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Sun, Yuhua; Wang, Qiuyan; Yan, Yang; Yan, Tongjiang; Li, Hui

doi:https://doi.org/10.1155/2020/6963487

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 6963487 | https://doi.org/10.1155/2020/6963487

Linear Complexity of Generalized Cyclotomic Sequences of Order 4 over

Yuhua Sun,¹Qiuyan Wang,^2,3Yang Yan,⁴Tongjiang Yan,¹and Hui Li⁵

Academic Editor: Massimiliano Zanin

Received12 Jul 2019

Accepted17 Dec 2019

Published11 Mar 2020

Abstract

Generalized cyclotomic sequences of period have several desirable randomness properties if the two primes p and q are chosen properly. In particular, Ding deduced the exact formulas for the autocorrelation and the linear complexity of these sequences of order 2. In this paper, we consider the generalized sequences of order 4. Under certain conditions, the linear complexity of these sequences is developed over a finite field . The results show that, in many cases, they have high linear complexity.

1. Introduction

Let l be a prime number and denote a finite field with l elements. A sequence is called to be n-periodic if for all . Periodic sequences with certain properties are widely used in software testing, radar systems, stream ciphers, and so on. For cryptography applications, the linear complexity is an important factor. It is defined to be the length of the shortest linear feedback shift register which generates this sequence. For cyclotomic sequences, many researchers are devoted to studying their random properties [1–8]. The generalized cyclotomic sequences have been described and studied [9–11, 12, 13] for the past decades. A number of periodic sequences with attractive random properties have been constructed [6, 14–20].

Let p and q be two distinct odd primes with . Define and . The Chinese reminder theorem guarantees that there exists a common primitive root of both p and q. Let x be an integer satisfying

Whiteman proved that [21]where denotes the set of all invertible elements of the residue class ring .

The generalized cyclotomic classes of order d with respect to n are defined by [21]where the multiplication is that of . Clearly, the cosets depend on the choice of the common primitive root if . It is not hard to prove that [21]where denotes the empty set. Define

Then,

Let S be a nonempty subset of and

We define the binary sequence of period n as follows:where and . For and , the linear complexity of these sequences over has been calculated by Ding [16] with and Hu et al. [20] with . Furthermore, for and , Ding [9] determined the linear complexity of the two-prime sequences over a finite field and used these sequences to construct several classes of cyclic codes over with optimal or almost optimal property, where . In this paper, we only consider the case , , and . Under the assumption that or , we calculate the linear complexity of these sequences over the finite field . The results show that, in many cases, these sequences have high linear complexity.

This paper is organized as follows. Section 2 presents basic notations and results of periodic sequences and the generalized cyclotomy [21]. In Section 3, we give an expression for the linear complexity of the generalized cyclotomic sequences over . In the last section, we present concluding remarks of this paper.

2. Preliminaries

Firstly, we give the definition and formula of linear complexity of periodic sequences over a finite field. See [22] or [23] for more details.

Let l be a prime number and be a periodic sequence over with period n, where for . The sequence can be viewed as a power seriesin the power series ring .

Let , thenwhere .

Definition 1. The polynomial is called the minimal polynomial of the periodic sequence over . The is called the linear complexity of the sequence over , which is denoted by .
Indeed, is the length of the shortest linear feedback shift register which generates the sequence .
If , then has n distinct roots in the algebraic closure of , where denotes the n-th primitive root of unity. It is easy to see thatIn order to determine the linear complexity of generalized cyclotomic sequences, we introduce generalized cyclotomy.
Let symbols be the same as in the introduction and . The generalized cyclotomic numbers of order 4 with respect to n is defined byBy the well-known theorem ([24], P. 128), there are exactly two representations of n in the form with and the sign of b indeterminate.
Let and be a fixed primitive root of p and q, respectively. For , let and be the integers given uniquely byDefine a and b to be integers satisfyingwhere denotes the Legendre symbol.
It is clear that and is one of the two representations of n. The following lemma shows that the generalized cyclotomic numbers of order 4 with respect to n depend uniquely on this representation.

Lemma 1 (see [19], Theorem IV.1.). Let be two distinct primes with the fixed primitive roots and , respectively. , and a and b are the integers defined in (14).
If , then in Table 1, , , , , and .
If , then in Table 2, , , , , and .

3. Generalized Cyclotomic Sequences of Order 4

Throughout this section, let p and q be two distinct odd primes with . Define and . Let l be a prime and satisfy .

The generalized cyclotomic sequence of order 4 of period n is defined bywhere and are defined by (3) and . Here, in this paper, we treat it as a sequence over a finite field , where .

Denote the multiplicative order of l modulo n. Let be an n-th primitive root of unity over . For the sequence defined by (15), we know

Define δ as follows:

Note that the generalized cyclotomic classes of order 2 are given by

Define . The following lemma has been proven in [9].

Lemma 2 (see [9], Lemma 3.13). If , then we haveHence, if and only if .
To compute the linear complexity of , we need to compute . For this purpose, we require a number of auxiliary results.

Lemma 3 (see [16], Lemma 5). Let m be the least common multiple of two positive integers and . The system of congruenceshas solutions if and only ifwhere means that a divides b. When condition (22) holds, the system of the congruences of (21) has only one solution modulo m.

