Congruence Equations Research Articles

On the High-Power Mean of the Generalized Gauss Sums and Kloosterman Sums

The main aim of this paper is to use the properties of the trigonometric sums and character sums, and the number of the solutions of several symmetry congruence equations to research the computational problem of a certain sixth power mean of the generalized Gauss sums and generalized Kloosterman sums, and to give two exact computational formulae for them.

Solving System of Linear Congruence Equations over some Rings by Decompositions of Modules

Solving System of Linear Congruence Equations over some Rings by Decompositions of Modules

Синтез уравнений линейчатых поверхностей с двумя криволинейными и одной прямолинейной направляющими

Ruled surfaces have long been known and are widely used in construction, architecture, design and engineering. And if from the technical point of view the developable surfaces are more attractive, then architecture and design successfully experiment with non-developable ones. In this paper are considered non-developable ruled surfaces with three generators, two of which are curvilinear ones. According to classification, such surfaces are called twice oblique cylindroids. In this paper has been proposed an approach for obtaining of twice oblique cylindroids by immersing a curve in a line congruence of hyperbolic type. Real directrixes of such congruence are a straight line and a curve. It has been proposed to use helical lines (cylindrical and conical ones) as a curvilinear directrix, and a helical line’s axis as the straight one. Then the congruence’s rectilinear ray will simultaneously intersect the helical line and its axis. Congruence parameters are the line’s pitch and the guide cylinder or cone’s radius. The choice of the curvilinear directrix is justified by the fact that the helical lines have found a wide application in engineering and architecture. Accordingly, the helical lines based surfaces can have a great potential. In this paper have been presented parametric equations of the considered congruences. The congruence equations have been considered from the point of view related to introducing a new curvilinear coordinate system. The obtained system’s coordinate surfaces and coordinate lines have been also studied in the paper. To extract the surface, it is necessary to immerse the curve in the congruence. To synthesize the equations has been used a constructive-parametric method based on the substitution of the immersed line’s parametric equations in the congruence equations according to a special algorithm. In the paper have been presented 5 examples for the synthesis of ruled surfaces equations such as the twice oblique cylindroid and their visualization. The method is universal and algorithmic, and therefore easily adaptable for the automated construction of surfaces with variable parameters of both the congruence and the immersed line.

Chinese remainder theorem-based two-in-one image secret sharing with three decoding options

Two-in-one image secret sharing (ISS) with more decoding options has more applicable scenarios than that with only one decoding option. Conventional two-in-one ISS methods with more decoding options mainly combine visual secret sharing (VSS) and polynomial-based or XOR-based ISS. They have the drawbacks of lossy recovery for (k,n) threshold, only two decoding options, high computation in recovering phase or large pixel expansion. In this article, we design a Chinese remainder theorem (CRT)-based two-in-one ISS with three decoding options: lossless recovery, grayscale stacking recovery and visual previewing capability. In the encoding phase, we encode a binary secret image by random grid (RG)-based VSS (RGVSS). Meanwhile, a grayscale secret image is encoded by CRT in the processing of encoding the binary secret image. In the decoding phase, we can decode the binary secret image with two decoding options: grayscale stacking recovery and visual previewing capability. The grayscale secret image can be losslessly decoded with one decoding option through solving a set of linear congruence equations with only modular operation. Furthermore, the shadow image has no pixel expansion. We show our effectiveness by means of theoretical analyses and experiments.

Source-location privacy full protection in wireless sensor networks

In many scenarios, the locations of monitored targets need to be reported by source nodes, but should remain anonymous in wireless sensor networks. Source-location privacy protection is an important research topic. Many schemes have been designed based on different adversarial models. In this paper, a scheme named Source-location Privacy Full Protection (SPFP) is proposed. We consider a more practical adversarial model — a smart adversary — which is a combination of global and local models. To defend against the new adversary, first, we design a lightweight message sharing scheme that is based on congruence equations. Second, each message is mapped to a set of shares. The short lengths of the shares enable them to be processed and transmitted in an energy-efficient manner. The correctness and security of the scheme are proved in theorems. In addition, the proposed message sharing scheme can tolerate the unreliability of the sensor nodes and provides a more reliable data transmission mechanism for networks. Third, the source node constructs a cloud around itself based on the shares and dummy packages to hide its location. The radio actions of the nodes in the cloud are carefully arranged to conceal the real shares from the adversaries and render the nodes in the cloud statistically indistinguishable. Last, a random routing algorithm is seamlessly integrated into our scheme to deliver the real shares from the fake source nodes to the sink node, where the original message is reconstructed based on the received shares. The simulation results illustrate that our scheme can provide adequate protection of source-location privacy with a slight increase in energy consumption.

