Gaussian Integers Research Articles

Porosity in Conformal Dynamical Systems

AbstractIn this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that their limit sets are porous in large (in the sense of category and dimension) subsets. We also provide natural geometric and dynamic conditions under which the limit set of a GDMS is upper porous or mean porous. On the other hand, we prove that if the limit set of a GDMS is not porous, then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction.We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterisation of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are uniformly upper porous and mean porous almost everywhere.We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

On the Generalized Bykovskiĭ Presentation of Steinberg Modules

Abstract We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskiĭ gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.

A New Class of Q-Ary Codes for the McEliece Cryptosystem

The McEliece cryptosystem is a promising candidate for post-quantum public-key encryption. In this work, we propose q-ary codes over Gaussian integers for the McEliece system and a new channel model. With this one Mannheim error channel, errors are limited to weight one. We investigate the channel capacity of this channel and discuss its relation to the McEliece system. The proposed codes are based on a simple product code construction and have a low complexity decoding algorithm. For the one Mannheim error channel, these codes achieve a higher error correction capability than maximum distance separable codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding.

Some Characteristics of Prime Cyclic Ideal On Gaussian Integer Ring Modulo

Gaussian modulo integer is a complex number a + ib where a,b ∊ ℤ n. Some of the characteristics of the prime ideal on Gaussian integer are a trivial ideal {0} is a prime ideal, and I ideal prime if and only if I almost prime ideal. These characteristics do not necessarily apply to modulo Gaussian integers. In this paper, we give some characteristics of the prime ideal on modulo Gaussian integer.

Montgomery Reduction for Gaussian Integers

Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

Construction of Gaussian Integer Periodic Complementary Sequence Set with Zero Correlation Zone

Based on aperiodic complementary sequence (ACS) sets and orthogonal matrices, the construction of Gaussian integer (GI) zero correlation zone (ZCZ) periodic complementary sequence (ZPCS) sets is proposed. The constructed GI ZPCS sets can achieve the theoretical bound, and the length of ZCZ can be chosen flexibly. The GI ZPCS sets obtained by this paper can be used in multi-carrier code division multiple access system to obtain higher spectrum efficiency and eliminate interference.

The second moment of symmetric square L-functions over Gaussian integers

We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog–Biro–Cherubini–Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.

Multi-Channel Modulo Samplers Constructed From Gaussian Integers

Recently, there is an increased interest in the study of modulo analog to digital converters (ADCs). These new systems can reconstruct a signal whose amplitude is much higher than the conventional ADC's dynamic range. Modulo ADCs are characterized by their modulo threshold and in the current literature, all existing works are limited to real-valued moduli. In this paper, we propose multi-channel modulo samplers with complex-valued moduli to sample a band-limited complex signal. Specifically, we discuss the construction of complex divisors from Gaussian integers and propose their efficient implementations. A memory-efficient, closed-form recovery algorithm is also proposed. Simulation results demonstrate that the proposed systems can provide stable reconstruction of a high dynamic range complex-valued signal at low sampling rates.

Novel Hybrid Public/Private Key Cryptography Based on Perfect Gaussian Integer Sequences

This paper proposes a novel hybrid public/private key cryptography scheme based on perfect Gaussian integer sequences (PGISs) of period $N=pq$ . First, a review study of construction degree-4 PGIS is addressed. We show that circular convolution over PGISs is a trapdoor one-way permutation function that enables simultaneous cipher encryption and digital signatures. To implement the proposed cipher encryption scheme, a private PGIS is assigned as the encryption key sequence for circular convolution with the plaintext to generate the ciphertext. The reverse decryption key sequence involves the time reflection and complex conjugation of the encryption sequence, which can be regenerated using a pair of public and private keys. The security level of the proposed scheme is the same as that of the Rivest-Shamir-Adleman (RSA) system; however, the capacity of a cryptosystem based on PGISs may outperform that of based on RSA, because abundant PGISs are available. Simulation results show that the approximation error when finite digits are used to represent the irrational coefficients of a normalized PGIS can be relatively small compared with the noise. This contributes to the simplicity of this scheme’s implementation. With the fast development of IoT (internet of things), the adaptation and applicability of the proposed scheme to IoT platforms are also addressed, where lightweight cryptographic functions are preferable due to the limited resources of IoT devices.

