Autocorrelation and Linear Complexity of Binary Generalized Cyclotomic Sequences with Period pq

Ding constructed a new cyclotomic class (V0 , V1). Based on it, a construction of generalized cyclotomic binary sequences with period pq is described, and their autocorrelation value, linear complexity, and minimal polynomial are confirmed. +e autocorrelation function CS(w) is 3-level if p ≡ 3mod4, and CS(w) is 5-level if p ≡ 1mod4. +e linear complexity LC(S)> (pq/2) if p ≡ 1mod 8, p> q + 1, or p ≡ 3mod4 or p ≡ − 3mod8. +e results show that these sequences have quite good cryptographic properties in the aspect of autocorrelation and linear complexity.


Introduction
Pseudorandom sequences with good cryptography properties have wide applications in CDMA, global positing systems, and stream ciphers. e 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 F l denote a finite field with l elements, where l is a prime power. A sequence S � s i is periodic if there exists a positive integer N such that s j+N � s j for all j ≥ 0.
Let S � s i be a periodic sequence over F l with period N. e periodic autocorrelation function of binary sequence S is defined by where 0 ≤ w ≤ N − 1. Autocorrelation function measures the amount of similarity between sequence S and a shift of S by w shifts. Only when the values of C S (w) distribute flat and low, sequence S is easy to distinguish from each time shifted version of itself. e 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 S, denoted by LC(S), is the least integer L of a linear recurrence relation over F l satisfied by S: where c 0 , c 1 , . . . , c L ∈ F l . e 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 LC(S) ≥ (N/2), where N is the least period of S, then S is considered to be good from the viewpoint of linear complexity. e minimal polynomial m(x) of S is and the linear complexity LC(S) of S is given by Sequences from cyclotomic and generalized cyclotomic are important families of pseudorandom sequences.
For an integer N ≥ 2, let Z N � 0, 1, . . . , N − 1 { } denote the residue class ring of integers modulo N and Z * N be the multiplicative group consisting of all invertible elements in i are called classical cyclotomic classes of order d with respect to N when N is prime and generalized cyclotomic classes of order d with respect to N when N is composite. e 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 Z * p , and then divided the residual class ring Z p to construct Gauss classical cyclotomic. Whiteman [2] divided the multiplicative group Z * pq and then divided the residual class ring Z pq to construct Whiteman generalized cyclotomic. Ding and Helleseth [3] divided the multiplicative group Z * Classical cyclotomic sequences include Legendre sequences, d-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 2 k .
Bai [9] determined the autocorrelation value of Ding generalized cyclotomic sequences with period pq 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 p m and confirmed they had high linear complexity. Edemskiy [13] constructed a kind of balanced binary generalized cyclotomic sequences with period p n+1 . Zhang et al. [14] determined the linear complexity of generalized cyclotomic sequences with period 2p m . Hu et al. [15] constructed generalized cyclotomic sequences with period p m+1 q n+1 and determined their linear complexity. Ke et al. [16] determined the linear complexity and the autocorrelation value of Ding generalized cyclotomic sequences with period 2p m . Chang et al. [17] constructed binary generalized cyclotomic sequences with period pqr and determined their linear complexity and minimal polynomial.
Ding [18] constructed a new cyclotomic class (V 0 , V 1 ) and obtained a kind of cyclic code from it. Liu and Chen [19] determined binary generalized cyclotomic sequences with period pq based on the new cyclotomic class (V 0 , V 1 ) and determined their autocorrelation value, linear complexity, and minimal polynomial.
In this paper, based on Ding's new cyclotomic class (V 0 , V 1 ), a simple construction of binary generalized cyclotomic sequences with period pq is constructed, and their autocorrelation value, linear complexity, and minimal polynomial are confirmed. e 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.
Let N � pq, where p and q are two distinct odd primes. Let g be the unique common primitive root of p and q. e existence and uniqueness of g are guaranteed by Lemma 1. Similarly, there exists a unique integer x which satisfies the following system of congruences: Let d � gcd(p − 1, q − 1) and e � (gcd(p − 1)(q − 1)/d). According to Whiteman [2], Whiteman generalized cyclotomic class of order d is It can be easily seen that en, Lemma 2 (see [7]). Let And a new binary generalized cyclotomic sequence S of order 2 is Let m � ord pq (2) and α be a primitive pqth root of unity (2) When p ≡ − 1 mod 8, q ≡ 3 mod 8 or p ≡ 3 mod 8, q ≡ − 1 mod 8, Lemma 3 (see [8]). Let And a binary generalized cyclotomic sequence S of order 2 is defined by en, Lemma 4 (see [9]). Let And a binary generalized cyclotomic sequence S of order 4 is defined by Lemma 5 (see [10]). Let And a binary generalized cyclotomic sequence S of order 2 k is defined by Let m � ord pq (2) and α be a pqth primitive root of unity in finite field F 2 m , If for all s, g s ≡ 2 mod pq is true, (2) When there exists s such that g s ≡ 2modpq, e following are Ding's new cyclotomic class (V 0 , V 1 ).
With the above preparations, a partition of Z * N is en, Let And a binary generalized cyclotomic sequence S with period pq constructed in [19] is Lemma 6 (see [19]). Let S � s i be the binary sequences defined. en, the autocorrelation of S is Lemma 7 (see [19]).

