Gaussian Integers Research Articles

The geometry of Gaussian integer continued fractions

The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship |ad−bc|=1 for a,b,c,d integers as it arises in the Farey tessellation of the hyperbolic plane H2 and the array of Ford circles in the upper-half of the real plane R2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition |αγ−βδ|=1 for α,β,γ,δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers.

Coprime Sensing via Chinese Remaindering Over Quadratic Fields—Part I: Array Designs

A coprime antenna array consists of two or more sparse subarrays featuring enhanced degrees of freedom (DOF) and reduced mutual coupling. This paper introduces a new class of planar coprime arrays, based on the theory of ideal lattices. In quadratic number fields, a splitting prime p can be decomposed into the product of two distinct prime ideals, which give rise to the two sparse subarrays. Their virtual difference coarray enjoys a quadratic gain in DOF, thanks to the generalized Chinese remainder theorem (CRT). To enlarge the contiguous aperture of the coarray, we present hole-free symmetric CRT arrays with simple closed-form expressions. The ring of Gaussian integers and the ring of Eisenstein integers are considered as examples to demonstrate the procedure of designing coprime arrays. With Eisenstein integers, our design yields a difference coarray that is a subset of the hexagonal lattice, offering a significant gain in DOF over the rectangular lattice, given the same physical areas. Maximization of CRT arrays in the aspect of essentialness and the superior performance in the context of angle estimation will be presented in the companion paper (Part II).

A model of discrete quantum computation

We present a model of discrete quantum computing focused on a set of discrete quantum states. For this, we choose the set that is the most outstanding in terms of simplicity of the states: the set of Gaussian coordinate states, which includes all the quantum states whose coordinates in the computation base, except for a normalization factor $$\sqrt{2^{-k}}$$ , belong to the ring of Gaussian integers $$\mathbb {Z}[i]=\{a+bi\ |\ a,b\in \mathbb {Z}\}$$ . We also introduce a finite set of quantum gates that transforms discrete states into discrete states and generates all discrete quantum states, and the set of discrete quantum gates, as the quantum gates that leave the set of discrete states invariant. We prove that the quantum gates of the model generate the expected discrete states and the discrete quantum gates of 2-qubits and conjecture that they also generate the discrete quantum gates of n-qubits.

Cryptanalysis of RSA-type cryptosystems based on Lucas sequences, Gaussian integers and elliptic curves

In this paper, we apply the continued fraction method to launch an attack on the three RSA-type cryptosystems when the private exponent d is sufficiently small. The first cryptosystem, proposed by Kuwakado, Koyama and Tsuruoka in 1995, is a scheme based on singular cubic curves y2=x3+bx2(modN) where N=pq is an RSA modulus. The second cryptosystem, proposed by Elkamchouchi, Elshenawy and Shaban in 2002, is an extension of the RSA scheme to the field of Gaussian integers using a modulus N=PQ where P and Q are Gaussian primes such that p=|P| and q=|Q| are ordinary primes. The third cryptosystem, proposed by Castagnos in 2007, is a scheme over quadratic field quotients with an RSA modulus N=pq based on Lucas sequences. In the three cryptosystems, the public exponent e is an integer satisfying the key equation ed−k(p2−1)(q2−1)=1. Our attack is applicable to primes p and q of arbitrary sizes and we do not require the usual assumption that p and q have the same bit size. Thus, this is an extension of our recent result presented at ACISP 2016 conference. Our experiments demonstrate that for a 513-bit prime p and 511-bit prime q, our method works for values of d of up to 520 bits.

Nearly perfect Gaussian integer sequences with arbitrary degree

Based on p-ary pseudorandom sequences, this study proposes a construction of degree-k Gaussian integer sequences of period N = k ( p m − 1 ) / ( p − 1 ) by utilising kth power residue symbol satisfying k | ( p − 1 ) , where p is an odd prime and positive integers m , k ≥ 2 . The periodic autocorrelation values are 0 at shifts τ ≢ 0 ( mod ( N / k ) ) of the resultant sequences. Specially, there is exactly one non-zero out-of-phase periodic autocorrelation value of the resultant sequences for k = 2 . The non-zero elements of the sequences are balanced and can be predefined flexibly. Moreover, the maximum energy efficiency of the proposed sequences is close to ( p − 1 ) / p for sufficiently large m.

