Abstract
Ding constructed a new cyclotomic class . Based on it, a construction of generalized cyclotomic binary sequences with period is described, and their autocorrelation value, linear complexity, and minimal polynomial are confirmed. The autocorrelation function is 3-level if , and is 5-level if . The linear complexity if , , or or . The results show that these sequences have quite good cryptographic properties in the aspect of autocorrelation and linear complexity.
1. Introduction
Pseudorandom sequences with good cryptography properties have wide applications in CDMA, global positing systems, and stream ciphers. The security of stream ciphers depends on the randomness of the key stream, which makes the construction of pseudorandom sequences to be an important research direction. Many researchers focused on cyclotomic sequences, which have good balance property. Linear complexity and autocorrelation are important criteria for measuring unpredictability of cyclotomic sequences.
Let denote a finite field with elements, where is a prime power. A sequence is periodic if there exists a positive integer such that for all .
Let be a periodic sequence over with period . The periodic autocorrelation function of binary sequence is defined bywhere .
Autocorrelation function measures the amount of similarity between sequence and a shift of by shifts. Only when the values of distribute flat and low, sequence is easy to distinguish from each time shifted version of itself. The autocorrelation function with the ideal distribution of values is two-valued, which is given as
Sequences with ideal autocorrelation functions have many applications in cryptography, coding, and other communication engineering.
Linear complexity of , denoted by , is the least integer of a linear recurrence relation over satisfied by :where . The linear complexity of a sequence is also defined to be the length of the shortest linear feedback shift register which can generate the sequence. It is an important criterion of randomness of sequences in stream ciphers. To resist the attack from Berlekamp–Massey algorithm, the sequences used in cipher systems should have large linear complexity. If , where is the least period of , then is considered to be good from the viewpoint of linear complexity.
The minimal polynomial of isand the linear complexity of is given by , where is the generating polynomial of , that is,
Sequences from cyclotomic and generalized cyclotomic are important families of pseudorandom sequences.
For an integer , let denote the residue class ring of integers modulo and be the multiplicative group consisting of all invertible elements in . A partition of is a family of sets satisfying for all and . Suppose is a multiplicative subgroup of and there exist elements such that for all ; then, are called classical cyclotomic classes of order with respect to when is prime and generalized cyclotomic classes of order with respect to when is composite. The sequences constructed by them are called classical cyclotomic sequences and generalized cyclotomic sequences, respectively. Gauss [1] first proposed the concept of cyclotomic, divided the multiplicative group , and then divided the residual class ring to construct Gauss classical cyclotomic. Whiteman [2] divided the multiplicative group and then divided the residual class ring to construct Whiteman generalized cyclotomic. Ding and Helleseth [3] divided the multiplicative group and then divided the residual class ring to construct the Ding generalized cyclotomic. The above three kinds of cyclotomic theories are the most representative and the most widely used cyclotomic theories.
Classical cyclotomic sequences include Legendre sequences, -degree residual sequences, and Hall sextic residue sequences. Damgaard [4] determined the autocorrelation value of Legendre sequences, and then Ding et al. [5] determined their linear complexity. Kim and Song [6] determined the linear complexity of Hall sextic residue sequences.
Ding [7] constructed Whiteman generalized cyclotomic sequences of order 2 and confirmed their linear complexity. And Ding [8] determined the autocorrelation value of Whiteman generalized cyclotomic sequences of order 2. Bai [9] constructed Whiteman generalized cyclotomic sequences of order 4 and determined their linear complexity. Yan et al. [10] extended Whiteman generalized cyclotomic sequences to the case of order .
Bai [9] determined the autocorrelation value of Ding generalized cyclotomic sequences with period of order 2. And Bai et al. [11] confirmed they had high linear complexity. Yan et al. [12] constructed Ding generalized cyclotomic sequences with period and confirmed they had high linear complexity. Edemskiy [13] constructed a kind of balanced binary generalized cyclotomic sequences with period . Zhang et al. [14] determined the linear complexity of generalized cyclotomic sequences with period . Hu et al. [15] constructed generalized cyclotomic sequences with period and determined their linear complexity. Ke et al. [16] determined the linear complexity and the autocorrelation value of Ding generalized cyclotomic sequences with period . Chang et al. [17] constructed binary generalized cyclotomic sequences with period and determined their linear complexity and minimal polynomial.
Ding [18] constructed a new cyclotomic class and obtained a kind of cyclic code from it. Liu and Chen [19] determined binary generalized cyclotomic sequences with period based on the new cyclotomic class and determined their autocorrelation value, linear complexity, and minimal polynomial.
In this paper, based on Ding’s new cyclotomic class , a simple construction of binary generalized cyclotomic sequences with period is constructed, and their autocorrelation value, linear complexity, and minimal polynomial are confirmed. The remainder of this paper is organized as follows. Section 2 proposes a construction of generalized cyclotomic binary sequences. Section 3 calculates the autocorrelation value, linear complexity, and minimal polynomial of the new sequences and compares our results with [19]. Section 4 concludes this paper.
2. Preliminaries
Lemma 1 (see [20]). Let be integers. The system of congruences,has solutions if and only if .
If the above condition is satisfied, the solution is unique modulo .
Let , where and are two distinct odd primes. Let be the unique common primitive root of and . The existence and uniqueness of are guaranteed by Lemma 1. Similarly, there exists a unique integer which satisfies the following system of congruences:
Let and . According to Whiteman [2], Whiteman generalized cyclotomic class of order is
It can be easily seen that for all and .