Theorem 1. Let S � s i be the new binary sequences defined in (44); then, the autocorrelation of S is
Proof. By the definition of S, e autocorrelation function of the new sequences
(2) When p ≡ − 1mod8, (3) When p ≡ 3mod8, (66) Proof. Let α be a pqth primitive root of unity in finite field F 2 m . en, It can be easily seen that Case 2. p ≡ − 1mod8: choose α such that S(α) � 0. en, Case 4. p ≡ − 3mod8. en, e linear complexity of the new sequences is 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 p ≡ 3 mod 4 in eorem 1. And the linear complexity of the (75) 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 p ≡ 3mod4 in eorem 1. And the linear complexity of the above sequence is equal to 32, which is consistent with the case p ≡ 3mod8 in eorem 2.
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 p ≡ 1mod4 in eorem 1. And the linear complexity of the above sequence is equal to 36, which is consistent with the case p ≡ − 3mod8 in eorem 2.  (77) Our corresponding new binary sequence of period 51 is as follows: 00010111011110110010111110011111010011011-1101110100.
By using Magma, the linear complexity of the above sequence is equal to 32, which is consistent with the case p ≡ 1mod8 in eorem 1, but the autocorrelation value of the above sequence is 4-level, which is consistent with the case p ≡ 1mod4 in eorem 2.

Comparisons of Results.
e comparisons of our results with [19] are listed in Tables 1 and 2. e comparisons show the following: (i) When p ≡ 3mod4, the autocorrelation C S (w) of the two sequences is unequal, but they are 3-level. When p ≡ 1mod4, the autocorrelation of the two sequences is unequal, but both of them are 5-level. (ii) When p ≡ 1mod4, p > q, or p ≡ 3mod4, the linear complexity of our new sequences is larger.

Conclusion
is paper presents a construction of generalized cyclotomic binary sequences with period pq based on Ding's new cyclotomic class (V 0 , V 1 ). And the autocorrelation value, linear complexity, and minimal polynomial of our new sequences are determined.
e autocorrelation function C S (w) is 5-level if p ≡ 1mod4. C S (w) is 3-level, and S has almost optimal autocorrelation if p ≡ 3mod4. e linear complexity LC(S) > (pq/2) if p ≡ 1mod8, p > q + 1, or p ≡ − 1 mod 8; LC(S) � pq − q or pq − 1 if p ≡ ± 3mod8, which is very close to the period. e results show that our Table 1: Comparison of autocorrelation C S (w).

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
e authors declare that they have no conflicts of interest.