Perfect Gaussian Integer Sequence Pairs from Cyclic Difference Set Pairs

Perfect Gaussian Integer Sequence Pairs from Cyclic Difference Set Pairs

Fourth power diophantine equations in Gaussian integers

In this paper, we examine a class of fourth power diophantine equations of the form $$x^4+kx^2y^2+y^4=z^2$$ and $$ax^4+by^4=cz^2$$ , in the Gaussian integers, where a and b are prime integers.

A fraction-free unit-circle zero location test for a polynomial with any singularity profile

ABSTRACTThis paper presents a fraction-free (FF) version of the Bistritz test to determine the zero location (ZL) of a polynomial with respect to the unit circle. The test has the property that when it is invoked on a polynomial with Gaussian or real integer coefficients, it is an efficient integer algorithm completed without fractions over the respective integral domain. The test is not restricted to integers but remains integer preserving (IP) in all possible encounters of abnormalities and singularities. We define a symmetric subresultant polynomial sequence (SSPS) for the Sylvester matrix of two symmetric polynomials. We then show that the sequence of polynomials produced by the FF test coincides with the SSPS of its first two polynomials when the test is normal and the SSPS is strongly nonsingular, or else its polynomials match the non-singular subresultant polynomial and pass over intermediate gaps of singular subresultants in an IP and efficient manner. This relationship (interesting in its own right) is used to show that the test is IP and normally attains integers of minimal size.

A structure theorem for multiplicative functions over the Gaussian integers and applications

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x2−y2 = n2 for some Gaussian integer n. The analog of this statement over Z remains open.

Construction of Period <inline-formula> <tex-math notation="LaTeX">$qp$ </tex-math> </inline-formula> PGISs With Degrees Equal to or Larger Than Four

The degree of a perfect Gaussian integer sequence (PGIS) is defined as the number of distinct nonzero Gaussian integers within one period of the sequence. This paper focuses on constructing PGISs with degrees equal to or larger than four and period of $N = qp$ , where $q$ and $p$ are distinct primes. The study begins with the partitioning of a ring $\mathbb {Z}_{N}$ into four subsets, after which degree-4 PGISs can be constructed from either the time or frequency domain. In these two approaches, nonlinear constraint equations are derived to govern the coefficients for the associative sequences to be perfect. By transforming nonlinear constraint equations into a system of linear equations, the construction of degree-4 PGISs becomes straightforward. To construct PGISs with degrees larger than four, further partitioning of $\mathbb {Z}_{N}$ should be carried out; here, two cases, the even period $N = 2p$ and the odd period $N = qp$ , are treated separately. We can adopt the Legendre sequences of the prime period $p$ to construct PGISs of period $2p$ with degrees larger than four. For the case of period $qp$ , we introduce the Jacobi symbols to partition $\mathbb {Z}_{N}$ into seven subsets and construct PGISs with more diverse degrees.

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a b , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a b through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.

Signal constellations employing multiplicative groups of Gaussian and Eisenstein integers for Enhanced Spatial Modulation

In this paper, we propose two new signal constellation designs employing Gaussian and Eisenstein Integers for Enhanced Spatial Modulation (ESM). ESM is a novel technique which was propounded by Cheng et al. The advantage of ESM over other Spatial Modulation (SM) schemes lies in its ability to enhance spectral efficiency while keeping the energy efficiency intact. This is done by activating either one or two antennas judiciously depending upon the required trade-off. In ESM, information radiated from the antennas depends upon index of the active transmit antenna combination(s) and also on the set of constellation points chosen, which may include points from multiple constellations. In this paper, we propose signal constellations based on multiplicative groups of Gaussian and Eisenstein integers. The set comprising of Gaussian and Eisenstein integers serves as primary and secondary constellation points for Gaussian Enhanced Spatial Modulation (GESM) scheme. The secondary constellation points are deduced from a single geometric interpolation from the primary constellation points. The Monte Carlo simulation results indicate that the proposed nonuniform constellations achieve impressive SNR gains compared to conventional constellation points used in the design of ESM. This new design has been described for MIMO employing 4 × 4 and 8 × 8 antenna configurations with only two active antennas. Furthermore, a closed form expression for the pairwise error probability (PEP) for the GESM scheme has been deduced. The PEP is utilized to determine the upper bound on the average bit error probability (ABEP). Our simulations indicate that the proposed GESM from Gaussian and Eisenstein integers scheme outperforms all the other variants of SM including conventional ESM by at least 2.5 dB at an average bit error ratio (ABER) of 10−5. Close correspondence between the theoretical analysis and the Monte Carlo simulation results are observed.