Exponential sums with automatic sequences

We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type $e_q(f(n))$, where $f$ is a rational fraction, in the P\'olya-Vinogradov range. This applies to Kloosterman sums, and may be used to study solubility of congruence equations over automatic sequences. We obtain this as consequence of a general result, stating that sums over automatic sequences can be bounded effectively in terms of two-point correlation sums over intervals.

New bounds on the decycling number of generalized de Bruijn digraphs

A set of vertices of a graph whose removal leaves an acyclic graph is referred as a decycling set, or a feedback vertex set, of the graph. The minimum cardinality of a decycling set of a graph [Formula: see text] is referred to as the decycling number of [Formula: see text]. For [Formula: see text], the generalized de Bruijn digraph [Formula: see text] is defined by congruence equations as follows: [Formula: see text] and [Formula: see text]. In this paper, we give a systematic method to find a decycling set of [Formula: see text] and give a new upper bound that improve the best known results. By counting the number of vertex-disjoint cycles with the idea of constrained necklaces, we obtain new lower bounds on the decycling number of generalized de Bruijn digraphs.

One kind sixth power mean of the three-term exponential sums

Abstract In this paper, we use the estimate for trigonometric sums and the properties of the congruence equations to study the computational problem of one kind sixth power mean of the three-term exponential sums. As a conclusion, we give an exact computational formula for it.

Solutions to some congruence equations via suborbital graphs.

We relate the connection between the sizes of circuits in suborbital graph for the normalizer of Gamma _0(m) in PSL(2,mathbb {R}) and the congruence equations arising from related group action. We give a number theoretic result which says that all prime divisors of 3u^2mp 3u+1 for any integer u must be congruent to 1pmod {3}.

CARDINALITY OF SETS ASSOCIATED TO CERTAIN DEGREE SEVEN POLYNOMIALS

Let be a function of two variables. Let be an integer and let ∑ , where the sum is taken over a complete set of residue modulo . The value of depends on the estimate of the cardinality of the following set { where and are the partial derivative of with respect to and . In this paper, we discuss the cardinality, of the set of solutions for congruence equations of some special binary forms. Firstly we need to obtain the p-adic sizes of common zeros of the partial derivative polynomials by using Newton polyhedron technique. The polynomial that we consider is in the form of . Keywords: -adic order, Newton Polyhedron, Indicator diagram, Cardinality

The weight enumerators of several classes of [formula omitted]-ary cyclic codes

Let p be an odd prime, and m and k be two positive integers with m≥3. Let h±1(x) and h±t(x) be the minimal polynomials of ±α−1 and ±α−t over Fp, respectively, where α is a primitive element of Fpm. Let C1,−1,±t, C±1,t,−t and C1,−1,t,−t be the cyclic codes over Fp of length pm−1 with parity-check polynomials h1(x)h−1(x)h±t(x), h±1(x)ht(x)h−t(x) and h1(x)h−1(x)ht(x)h−t(x), respectively. This paper determines the weight distributions of the cyclic codes C1,−1,±t, C±1,t,−t and C1,−1,t,−t for the parameter t satisfying some congruence equations.

On the decycling number of generalized Kautz digraphs

For n>d≥2, the generalized Kautz digraph GK(d,n) is defined by congruence equations as follows: V(GK(d,n))={0,1,2,⋯,n−1} and A(GK(d,n))={(x,y)|y≡−dx−i(modn),1≤i≤d}. A set of vertices of a graph whose removal leaves an acyclic graph is called a decycling set of the graph. The minimum size of a decycling set of a graph G is referred to as the decycling number of G. Let f(d,n) be the decycling number of the generalized Kautz digraph GK(d,n). In this paper, we study f(d,n) for all n≥d≥2. As a consequence, we obtain the upper bound of f(d,n) as follows: f(d,n)≤(12−d−12d2)n+d2(d−t+5)−2, where n≡t(mod(d+1)).

