Gaussian Integers Research Articles

A New Secure Authenticated Key Agreement Protocol in Gaussian Field

Key agreement protocols are a fundamental building block for ensuring authenticated and private communications between two parties over an insecure network. In this paper we propose an efficient and secure authenticated key agreement protocol based on DLP (Discrete Logarithm Problem) and Gaussian number field. The main purpose of this paper is to use the Gaussian integers; the set of all complex numbers ai b + with , ab Z ∈ in the Gaussian integers, to design new key agreement protocol that can help the system to be more secure. We show that our protocol meets the security attributes and strong against most of potential attacks. Also it provides most of desirable performance attributes.

New Perfect Gaussian Integer Sequences of Period <i>pq</i>

In this letter, by using Whiteman's generalized cyclotomy of order 2 over ℤ pq , where p,q are twin primes, we construct new perfect Gaussian integer sequences of period pq.

Dimension Estimates for Certain Sets of Infinite Complex Continued Fractions

We prove upper and lower estimates on the Hausdorff dimension of sets of infinite complex continued fractions with finitely many prescribed Gaussian integers. Particulary we will conclude that the dimension of theses sets is not zero or two and there are such sets with dimension greater than one and smaller than one.

MODUL FAKTOR YANG DIBENTUK DARI SUBMODUL Z^2 PADA MODUL R^2 ATAS GAUSSIAN INTEGERS

We prove that ℝ2 is module over Gaussian Intergers and the set of all coset of submodule in module ℝ2 over Gaussian Integers is a quotient module. We find the proof by showing that ℝ2 is both a right module and a left module over Gaussian Integers and showing that the set of all coset of submodule in module ℝ2 is both a right module and a left module over Gaussian Integers.

Gaussian Integer Sequences With Ideal Periodic Autocorrelation Functions

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if the out-of-phase value of the periodic autocorrelation function is equal to zero. This paper presents two novel classes of perfect sequences constructed using two groups of base sequences. The nonzero elements of these base sequences belong to the set {±1, ±j}. A perfect sequence can be obtained by linearly combining these base sequences or their cyclic shift equivalents with arbitrary nonzero complex coefficients of equal magnitudes. In general, the elements of the constructed sequences are not Gaussian integers. However, if the complex coefficients are Gaussian integers, then the resulting perfect sequences will be Gaussian integer perfect sequences (GIPSs). In addition, a periodic cross-correlation function is derived, which has the same mathematical expression as the investigated sequences. Finally, the maximal energy efficiency of the proposed GIPSs is investigated.

The Prouhet-Tarry-Escott problem for Gaussian integers

Given natural numbers $n$ and $k$, with $n>k$, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of $\Z$, say $X=\{x_1,...,x_n\}$ and $Y=\{y_1,...,y_n\}$, such that \[x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for $i=1,...,k$. Many partial solutions to this problem were found in the late 19th century and early 20th century. When $n=k-1$, we call a solution $X=_{n-1}Y$ ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the Gaussian integers. This paper extends the framework of the problem to this setting. We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers.

Cyclotomic matrices over the Eisenstein and Gaussian integers

We classify all cyclotomic matrices over the Eisenstein and Gaussian integers, that is, all Hermitian matrices over the Eisenstein and Gaussian integers that have all their eigenvalues in the interval [−2,2].

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.

Quantum codes from codes over Gaussian integers with respect to the Mannheim metric

In this paper, some nonbinary quantum codes using classical codes over Gaussian integers are obtained. Also, some of our quantum codes are better than or comparable with those known before, (for instance $[[8,2,5]]_{4+i}$).

The Tarry-Escott problem of degree two

The optimum reduced solutions of degree two of the Tarry-Escott problem over a ring of Gaussian integers and over a ring of polynomials over a finite field are completely determined.

A mild generalization of Eisenstein’s criterion

We state and prove a mild generalization of Eisenstein’s Criterion for a polynomial to be irreducible, correcting an error that Eisenstein made himself.

Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Let ℤn[i] be the ring of Gaussian integers modulo n. We construct for ℤn[i] a cubic mapping graph Γ(n) whose vertex set is all the elements of ℤn[i] and for which there is a directed edge from a ∈ ℤn[i] to b ∈ ℤn[i] if b = a3. This article investigates in detail the structure of Γ(n). We give suffcient and necessary conditions for the existence of cycles with length t. The number of t-cycles in Γ1(n) is obtained and we also examine when a vertex lies on a t-cycle of Γ2(n), where Γ1(n) is induced by all the units of ℤn[i] while Γ2(n) is induced by all the zero-divisors of ℤn[i]. In addition, formulas on the heights of components and vertices in Γ(n) are presented.

The Complement Graph for Gaussian Integers Modulo n

This article is a continuation for the work done in [1, 2] on the zero divisor graph for the ring of Gaussian integers modulo n. It investigates when the complement graph of the zero divisor graph for the Gaussian integers modulo n connected, planar, regular, or Eulerian. The girth and diameter were also studied.

On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n

Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several properties of the line graph of Γ(ℤn[i]), L(Γ(ℤn[i])) are studied. It is determined when L(Γ(ℤn[i])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of L(Γ(ℤn[i])) is given when n is a power of a prime. On the other hand, several graph invariants for Γ(ℤn[i]) are also determined.

Triangular Numbers, Gaussian Integers, and KenKen

SummaryLatin squares form the basis for the recreational puzzles sudoku and KenKen. In this article we show how useful several ideas from number theory are in solving a KenKen puzzle. For example, the simple notion of triangular number is surprisingly effective. We also introduce a variation of KenKen that uses the Gaussian integers in order to illustrate the concept of unique factorization.

Primality Testing Using Complex Integers and Pythagorean Triplets

Prime integers and their generalizations play important roles in protocols for secure transmission of information via open channels of telecommunication networks. Generation of multidigit large primes in the design stage of a cryptographic system is a formidable task. Fermat primality checking is one of the simplest of all tests. Unfortunately, there are composite integers (called Carmichael numbers) that are not detectable by the Fermat test. In this paper we consider modular arithmetic based on complex integers; and provide several tests that verify the primality of real integers. Although the new tests detect most Carmichael numbers, there are a small percentage of them that escape these tests.

On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.

The cubic mapping graph for the ring of Gaussian integers modulo n

The article studies the cubic mapping graph Γ(n) of ℤ n [i], the ring of Gaussian integers modulo n. For each positive integer n > 1, the number of fixed points and the in-degree of the elements \(\overline 1 \) and \(\overline 0 \) in Γ(n) are found. Moreover, complete characterizations in terms of n are given in which Γ2(n) is semiregular, where Γ2(n) is induced by all the zero-divisors of ℤ n [i].

Numerical approximation of The Masser-Gramain constant to four decimal digits: $\delta = 1.819...$

We prove that the constant studied by Masser, Gramain, and Weber, satisfies 1.819776 < delta < 1.819833, and disprove a conjecture of Gramain. This constant is a two-dimensional analogue of the Euler-Mascheroni constant; it is obtained by computing the radius rk of the smallest disk of the plane containing k Gaussian integers. While we have used the original algorithm for smaller values of k, the bounds above come from methods we developed to obtain guaranteed enclosures for larger values of k.

THE DIGRAPH OF THE SQUARE MAPPING ON QUOTIENT RINGS OVER THE GAUSSIAN INTEGERS

In this work, we investigate the structure of the digraph [Formula: see text] associated with the square mapping on the ring of Gaussian integers by using the exponent of the unit group modulo γ. The formula for the fixed points of [Formula: see text] is established. Some connections of the lengths of cycles with the exponent of the unit group modulo γ are presented. Furthermore, we study the maximum distance from the cycle on each component.