Cyclotomic Sequences Research Articles

Design sequences with high linear complexity over finite fields using generalized cyclotomy

Based on the generalized cyclotomy theory, we design some classes of sequences with high linear complexity over the finite fields. First, we construct a new class of sequence from some generalized cyclotomic sequences of different orders with different prime powers period. Then we obtain the discrete Fourier transform, defining pairs and the linear complexity of the new sequences. Finally, we study the linear complexity of a special class of q−ary (q prime) sequences.

Cyclic codes from two-prime generalized cyclotomic sequences of order 6

Cyclic codes have wide applications in data storage systems and communication systems. Employing binary two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field $\mathrm{GF}(q)$ and give their generator polynomials. And we also calculate the minimum distance of some cyclic codes and give lower bounds on the minimum distance for some other cyclic codes.

Linear complexity of generalized cyclotomic binary sequences of order 2d and length 2pm

In this paper, we construct two new cyclotomic binary sequences of order [Formula: see text] and length [Formula: see text] which include the sequences constructed by Ke et al. in 2012. We also compute their linear complexity. The results show that these sequences have good linear complexity.

Cyclic codes from the second class two-prime Whiteman’s generalized cyclotomic sequence with order 6

Let n1=df+1$n_{1}=df+1$ and n2=dfź+1$n_{2}=df^{\prime }+1$ be two distinct odd primes with positive integers d,f,fź$d, f, f^{\prime }$ and gcd(f,fź)=1$\gcd (f,f^{\prime })=1$. In this paper, we compute the linear complexity and the minimal polynomial of the two-prime Whiteman's generalized cyclotomic sequence of order d=6$d=6$ over GF(q)$\text {GF}(q)$, where q=pm$q=p^{m}$ and p is an odd prime and m is an integer. We employ this sequence of order 6 to construct several classes of cyclic codes over GF(q)$\text {GF}(q)$ with length n1n2$n_{1}n_{2}$. We also obtain lower bounds on the minimum distance of these cyclic codes.

On the linear complexity of new generalized cyclotomic binary sequences of order two and period pqr

Periodic sequences over finite fields, constructed by classical cyclotomic classes and generalized cyclotomic classes, have good pseudorandom properties. The linear complexity of a period sequence plays a fundamental role in the randomness of sequences. Let p, q, and r be distinct odd primes with gcd(p−1, q−1)=gcd(p−1, r−1)=gcd(q−1, r−1)=2. In this paper, a new class of generalized cyclotomic sequence with respect to pqr over GF(2) is constructed by finding a special characteristic set. In addition, we determine its linear complexity using cyclotomic theory. Our results show that these sequences have high linear complexity, which means they can resist linear attacks.

On the linear complexity of Hall’s sextic residue sequences over $${ GF}(q)$$ G F ( q )

In this paper, we derive the linear complexity of Hall’s sextic residue sequences over the finite field of odd prime order. The order of the field is not equal to a period of the sequence. Our results show that Hall’s sextic residue sequences have high linear complexity over the finite field of odd order. Also we estimate the linear complexity of series of generalized sextic cyclotomic sequences. The linear complexity of these sequences is larger than half of the period.

Linear complexity of generalised cyclotomic quaternary sequences of length 2 p m +1 q n +1

Sequences with high linear complexity play a fundamental part in cryptography. In this study, the authors construct general forms of Whiteman's generalised cyclotomic quaternary sequences with period 2p m+1 q n+1 of order two over 𝔽4 and give the linear complexity of the proposed sequences. The conclusions reveal that such sequences have good balance property and high linear complexity.

On the k-error linear complexity of generalised cyclotomic sequences

Generalised cyclotomic binary sequences are divided into Whiteman-generalised cyclotomic and Ding-generalised cyclotomic. Firstly, two classes of error generalised cyclotomic sequences with length pq over Z

On the k-error linear complexity of generalised cyclotomic sequences

Generalised cyclotomic binary sequences are divided into Whiteman-generalised cyclotomic and Ding-generalised cyclotomic. Firstly, two classes of error generalised cyclotomic sequences with length pq over Zpq are constructed; secondly, k-error linear complexity of generalised cyclotomic sequences are considered, using the trace representation and cyclotomic theory, respectively. It is shown that k-error linear complexity of generalised cyclotomic sequences do not exceed p + q-1 which is much less than the (zero-error) linear complexity.

Linear Complexity of New Generalized Cyclotomic Sequences of Order Two with Odd Length

Linear Complexity of New Generalized Cyclotomic Sequences of Order Two with Odd Length

Linear complexity of Ding-Helleseth sequences of order 2 over GF(l)

Generalized cyclotomic sequences have good pseudo-random properties and have been widely used as keystreams in private-key cryptosystems. In this paper, the linear complexity and minimal polynomials of Ding-Helleseth sequences of order 2 have been determined over a finite field GF(l). Results show that these sequences have high linear complexity.