Extended Closed-Form Expressions for the Robust Symmetrical Number System Dynamic Range and an Efficient Algorithm for Its Computation

The robust symmetrical number system (RSNS) is a number theoretic transform based on N ≥ 2 sequences that can extract the maximum amount of information from symmetrical folding waveforms. The sequences, based on coprime moduli, exhibit an integer gray code property making the RSNS well suited for many applications that benefit from an inherent error detection and correction capability, such as analog-to-digital converters, direction finding arrays, and radar waveform design. To use the RSNS, it is necessary to know the greatest length of combined sequences without ambiguities, called the dynamic range M, for which only a few closed-form expressions currently exist. In this paper, an efficient algorithm for computing M and its position within the combined set of sequences is presented and shown to be independent of the size of the moduli. The algorithm is used to generate the equations for several groups of additional moduli arrangements. Closed-form expressions for M are conjectured and proved using the obtained congruence equations that define the ambiguity locations.

On the Sixth Power Mean Value of the Generalized Three-Term Exponential Sums

The main purpose of this paper is using the estimate for trigonometric sums and the properties of the congruence equations to study the computational problem of one kind sixth power mean value of the generalized three-term exponential sums and give an exact computational formula for it.

CONGRUENCE EQUATIONS OF axi+ byj≡ c AND axi+ byj+ dzt≡ c(modp) WHEN p=2q+1 WITH p AND q ODD PRIMES

Let p and q be odd primes with p=2q+1. We study the number of solutions of congruence equations <TEX>$ax^i\;+\;by^j\;{\equiv}\;c$</TEX> (mod p) and a<TEX>$ax^i\;+\;by^j\;+\;dz^t\;{\equiv}\;c(modp)$</TEX>

Covering groups and the standard elementary particle model

We determine all Lie groups compatible with the gauge structure of the Standard Elementary Particle Model (SM) and their representations. The groups are specified by congruence equations of quantum numbers. By comparison with the experimental results, we single out one Lie group G sm and show that this choice implies certain old and new correlations between the quantum numbers of the SM quantum fields as well as some hitherto unknown group theoretical properties of the Higgs mechanism.

EXACT SOLUTION OF LINEAR EQUATIONS ON DISTRIBUTED-MEMORY MULTIPROCESSORS

ABSTRACT We present two new parallel algorithms for exact (error-free) solution of a system of linear equations on a distributed-memory multiprocessor. The exact solution is obtained using the congruence technique which consists of two steps: First, the system of linear equations is converted to systems of linear congruence equations with respect to several prime moduli, and each of these systems is solved on a separate processor. Then, these solutions are combined using the mixed-radix conversion algorithm to obtain the exact solution. The first step is completely (embarrassingly) parallel with no communication requirements among the processors. We improve our previous work and describe two efficient parallel algorithms for the second step. We present the results of our experiments on an Intel iPSC/860 with 8 processors. A linear system of dimension 128 with integer entries as large as 10577 is solved in about 195 seconds on 8 processors with an efficiency of 99.76%.

Linear congruence equations for the solutions of the N-Queens problem

We demonstrate a method using linear congruence equations to generate solutions to the N-Queens problem. There are only a few papers in the literature generating solutions for every N. Our method generates solutions for every N, and the number of solutions produced by our method is larger than the number of solutions given in these papers.

Design of large diameter cast-in-place bored piles using a ‘pilot’ pile and congruence equations : Lizzi, F Ground Engng, V14, N4, May 1981, P24–P33

The author reports on the results of a test programme carried out since earlier work which was described in March 1980 ground engineering. The expeprimental approach is based on the use as a test instrument, of a small diameter cast-in-place pile, constructed on the site with the same technology and the same material as the large diameter pile to be designed. This small pile, referred to as the 'pilot' pile, which is instrumented along the shaft and at the toe, is subjected to a direct load test up to failure. The data collected from the instrumentation is used for the design of the large diameter pile. The purpose of the present paper is: to restate the problem and to clarify some of the points which were not clearly expressed in the previous paper; to report the results of a later test programme and to indicate the conclusions reached. (TRRL)