Cyclotomic Classes Research Articles

Constructions of Strongly Regular Cayley Graphs Using Even Index Gauss Sums

AbstractIn this paper, generalizing the result in [9], I construct strongly regular Cayley graphs by using union of cyclotomic classes of and Gauss sums of index w, where is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.

Strongly regular Cayley graphs, skew Hadamard difference sets, and rationality of relative Gauss sums

In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields. Our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White (2002) [23] and several subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.

Quaternary cryptographic bent functions and their binary projection

The nonlinearity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, bent functions have been generalized to the alphabet ℤq = ℤ/qℤ and also studied by several authors. This generalization maps the classical definition of binary bent functions to generalized bent functions. This paper propose a new approach for generalized bent functions over ℤ4 and their associated derived boolean functions. We present in this paper a new construction of a family of m-variable quaternary (ℤ4 = ℤ/4ℤ = {0,1,2,3}-valued) bent functions over Galois ring and cyclotomic classes using an intern function defined on a particular partition of the Teichmuller set. Using a particular binary projection map we obtain a family of 2m-variable boolean bent functions and a family of 2m + 1-variable boolean plateaued functions of amplitude 2m + 1 with nonlinearity equal to 4m − 2m.

Some results on strongly regular graphs from unions of cyclotomic classes

In this paper, we construct some families of strongly regular graphs on finite fields by using unions of cyclotomic classes and index 2 Gauss sums. New infinite families of strongly regular graphs are found.

Perfect Gaussian Integer Sequences of Odd Prime Length

A Gaussian integer is a complex number whose real and imaginary parts are both integers. A Gaussian integer sequence is called perfect (odd perfect) if the out-of-phase values of the periodic (odd periodic) autocorrelation function are equal to zero. In this letter, for any odd prime p, using the cyclotomic classes of order 2 and 4 with respect to GF(p), we propose perfect and odd perfect Gaussian integer sequences of length p. Several examples are also given.

Linear complexity of binary sequences derived from Euler quotients with prime-power modulus

We extend the definition of binary threshold sequences from Fermat quotients to Euler quotients modulo pr with odd prime p and r⩾1. Under the condition of 2p−1≢1(modp2), we present exact values of the linear complexity by defining cyclotomic classes modulo pn for all 1⩽n⩽r. The linear complexity is very close to the period and is of desired value for cryptographic purpose. We also present a lower bound on the linear complexity for the case of 2p−1≡1(modp2).

Gauss periods and codebooks from generalized cyclotomic sets of order four

Let p, q be distinct primes with gcd(p ? 1, q ? 1) = 4. Let D 0, D 1, D 2, D 3 be Whiteman's generalized cyclotomic classes, satisfying the multiplicative group $${{\mathbb Z}^*_{pq}=D_0\cup D_1\cup D_2\cup D_3}$$ . In this paper, we give formulas of Gauss periods: $${\sum_{i\in D_0\cup D_2}\zeta^i}$$ and $${\sum_{i\in D_0}\zeta^i}$$ , where $${\zeta}$$ is a pqth primitive root of unity. As an application, we get the maximum cross-correlation amplitudes of three codebooks from generalized cyclotomic sets of order four and supply conditions on p and q such that they nearly meet the Welch bound.

Constructions of strongly regular Cayley graphs using index four Gauss sums

We give a construction of strongly regular Cayley graphs on finite fields $\mathbb{F}_{q}$ by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.

Strongly regular graphs from unions of cyclotomic classes

We give two constructions of strongly regular Cayley graphs on finite fields Fq by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.

Cyclotomic constructions of skew Hadamard difference sets

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of F q of order N = 2 p 1 m , where p 1 is a prime and m a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.

$k$-Fold Cyclotomy and Its Application to Frequency-Hopping Sequences

For an integer k ≥ 1, let , 1≤ qi ≤ k,be prime powers such that qi - M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> f + 1 for some integers M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> and f. In this paper, the k-fold cyclotomy of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">qk</sub> × ⋯ × F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">qk</sub> as a nontrivial generalization of the conventional cyclotomy (k = 1 case) and its application to frequency-hopping sequences (FHSs) are presented, where F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> is the finite field with q elements. First, the definitions of k-fold cyclotomic classes and k-fold cyclotomic numbers are given. And then, their basic properties including k-fold diagonal sums are derived. Based on them, new optimal FHS sets of length N and frequency set size M or M + 1 with respect to the Peng-Fan bound are constructed for a product N of distinct odd primes and a di visor M of N - 1. Furthermore, new optimal FHSs of length N and frequency set size M with respect to the Lempel-Greenberger bound are constructed when N has at least one prime factor which is 3 modulo 4 and (N - 1)/M is an even integer. Our constructions give several new optimal parameters not covered in the literature, which are summarized in Table I.

