A Kind of Quaternary Sequences of Period and Their Linear Complexity

Abstract

Sequences with high linear complexity have wide applications in cryptography. In this paper, a new class of quaternary sequences over with period is constructed using generalized cyclotomic classes. Results show that the linear complexity of these sequences attains the maximum.

1. 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. The 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 denote a finite field with l elements, where l is a prime power. A sequence is periodic if there exists a positive integer T such that for all . Let be a periodic sequence over . The linear complexity of S, denoted by , is the least integer L of a linear recurrence relation over satisfied by S:where and . By B-M algorithm [2], if (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 (code-division 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 [3–5]. 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 [10–15].

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 constructed by the inverse Gray mapping, was studied in [18]. A family of quaternary sequences of period over 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 with period 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]. The linear complexity of these sequences is computed, and the results show that the proposed sequences have high linear complexity.

This paper is organized as follows. In Section 2, the periodic sequence S with period is given. Section 3 determines the linear complexity of the constructed sequence. Finally, we give some remarks on this paper.

2. Preliminaries

For a positive integer , use to denote the ring with integer addition modulo a and integer multiplication modulo a. Usually, we use to denote all invertible elements of , i.e., all elements b in satisfying . Obviously, the group has cardinality , where denotes the Euler function.

For a subset and an element , definewhere addition and multiplication refer to those in .

Let p and q be two distinct odd primes. Let m and n denote two positive integers. Suppose that is a primitive element of . Then, is a primitive root of for [22]. Without loss of generality, assume is an odd integer. It is known that is also a primitive root of [22]. Obviously, is a common primitive root of and for all . By the same argument, there exists an integer such that is a common primitive root of and for any .

Lemma 1 (see [23]). Let be positive integers. For a set of integers , the system of congruenceshas solutions if and only ifIf (4) is satisfied, the solution is unique modulo .

Let be the unique solution of the following congruence equations:

Lemma 1 guaranteed the existence and uniqueness of the common primitive root of , , , and . Similarly, there exists a unique integer y satisfying the following system of congruences:

Assume that and . Then, is the least positive integer that satisfies ([9], Lemma 2), i.e., . In the sequel, let i and j be two integers with and . The generalized cyclotomic classes with respect to , similar to Ding–Helleseth’s generalized cyclotomic classes ([9]), are defined as follows:

By Lemma 7 in [8], we get . Let

Similarly, we have . For abbreviation, denote and for . With the above preparations, we get a partition of as follows:

Let be the finite field with 4 elements, where α satisfies . A class of quaternary sequence can be given by allocating each elements of to each generalized cyclotomic class with respect to . To ensure the constructed sequence has high linear complexity, we should technologically do with it.

Let be a set of four tuples over , and the elements in these tuples are pairwise distinct. A quaternary generalized cyclotomic sequence of period is defined aswhere and if and and if . It is easily seen that the sequence is balanced.

3. 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 be a periodic sequence over the finite field of period N. We first recall the definition of linear complexity of periodic sequences that is given in Section 1. The linear complexity of S over , denoted by , is the smallest positive integer L satisfying the following linear recurrence relation:where and . The polynomialassociated 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. The following lemma gives a method to compute the linear complexity of S by using the generating polynomial .

Lemma 2 (see [24]). Let S be a sequence over of period N. Then, the minimal polynomial of S isand the linear complexity of S is given bywhere is the generating polynomial of S.

Lemma 3 (Lemma 2, [10] and Lemma 1, [20]). Let notations be defined as above. Then, for and , we havewhere

For the generalized cyclotomic classes and corresponding to and , we have the following lemma.

Lemma 4 (Lemma 1, [25]). For and , we havewhere

Lemma 5 (see [9]). if and only if for and .

Lemma 6 (see [22]). Let symbols be the same as before. Then, we have

Lemma 7. Let symbols be the same as before. Then, we have

Proof. We only prove the first part of this lemma.

Sufficiency. Since , then . If , by Lemma 6, . Since , we get .

Necessity. If , by the definitions of y in (6) and in (7), we know . It follows from Lemma 6 that .
By the method analogous to that used above, we can get the second conclusion of this lemma.
Let (8). Assume that β is a primitive root of unity in . It can be easily checked thatBy Lemma 2, in order to determine the linear complexity of S, we need to determine over . By (22), we should check whether , , is a root of . If it is a root of , we need to verify whether it is a multiple root of .
Recall that and for . Definefor , , and . Let a and b be two integers with and . For , suppose with It follows from Lemma 4 thatSimilarly, we haveCombining (24)–(26), we haveLetThen,We first computewhere and . The computation is divided into the following cases. Case 1: and . With simple derivation, we have Case 2: . Then, where is a primitive root of unity. Case 3: or . Let , then . It follows from Lemma 4 that Case 4: and . Let , then is a qth primitive root of unity. Hence, we obtain Case 5: . By Lemma 4, we get where .From the above discussions, we have proved the first part of the following lemma.

Lemma 8. For with , we havewhere is a primitive root of unity and β is a primitive root of unity.

Proof. The proof of the second conclusion of this lemma is similar to the first part and we omit it.

Lemma 9. For with , we obtainwhere and .

Proof. Because (38)–(41) can be proved in a similar way, here we only prove (38). By notations and (25), we getwhere with .
If , for each , it can be easily seen that . Thus,If , then is a pth primitive root of unity andwhere .
If , we have and . By Lemma 3, we knowTherefore,where and
In the following, we will determine the terms with as coefficients in (27), respectively.
First, we compute the terms with a as coefficient.
It follows from Lemmas 8 and 9 thatSimilarly, we compute the terms with b as coefficients:The terms with c as coefficient areThe terms with d as coefficient areIt can be easily checked thatNext, we determine according to the values of p and q, where with .(1)If and , by Lemmas 6 and 7 we know , , and . Hence,(2)If and , then , , and . Hence,(3)If and , then , , and . Hence,(4)If and , then , , and . Hence,From the choice of e, we know if and if . By Lemma 2 and the above discussions, we obtain .

Theorem 1. Let be the quaternary sequence defined by (11). Then, the linear complexity of S is .

Example 1. Let and . Then,It can be checked by Magma that and .

Example 2. Let and . Then,It can be checked by Magma that and .

Remark 1. For , let with In the case that if and the case that if , we know for . Hence, we need to check if is a multiple root of . This means we should check if is a root of the derivation polynomial of the generating polynomial of S. By definitions, we haveandrespectively.
For the fixed a and b with and by the following equations,we know there are at least many satisfying . Hence, will have at mostroots for . By Lemma 2, we obtain

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the National Science Foundation of China (nos. 61602342 and 11701553), Natural Science Foundation of Tianjin (grant no. 18JCQNJC70300), Science and Technology Development Fund of Tianjin Education Commission for Higher Education (no. 2018KJ215, 2017KJ237), Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (nos. SX201804 and SX201904), China Scholarship Council (no. 201809345010), and Foundation of Science and Technology on Information Assurance Laboratory (no. 61421120102162112007).