Define two sets
Then,
Lemma 2 (see [7]). LetAnd a new binary generalized cyclotomic sequence of order 2 isLet and be a primitive th root of unity in finite field , . Then,(1)When , or , ,(2)When , or , ,(3)When , or , ,(4)When , or , ,(5)When , or , ,(6)When , or , ,
Lemma 3 (see [8]). LetAnd a binary generalized cyclotomic sequence of order 2 is defined byThen,(1)When is even,(2)When is odd,
Lemma 4 (see [9]). LetAnd a binary generalized cyclotomic sequence of order 4 is defined by(1)If or , then If or , then(2)When , or , ,
Lemma 5 (see [10]). LetAnd a binary generalized cyclotomic sequence of order is defined byLet and be a th primitive root of unity in finite field ,Then,(1)If for all , is true,(2)When there exists such that , where .The following are Ding’s new cyclotomic class .
Assume ; letWith the above preparations, a partition of isThen,LetAnd a binary generalized cyclotomic sequence with period constructed in [19] is
Lemma 6. (see [19]). Let be the binary sequences defined. Then, the autocorrelation of is
Lemma 7. (see [19]). Let . Then,(1)When ,(2)When ,(3)When ,(4)When ,Now, letA new binary cyclotomic sequence with period is defined by
3. Main Results
3.1. Autocorrelation of Our New Sequences
Let
Lemma 8 (see [19]). Let and be the sets defined above; then,(1) if and only if (2) if and only if
Lemma 9 (see [19]). Let be the sets defined above; then, if and only if .
Lemma 10. Let ; then,(1)(2)(3)(4)(5)(6)(7)(8)(9)
Proof. For the proof of (1) and (6)‒(9), one can refer to in [19]; we just prove (2)‒(5).
Theorem 1. Let be the new binary sequences defined in (44); then, the autocorrelation of is
Proof. By the definition of ,Let ; then,The autocorrelation function of the new sequences is 5-level if . is 3-level.
3.2. Linear Complexity and Minimal Polynomial of Our New Sequences
Lemma 11. (see [19]). Let and be the sets defined above; then,Denote
Assume and is a th primitive root of unity in finite field . According to the Blahut theorem, the linear complexity of sequence is
Lemma 12. Let ; then,
Proof. Let ; then, by Lemma 11,Let ,Then, by Lemma 11,Let ; then,When runs through runs through , and , the set takes on each element in exactly times. Therefore,Let ; then,When runs through runs through , and , the set takes on each element in exactly times. Therefore,Let ; then,
Lemma 13. Let and be the sets defined above; then, if and only if .
Proof. Since the characteristic of finite field is 2, .
Let ; then, .Therefore, .
Let ; then, .Therefore, .
Since , if and only if .
Theorem 2. Let . Then,(1)When ,(2)When ,(3)When ,(4)When ,
Proof. Let be a th primitive root of unity in finite field Then,Define . It can be easily seen that
Case 1. : choose such that . Then,
Case 2. : choose such that . Then,
Case 3. . Then,
Case 4. . Then,The linear complexity of the new sequences is if , , or ; or if , which is very close to period .
The following are some examples.
Example 1. Let and . Then,Our corresponding new binary sequence of period 21 is as follows: 000101100110110101111.
By using Magma, the autocorrelation value of the above sequence is 3-level, which is consistent with the case in Theorem 1. And the linear complexity of the above sequence is equal to 14, which is consistent with the case in Theorem 2.
Example 2. Let and . Then,Our corresponding new binary sequence of period 33 is as follows: 001100111110110101110100100111101.
By using Magma, the autocorrelation value of the above sequence is 3-level, which is consistent with the case in Theorem 1. And the linear complexity of the above sequence is equal to 32, which is consistent with the case in Theorem 2.
Example 3. Let and . Then,Our corresponding new binary sequence of period 39 is as follows: 001101111101100100111100100110111110110.
By using Magma, the autocorrelation value of the above sequence is 5-level, which is consistent with the case in Theorem 1. And the linear complexity of the above sequence is equal to 36, which is consistent with the case in Theorem 2.
Example 4. Let and . Then,Our corresponding new binary sequence of period 51 is as follows: 000101110111101100101111100111110100110111101110100.
By using Magma, the linear complexity of the above sequence is equal to 32, which is consistent with the case in Theorem 1, but the autocorrelation value of the above sequence is 4-level, which is consistent with the case in Theorem 2.
3.3. Comparisons of Results
The comparisons of our results with [19] are listed in Tables 1 and 2.
The comparisons show the following:(i)When , the autocorrelation of the two sequences is unequal, but they are 3-level. When , the autocorrelation of the two sequences is unequal, but both of them are 5-level.(ii)When , , or , the linear complexity of our new sequences is larger.
4. Conclusion
This paper presents a construction of generalized cyclotomic binary sequences with period based on Ding’s new cyclotomic class . And the autocorrelation value, linear complexity, and minimal polynomial of our new sequences are determined. The autocorrelation function is 5-level if . is 3-level, and has almost optimal autocorrelation if . The linear complexity if , , or ; or if , which is very close to the period. The results show that our new sequences have quite good cryptographic properties in the aspect of autocorrelation and linear complexity.
Data Availability
All the data used to support the findings of this study are included in Section 3.2 of this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 61902304) and the National Statistical Science Project (2019LY36).