Linear Complexity of Quaternary Sequences Generated Using Generalized Cyclotomic Classes Modulo 2p

Let p be an odd prime number. We define a family of quaternary sequences of period 2p using generalized cyclotomic classes over the residue class ring modulo 2p. We compute exact values of the linear complexity, which are larger than half of the period. Such sequences are 'good' enough from the viewpoint of linear complexity.

A new class of bent and hyper-bent Boolean functions in polynomial forms

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ (n = 2m) having the form $${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$$ where o(s i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 n ? 1 which contains s i and whose coefficients a and b are, respectively in $${F_{2^{o(s_1)}}}$$ and $${F_{2^{o(s_2)}}}$$ . Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s 1 = 2 m ? 1 and $${s_2={\frac {2^n-1}3}}$$ , where $${a\in\mathbb{F}_{2^{n}}}$$ (a ? 0) and $${b\in\mathbb{F}_{4}}$$ provide a construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s 1 is of the form r(2 m ? 1) where r is co-prime with 2 m + 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.

Linear complexity and autocorrelation values of a polyphase generalized cyclotomic sequence of length pq

A construction of a family of generalized polyphase cyclotomic sequences of length pq is presented in terms of the generalized cyclotomic classes modulo pq. Their linear complexity and corresponding minimal polynomials are deduced. Some upper bounds on periodic and aperiodic autocorrelation values of resulting sequences are also estimated by using certain exponential sums.

Large families of pseudo-random subsets formed by generalized cyclotomic classes

Recently, Dartyge and Sarkozy defined the measures, i.e., the well- distribution measure and the correlation measure of order k, of pseudo-randomness of subsets of the set {1, 2, . . . , N}, and they presented several examples for subsets with strong pseudo-random properties when N is a prime number. In this article, we present a construction of pseudo-random subsets for N = pq and give some partial results on the pseudo-random measures.

QUALIFIED DIFFERENCE SETS FROM UNIONS OF CYCLOTOMIC CLASSES

AbstractQualified difference sets (QDS) composed of unions of cyclotomic classes are discussed. An exhaustive computer search for such QDS and modified QDS that also possess the zero residue has been conducted for all powers n=4,6,8 and 10. Two new families were discovered in the case n=8 and some new isolated systems were discovered for n=6 and n=10.

Remarks on a cyclotomic sequence

We analyse a binary cyclotomic sequence constructed via generalized cyclotomic classes by Bai et al. (IEEE Trans Inforem Theory 51: 1849---1853, 2005). First we determine the linear complexity of a natural generalization of this binary sequence to arbitrary prime fields. Secondly we consider k-error linear complexity and autocorrelation of these sequences and point out certain drawbacks of this construction. The results show that the parameters for the sequence construction must be carefully chosen in view of the respective application.

Difference systems of sets and cyclotomy

Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. A method for the construction of DSS from partitions of cyclic difference sets was introduced in [V.D. Tonchev, Difference systems of sets and code synchronization, Rend. Sem. Mat. Messina, Ser. II, t. XXV 9 (2003) 217–226] and applied to cyclic difference sets ( n , ( n - 1 ) / 2 , ( n - 3 ) / 4 ) of Paley type, where n ≡ 3 ( mod 4 ) is a prime number. This paper develops similar constructions for prime numbers n ≡ 1 ( mod 4 ) that use partitions of the set of quadratic residues, as well as more general cyclotomic classes.

On the k-error linear complexity of cyclotomic sequences

Exact values and bounds on the k-error linear complexity of p-periodic sequences which are constant on the cyclotomic classes are determined. This family of sequences includes sequences of discrete logarithms, Legendre sequences and Hall's sextic residue sequence.

A New Method for Constructing Williamson Matrices

For every prime power q ? 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x2+4y2 for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q2.