Linear complexity of binary generalized cyclotomic sequences over GF([formula omitted

Periodic sequences over finite fields have been used as key streams in private-key cryptosystems since the 1950s. Such periodic sequences should have a series of cryptographic properties in order to resist many attack methods. The binary generalized cyclotomic periodic sequences, constructed by the cyclotomic classes over finite fields, have good pseudo-random properties and correlation properties. In this paper, the linear complexity and minimal polynomials of some generalized cyclotomic sequences over GF(q) have been determined where q=pm and p is an odd prime. Results show that these sequences have high linear complexity over GF(q) for a large part of odd prime power q, which means they can resist the linear attack method.

Multi-Rate Representation of Generalized Cyclotomic Sequences of Any Odd Period

Multi-Rate Representation of Generalized Cyclotomic Sequences of Any Odd Period

Gauss periods and cyclic codes from cyclotomic sequences of small orders

Let p = ef + 1 be an odd prime with positive integers e and f. In this paper, we calculate the values of Gauss periods of order e = 3, 4, 6 over a finite field GF(q), where q is a prime with q≠p. As applications, several cyclotomic sequences of order e = 3, 4, 6 are employed to construct a number of classes of cyclic codes over GF(q) with prime length. Under certain conditions, the linear complexity and reciprocal minimal polynomials of cyclotomic sequences are calculated, and the lower bounds on the minimum distances of these cyclic codes are obtained.

Linear complexity of binary cyclotomic sequences of order 6

The binary cyclotomic periodic sequences, constructed by the cyclotomic cosets over finite fields, have good pseudo-random properties. In this paper, we determine the linear complexity of binary cyclotomic sequences of order 6. Our results show that such sequences have large linear complexity so that they can resist the linear attack method.

The linear complexity of generalized cyclotomic sequences with period $$2p^n$$ 2 p n

We found the linear complexity of generalized cyclotomic sequences with period $$2p^n$$ 2 p n based on quadratic, biquadratic and partially sextic residues. Our method is based on the generating the polynomial of the classical cyclotomic sequences, which allows us to obtain some well-known results and also some new results.

Linear complexity and correlation of a class of binary cyclotomic sequences

Let $$p_1,p_2,\ldots ,p_n$$ be distinct odd primes and let $$e_1,e_2,\ldots ,e_n$$ be positive integers. Based on cyclotomic classes proposed by Ding and Helleseth (Finite Fields Appl 4:140–166, 1998), a binary cyclotomic sequence of period $$p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}$$ is defined and denoted by $$\underline{s_\Upsilon }$$ . The linear complexity of $$\underline{s_\Upsilon }$$ is determined and is proved to be greater than or equal to $$(p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}-1)/2$$ . The autocorrelation function of $$\underline{s_\Upsilon }$$ is explicitly computed. Let $$\ell \in \{1,2,\ldots ,n\}$$ . We also explicitly compute the crosscorrelation function of $$\underline{s_\Upsilon }$$ and the Legendre sequence $$\underline{L_{p_\ell }}$$ with respect to $$p_\ell $$ . It is shown that $$\underline{s_\Upsilon }$$ and $$\underline{L_{p_\ell }}$$ have two-level or three-level crosscorrelation, and all their two-level crosscorrelation functions are determined.

On the -error Linear Complexity of Generalized Cyclotomic Sequence with Length pm

摘要: 周期为pm的广义割圆序列具有很高的线性复杂度。该文通过改变序列的特征集，构造了一类周期相同的错误序列，确定了序列的k-错线性复杂度。结果表明，该类序列的(p1)/2-错线性复杂度不超过pm1，这比该序列的线性复杂度低得多。因此，该类序列没有达到足够的安全作为密钥流生成器。 关键词: 密码学 / 流密码 / 伪随机序列 / 广义割圆类 / k-错线性复杂度

Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3)

In this paper, we always assume that $p=6f+1$ is a prime. First, we calculate the values of exponential sums of cyclotomic classes of orders 3 and 6 over an extension field of GF(3). Then, we give a formula to compute the linear complexity of all $p^{n+1}$-periodic generalized cyclotomic sequences of order 6 over GF(3). After that, we compute the linear complexity and the minimal polynomial of a $p^{n+1}$-periodic, balanced and generalized cyclotomic sequence of order 6 over GF(3), which is analogous to a generalized Sidelnikov's sequence. At last, we give some BCH codes with prime length $p$ from cyclotomic sequences of orders three and six.

On the Linear Complexity of Quaternary Cyclotomic Sequences with the Period 2<i>pq</i>

On the Linear Complexity of Quaternary Cyclotomic Sequences with the Period 2<i>pq</i>