Linear complexity of generalized cyclotomic sequences of order 4 over F l

Generalized cyclotomic sequences of period pq 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 of order 4 is developed over a ﬁnite ﬁeld F l . Results show that in many cases they have high linear complexity.


Introduction
Let l be a prime number and F l denote a finite field with l elements. A sequence λ ∞ � λ 0 λ 1 · · · λ n− 1 · · · is called to be nperiodic if λ i � λ i+n for all i ≥ 0. 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][2][3][4][5][6][7][8].
Let p and q be two distinct odd primes with gcd(p − 1, q − 1) � d. Define n � pq and e � (p − 1) (q − 1)/d. e Chinese reminder theorem guarantees that there exists a common primitive root g of both p and q. Let x be an integer satisfying x ≡ g(mod p), x ≡ 1(mod q). (1) Whiteman proved that [21] Z * n � g i x j : i � 0, 1, . . . , e − 1; j � 0, 1, . . . , d − 1 , (2) where Z * n denotes the set of all invertible elements of the residue class ring Z n . e generalized cyclotomic classes D j (0 ≤ j ≤ d − 1) of order d with respect to n are defined by [21] D j � g i x j : i � 0, 1, . . . , e − 1 , where the multiplication is that of Z n . Clearly, the cosets D j depend on the choice of the common primitive root g if d ≥ 3. It is not hard to prove that [21] Z * en, Let S be a nonempty subset of 0, 1, We define the binary sequence λ ∞ � λ 0 λ 1 · · · λ n− 1 · · · of period n as follows: where i ≥ 0 and ρ ∈ 0, 1 { }. For ρ � 0 and S � 0 { }, the linear complexity of these sequences over F 2 has been calculated by Ding [16] with d � 2 and Hu et al. [20] with d � 4. Furthermore, for S � 0, 2, . . . , d − 2 { } and ρ � 0, Ding [9] determined the linear complexity of the two-prime sequences over a finite field F l and used these sequences to construct several classes of cyclic codes over F l with optimal or almost optimal property, where gcd(l, n) � 1. In this paper, we only consider the case ρ � 1, d � 4, and S � 0, 1 { }. Under the assumption that (n − 1)/4 ≡ 0(mod l) or l ∉ D 0 , we calculate the linear complexity of these sequences over the finite field F l . e results show that, in many cases, these sequences have high linear complexity. is 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 F l . In the last section, we present concluding remarks of this paper.

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 λ ∞ � λ 0 λ 1 · · · λ n− 1 · · · be a periodic sequence over F l with period n, where λ i ∈ F l for i ≥ 0. e sequence λ ∞ can be viewed as a power series in the power series ring where is called the linear complexity of the sequence λ ∞ over F l , which is denoted by L l (λ).
Indeed, L l (λ) is the length of the shortest linear feedback shift register which generates the sequence λ ∞ .
If gcd(n, l) � 1, then 1 − x n has n distinct roots ζ i n (0 ≤ i ≤ n − 1) in the algebraic closure Ω l of F l , where ζ n denotes the n-th primitive root of unity. It is easy to see that In order to determine the linear complexity of generalized cyclotomic sequences, we introduce generalized cyclotomy.
Let symbols be the same as in the introduction and e generalized cyclotomic numbers of order 4 with respect to n is defined by By the well-known theorem ( [24], P. 128), there are exactly two representations of n in the form n � a 2 + 4b 2 with a ≡ 1(mod 4) and the sign of b indeterminate.
Let g 1 and g 2 be a fixed primitive root of p and q, respectively. For i � 1, 2, let x i and y i be the integers given uniquely by Define a and b to be integers satisfying where (÷) denotes the Legendre symbol. It is clear that a ≡ 1(mod 4) and n � a 2 + 4b 2 is one of the two representations of n. e following lemma shows that the generalized cyclotomic numbers of order 4 with respect to n depend uniquely on this representation.
e generalized cyclotomic sequence λ ∞ of order 4 of period n is defined by where D 0 and D 1 are defined by (3) and P � pZ * q . Here, in this paper, we treat it as a sequence over a finite field F l , where gcd(l, n) � 1.
Denote ord n (l) the multiplicative order of l modulo n. Let ζ n be an n-th primitive root of unity over F l ord n (l) . For the sequence λ ∞ defined by (15), we know Define δ as follows: Note that the generalized cyclotomic classes of order 2 are given by Define η 0 � i∈C 0 ζ i n . e following lemma has been proven in [9].
For the case l ∉ D 0 , the linear complexity of the sequence λ ∞ defined by (15) can also be determined in the following theorem. □ Theorem 2. If l ∉ D 0 , then for the sequence λ ∞ defined in (15), we have Proof. If l ∉ D 0 , then there exists i ∈ 1, 2, 3 { } such that i ∈ D i . No matter what i is, there exists j ∈ 1, 2, 3 { } satisfying the congruence equation ij ≡ 2(mod 4). en, l j ∈ D 2 and Λ l j ζ n � Λ ζ l j n � 1 − Λ ζ n .
For j ∈ 0, 1, 2, 3 { }, define If l ∈ D 0 , it can be easily proved that d j (x) ∈ F l [x] for all j.
We have Lemma 8 (see [20], Lemma 3.3). Let notations be the same as before. en,