Lemma 4. For , then .

Proof. To prove this lemma, we need to prove for the integer x defined by (1).
By the generalized Chinese reminder theorem, there exists an integer s with such thatThat is, the integer s satisfiesThis s is unique. Hence, and .
Because is an n-th primitive root of unity over , we haveBy the definition of , we haveTogether with (25), we obtainDefine .

Lemma 5. Let symbols be the same as before. Then,

Proof. By Lemma 4, and if . If , then since . Hence, we obtainIf , by Lemma 4, and . By (25), we haveIf , by Lemma 4, and . By (25) and (27), we obtainIf , by Lemma 4, and . By (27), we haveIf , then . Then, by (25), we knowWhen s ranges over , it can be checked that takes on each element of exactly times. It follows from (25) thatIf , then . Then,When s ranges over , takes on each element of exactly times. It follows from (25) that

Lemma 6 (see [20], Lemma 3.3). Let notations be the same as before. Then,(1) if and only if (2) if and only if

Lemma 7 (see [21], Lemmas 2 and 4). For each ,

Define

Theorem 1. Let . Then, the linear complexity of the sequence defined by (15) is given as follows:(1)When and , we have(2)When and , we have(3)When and or and , we havewhere a and b are the integers defined in (14) and δ is defined in (18).

Proof. By definition, we haveWe first prove the conclusions for the case that . In this case, by Lemma 6, and b must be even. By Table 2, in Lemma 1 and Lemma 7, we haveFrom , we know . Hence,Whence,Note that . By Lemma 2, we have . It follows thatSimilarly, we haveBy Lemma 5, (45), and (47), we know when , there are exactly half of with such that . When , for all . Then, the desirable results on the linear complexity of the sequence s follow from (11), (17), and Lemma 5.
Now, we prove the conclusions for the case that . In this case, by Lemma 6, . It follows from Table 1 in Lemmas 1 and 7 thatFrom , we have . Hence,Whence,Similarly, we getSince and , we get or . Hence, . Together with , we obtain l which is an odd prime. By the representation with , we haveTherefore, if and only if . Since l is odd, we obtain if and only if l divides one of and . By Lemma 5, if , there are exactly a’s such that for . If , for all . Then, the desirable conclusions on the linear complexity of the sequence follow from (11), (17), (46), and Lemma 5.
For the case , the linear complexity of the sequence defined by (15) can also be determined in the following theorem.

Theorem 2. If , then for the sequence defined in (15), we have

Proof. If , then there exists such that . No matter what i is, there exists satisfying the congruence equation . Then, andThis implies .
Similarly, and . Hence, for all . Combining (11) and Lemma 5, we have the desirable results.
For , defineIf , it can be easily proved that for all j.
Let , then . We have

Lemma 8 (see [20], Lemma 3.3). Let notations be the same as before. Then,(1) if and only if (2) if and only if

Lemma 9 (see [20], Lemma 3.5). Let . Then, there are exactly two representations over the integer ring :where one of b and is divided by 4 and another is exactly divided by 2.

Lemma 10 (see [20], Corollary 3.9). Let . Fix a common primitive root of p and q. Then, if and only if the generalized cyclotomic numbers in Lemma 1 depend on the decomposition with ; if and only if the generalized cyclotomic numbers depend on the decomposition with , where means that and .
After the preparations above, we are ready to compute the linear complexity and minimal polynomials of the sequence defined in (15) over .

Corollary 1. Let and . Then, and the linear complexity and minimal polynomials of the sequence defined by (15) are given as follows:(1)When , then(2)When , then

Proof. If , then and . This contradicts with . Hence, . Also, in this case, by Lemma 8, .
By (45), (47), and Lemma 10, if , i.e., , we haveand if , i.e., , we obtainThen, the desired conclusions on the linear complexity and the minimal polynomial of the sequence follow from (11) and Lemma 5.
If , then , and by Lemma 8, . Hence, by Theorem 2, we have the following corollary.

Corollary 2. Let and . Then, for the sequence defined by (15), we have

Example 1. Let and . Then, and . If , Magma program shows that and . If , then and .

Example 2. Let and . Then, and . If , it can be easily checked that and . If , then and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Q. Wang was supported by the National Science Foundation of China under grant no. 61602342, Natural Science Foundation of Tianjin under grant no. 18JCQNJC70300, the Science & Technology Development Fund of Tianjin Education Commission for Higher Education under grant no. 2018KJ215, the Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) under grant nos. SX201804 and SX201904, the China Scholarship Council (no. 201809345010), and NSFC (nos. 61972456, 61802281, and 2017KJ237). Y. Sun was supported by the National Natural Science Foundation of China (nos.61902429 and 11775306), Shandong Provincial Natural Science Foundation of China (nos. ZR2017MA001 and ZR2019MF070), Fundamental Research Funds for the Central Universities (nos. 19CX02058A and.17CX02030A), the Open Research Fund from Shandong Provincial Key Laboratory of Computer Networks under grant no. SDKLCN-2018-02, Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (nos. SX201702 and. SX201806), the Projects of International Cooperation and Exchanges NSFC-RFBR (no. 61911530130), and International Cooperation Exchange Fund of China University of Petroleum (no. UPCIEF2019020).

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Copyright © 2020 Yuhua Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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