Selectivity in definite quaternion algebras

We study the embedding problem for commutative orders into Eichler orders in definite quaternion algebras over the rationals by using methods from the theory of quadratic forms, specifically, Minkowski–Siegel's formula for the representation mass. We characterize when the Gaussian or Eisenstein integers embed into some but not all classes in the genus of an Eichler order. We also give an application to the theory of supersingular elliptic curves.

Hyperbolic geometry and moduli of real curves of genus three

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to maximal real quartic curves, we obtain a more explicit description. We construct a Coxeter diagram that encodes the geometry of this component.

Wide Containers in Gaussian Networks

The w-wide diameter of a graph is a generalisation of its diameter and connectivity, and is useful for measuring the reliability of an interconnection network. The Gaussian networks, which are defined via quotient rings of the Gaussian integers, have been shown to satisfy many good properties of an interconnection network. In this paper, we will establish the wide diameters of the Gaussian networks and show that they are at most three more than the diameter.

Perfect Gaussian Integer Sequences of Period $p^{k}$ With Degrees Equal to or Less Than $k+1$

This paper presents the construction of perfect Gaussian integer sequence (PGIS) of period $N=p^{k}$ with degrees equal to or less than $k+1$ , where $p$ is a prime number and $k\geq 2$ . The study begins with the partitioning of $\mathbb {Z}_{N}$ into $k+1$ subsets, from which $k+1$ base sequences are defined to construct degree- $(k+1)$ PGISs. The $k$ constraint equations derived from matching the ideal periodic autocorrelation function criterion in the time domain are nonlinear. We propose a decomposition method that can transform these $k$ nonlinear equations into $2k-2$ linear equations with $2k-2$ variables. The number of nonlinear constraint equations derived from the frequency domain is $k+1$ , where the $k+1$ sequence coefficients can be derived sequentially by an iteration algorithm. These two proposed schemes simplify the construction of the degree- $(k+1)$ PGIS. PGISs having less than $k+1$ degrees can be constructed in two steps. First, some sequence coefficients are assigned a zero value or one or more pairs of two consecutive base sequences are combined into one. The second step involves the adjustment of new sequence coefficients for the associated sequence to be perfect. We show that there exist at least $2k+(({k(k-1)(k-2)})/{6})$ distinct PGIS patterns of period $N=p^{k}$ with degrees equal to or less than $k+1$ .

