we know that is the maximal ideal of ring R. The natural homomorphism will be denoted by “-” and the image in by .
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 ). Let be a nonconstant polynomial. If is a root of polynomial , then there exists a polynomial such that
Moreover, if is also the root of and , thenwhere .
Lemma 8. (see ). Let be a nonconstant polynomial. If for all , then there exists polynomial such that
If for all , then there exists polynomial such that
Theorem 3. Let be an odd prime. Let be a generalized cyclotomic quaternary sequence of period defined in (10). Then, the linear complexity of over is given by
Proof. With notations defined as before, let be of order and . It is easy to know that . Then, . Hence, when , we have .
By the definition of the connection polynomial and linear complexity over , we have .
Without loss of generality, assume that ; then, there exists a connection polynomial , and it satisfieswhere the degree of is less than .
By the definition of γ, we have and . Then, in the ring , holds.
By (4), we getIf , and by Lemma 1, we obtainAccording to (41), we have for all . Then, , which implies that there exists a polynomial such thatMoreover, it is easily seen thatwhere , , and is Euler function. Then,Thus, and ; this means thatHence, when , we haveIn other words, when , we have . Then, by Lemma 8, we get . But, this means that , which is contrary to our assumption. The proof is thus completed.
Theorem 4. Let be an odd prime. Let be a generalized cyclotomic quaternary sequence of period defined in (5). Then, the linear complexity of over is given by
Proof. The proof of this theorem is similar to that of Theorem 3. Thus, we omit it here.
This paper presented new classes of quaternary generalized cyclotomic sequences with order 4 and period , where p is an odd prime and . We computed their linear complexity over finite field as well as . The results indicated that the sequences proposed in this paper possess high linear complexities. Autocorrelation is another important measure of a sequence. For quaternary sequence s with even period over , the optimal maximum out-of-phase autocorrelation magnitude, denoted by , has been shown to be 2 . For the new proposed sequence, when the period is , Edemskiy has proved that if and if , respectively. However, for the general case, the autocorrelation properties of the new proposed sequences are not known. Hence, as a further work, it is valuable to analyze the autocorrelation values of the new proposed sequences.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research was supported by the National Natural Science Foundation of China (nos. 61772292 and 61772476), the Natural Science Foundation of Fujian Province (no. 2019J01273), and Fujian Normal University Innovative Research Team (no. IRTL1207).
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