Dimension of the intersection of certain Cantor sets in the plane

In this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.

Gaussian integers partition in power-free numbers product

Let $\mathrm{g}_1 (\alpha)$ be the number of Gaussian integer $\alpha$ representation in a product of square-free factors. Let $\mathrm{g}_2 (\alpha)$ be the number of Gaussian integer $\alpha$ representation in a product of power-free factors. In this paper we consider their summatory functions $\sum_{N(\alpha)\le x} \mathrm{g}_1 (\alpha)$ and $\sum_{N(\alpha)\le x} \mathrm{g}_2 (\alpha)$ and obtain asymptotic formulas for them. Also, we prove analogue of K\'{a}tai-Subbarao theorem to study the distribution of $\mathrm{g}_2 (\alpha)$ in increasing norm order case.

The standard form of matrices over the ring of Gaussian integers with respect to (z,k)-equivalence

The standard form of matrices over the ring of Gaussian integers with respect to (z,k)-equivalence

A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

Cauchy Problems for Discrete Holomorphic Functions

We solve Cauchy problems for discrete holomorphic functions defined on the Gaussian integers, which leads to the existence of discrete holomorphic functions with arbitrarily fast growth. This proves that certain classes of functions are closed in the sense of mathematical morphology.

Prime geodesic theorem for the Picard manifold

Let Γ=PSL(2,Z[i]) be the Picard group and H3 be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient Γ∖H3, called the Picard manifold, obtaining an error term of size O(X3/2+θ/2+ϵ), where θ denotes a subconvexity exponent for quadratic Dirichlet L-functions defined over Gaussian integers.

Bykovskii-Type Theorem for the Picard Manifold

Abstract We generalise a result of Bykovskii to the Gaussian integers and prove an asymptotic formula for the prime geodesic theorem in short intervals on the Picard manifold. Previous works show that individually the remainder is bounded by $O(X^{13/8+\epsilon })$ and $O(X^{3/2+\theta +\epsilon })$, where $\theta$ is the subconvexity exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral exponential sum and a smooth explicit formula, we obtain an improvement for both of these exponents. Moreover, by assuming two standard conjectures on $L$-functions, we show that it is possible to reduce the exponent below the barrier $3/2$ and get $O(X^{34/23+\epsilon })$ conditionally. We also demonstrate a dependence of the remainder in the short interval estimate on the classical Gauss circle problem for shifted centres.

A Novel Image Representation Method Under a Non-Standard Positional Numeral System

Image representation is an active research area in the field of image processing. This paper proposes a novel image representation method under a non-standard positional numeral system, wherein complex numbers are used as bases in such non-standard positional numeral system. It is different with binary code and decimal code, where the digit 2 is used as the base in the binary system, and the digit 10 is used as the base in the decimal system. In the proposed image representation method, a two-dimensional image is transformed into a one-dimensional 0 $\sim$ 1 sequence, and its Gaussian integer is calculated based on the derived one-dimensional 0 $\sim$ 1 sequence. On the contrary, the original two-dimensional image can be recovered from its Gaussian integer. When images are represented as Gaussian integers, the classical geometrical operations are introduced into image processing. Then, the relationship of different images is established by the methods of plane geometry, i.e., the addition, subtraction, multiplication, division, conjugate, and inverse operations of images are defined. The experimental results show that the selection of complex number as base in the positional numeral system is a very special coding method, a given digital image is effectively converted to a Gaussian integer by using the proposed image representation method, and the image arithmetic is also successfully achieved. In addition, three applications based on the proposed image representation method including image camouflage, image sharing, and image scrambling are selected to demonstrate that the new image representation has good and potential practical applicability in the field of secure encryption of digital images.

Simulation of Energy Efficient Clustering Technique in a Gaussian Network

In Wireless sensor network (WSN) the sensor nodes are randomly distributed in the network. As the sensor nodes are adhoc in nature the geographical location must be assessed earlier and for data transmission. Maintenance of the energy level in each sensor node is very challengeable. In this paper a Gaussian network is deployed with four virtual grids and cluster head is obtained using the Gaussian integer. The traditional DSDV protocol is modified and simulated in Network Simulator. The energy efficiency of modified DSDV with regard to increase in number of nodes in compared with traditional AODV protocol and found a better performance of the modified DSDV protocol.

CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES

Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.