Complex Linear Physical-Layer Network Coding

This paper presents the results of a comprehensive investigation of complex linear physical-layer network coding (PNC) in two-way relay channels. In this system, two nodes A and B communicate with each other via a relay R. Nodes A and B send complex symbols, $w_{A}$ and $w_{B}$ , simultaneously to relay R. Based on the simultaneously received signals, relay R computes a linear combination of the symbols, $w_{N}=\alpha w_{A}+\beta w_{B}$ , as a network-coded symbol and then broadcasts $w_{N}$ to nodes A and B. Node A then obtains $w_{B}$ from $w_{N}$ and its self-information $w_{A}$ by $w_{B}=\beta ^{-1}(w_{N}-\alpha w_{A})$ . Node B obtains $w_{B}$ in a similar way. A critical question at relay R is as follows: “given channel gain ratio $\eta = h_{A}/h_{B}$ , where $h_{A}$ and $h_{B}$ are the complex channel gains from nodes A and B to relay R, respectively, what is the optimal coefficients $(\alpha ,\beta )$ that minimizes the symbol error rate (SER) of $w_{N}=\alpha w_{A}+\beta w_{B}$ when the relay attempts to detect $w_{N}$ in the presence of noise?” Our contributions with respect to this question are as follows: 1) we put forth a general Gaussian-integer formulation for complex linear PNC in which $\alpha ,\beta ,w_{A}, w_{B}$ , and $w_{N}$ are the elements of a finite field of Gaussian integers, that is, the field of $\mathbb {Z}[i]/q$ , where $q$ is a Gaussian prime. Previous vector formulation, in which $w_{A}$ , $w_{B}$ , and $w_{N}$ were represented by 2-D vectors and $\alpha $ and $\beta $ were represented by $2\times 2$ matrices, corresponds to a subcase of our Gaussian-integer formulation, where $q$ is real prime only. Extension to the Gaussian prime $q$ , where $q$ can be complex, gives us a larger set of signal constellations to achieve different rates at different values of SNR; and 2) we show how to divide the complex plane of $\eta $ into different Voronoi regions, such that the $\eta $ within each Voronoi region shares the same optimal PNC mapping $(\alpha _{\mathrm{ opt}},\beta _{\mathrm{ opt}})$ . We uncover the structure of the Voronoi regions that allows us to compute a minimum-distance metric that characterizes the SER of $w_{N}$ under optimal PNC mapping $(\alpha _{\mathrm{ opt}},\beta _{\mathrm{ opt}})$ . Overall, the contributions in 1) and 2) yield a toolset for a comprehensive understanding of complex linear PNC in $\mathbb {Z}[i]/q$ . We believe investigation of linear PNC beyond $\mathbb {Z}[i]/q$ can follow the same approach.

DFT Computation Using Gauss-Eisenstein Basis: FFT Algorithms and VLSI Architectures

A joint numerical representation based on both Gaussian and Eisenstein integers is proposed. This Gauss-Eisenstein representation maps complex numbers into four-tuples of integers with arbitrarily high precision. The representation furnishes the computation of the 3-, 6-, and 12-point discrete Fourier transform (DFT) at any desired accuracy. The associated fast algorithms based on the Gauss-Eisenstein integers are error-free up to the final reconstruction step, which can be realized in hardware as a multiplierless implementation. The introduced methods are compared with competing algorithms in terms of arithmetic complexity. We propose three FRS architectures based on the following methods: Dempster-McLeod representation, expansion factor, and addition aware quantization. The Gauss-Eisenstein 12-point DFT is physically realized on a Xilinx Virtex 6 FPGA device with maximum clock frequency of 302 MHz for the expansion factor FRS with real-time throughput of 3:62 × 109 coefficients/s. The FPGA verified digital designs were synthesized, mapped, placed and finally routed for 0:18mm CMOS technology assuming a 1.8 V DC supply employing Austria Micro Systems (AMS) standard-cell library (hitkit version 4.11). The routed ASIC is predicted to operate at a maximum frequency of 505 MHz for the expansion factor FRS with potential real-time throughput of 6:06 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">9</sup> coefficients/s.

A Further Stroll into the Eisenstein Primes

If one imagines the Eisenstein primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture.If the frog's journey terminates for a given hop size, it implies that a prime-free “moat” greater than the hop size completely surrounds the origin.In the earlier MONTHLY article “A Stroll Through the Gaussian Primes,” Ellen Gethner, Stan Wagon, and Brian Wick [3] explored this problem in the Gaussian primes and by computational methods proved the existence of a -moat. Additionally they proved that prime-free neighborhoods of arbitrary radius k surrounding a Gaussian prime exist.In their concluding remarks, Gethner et al. note that “Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1.”This paper takes up that challenge by examining similar questions about walks to infinity in the ring of Eisenstein integers. We prove that prime-free neighborhoods of arbitrary radius k surrounding an imaginary quadratic prime exist by directly generalizing Gethner's proof regarding prime-free neighborhoods surrounding a Gaussian prime. By computational means we establish the existence of a -moat in the Eisenstein primes and provide an estimate for the bounds of any k-moat. A subtle difference between the structure of Eisenstein and Gaussian integers results in our estimate being significantly different from our initial expectation.

Power sums over finite commutative unital rings

In this paper we compute the sum of the k-th powers of all the elements of a finite commutative unital ring, thus generalizing known results for finite fields, the rings of integers modulo n or the ring of Gaussian integers modulo n. As an application, we focus on quotient rings of the form (Z/nZ)[x]/(f(x)) for a polynomial f∈Z[x].