Elliptic and hyperelliptic equations

This chapter is of an auxiliary nature, being mainly concerned with the relationship between bounds for linear forms in the logarithms of algebraic numbers in different (archimedean and non-archimedean) metrics. This material will later be used in the analysis of Thue and Thue-Mahler equations. Elliptic and hyperelliptic equations, and equations of hyperelliptic type, will be analysed using direct bounds for linear forms in the logarithms of algebraic numbers, and the necessary results are stated without proof at the end of the chapter. *** DIRECT SUPPORT *** A00I6B17 00002

The positive integer points and integral points of elliptic curves are very important in the theory of number and arithmetic algebra, it has a wide range of applications in cryptography and other fields. There are some results of positive integer points of elliptic curve y2 = x3 + ax + b, a, b ∈ Z In 1987, D. Zagier submit the question of the integer points on y2 = x3 − 27x + 62, it count a great deal to the study of the arithmetic properties of elliptic curves.In 2009, Zhu H L and Chen J H solved the problem of the integer points on y2 = x3 − 27x + 62 by using algebraic number theory and P-adic analysis method.In 2010, By using the elementary method, Wu H M obtain all the integral points of elliptic curves y2 = x3 − 27x − 62.In 2015, Li Y Z and Cui B J solved the problem of the integer points on y2 = x3 − 21x − 90 By using the elementary method.In 2016, Guo J solved the problem of the integer points on y2 = x3 + 27x + 62 by using the elementary method.In 2017, Guo J proved that y2 = x3 − 21x + 90 has no integer points by using the elementary method.Up to now, there is no relevant conclusions on the integral points of elliptic curves y2 = x3 + 3x − 14, which is the subject of this paper.By using congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 3x − 14 has only one integer point: (x, y) = (2, 0).

In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q. How to deal with standard Weierstras equations is well understood, but comparatively little is known about elliptic equations of different type. We give a detailed analysis of the general situation and subsequently apply the elliptic logarithm method to a variety of unusual elliptic equations, notably to some equations directly related to Krawtchouk polynomials. 1991 Mathematics subject classification: 11D25, 11G05, 11Y50, 12D10

We consider approximation algorithms for the metric labeling problem. This problem was introduced in a recent paper by Kleinberg and Tardos [20], and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case where k is the number of labels and a 2-approximation for the uniform metric case. More recently, Gupta and Tardos [15] gave a 4-approximation for the truncated linear metric, a natural non-uniform metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce a natural integer programming formulation and show that the integrality gap of its linear relaxation either matches or improves the ratios known for several cases of the metric labeling problem studied until now, providing a unified approach to solving them. In particular, we show that the integrality gap of our LP is bounded by O(log k log log k) for general metric and 2 for the uniform metric thus matching the ratios in [20]. We also develop an algorithm based on our LP that achieves a ratio of 2 + √2 s 3.414 for the truncated linear metric improving the ratio given in [15]. Our algorithm uses the fact that the integrality gap of our LP is 1 on a linear metric. We believe that our formulation has the potential to provide improved approximation algorithms for the general case and other useful special cases. Finally, our formulation admits general non-metric distance functions. This leads to a non-trivial approximation guarantee for a non-metric case that arises in practice [21], namely the truncated quadratic distance function. We note here that there are non-metric distance functions for which no bounded approximation ratio is achievable.

In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area means finding the majority of points on the elliptic curve and minimizing the time to compute the solution in contrast with the necessary time to compute the inverse solution. We can compute the positive solution of PDE (partial differential equation) like oscillations of f(s)/s around the principal eigenvalue λ1 of -Δ in H 0 1 (Ω).Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves.

In this paper, we studied the elliptic curve y2 = (x – a)(x2 + ax + r), a, r ∈ Z+ in the case of a = 6,r = 7 which has no relevant conclusions up to now. And we obtained the conclusion of the integer points on Elliptic y2 = (x – 6)(x2 + 6x + 7), elliptic curve in title has only one integer point (x, y) = (6,0) were proved with the help of the elementary methods such as congruence and Legendre symbol.

By using elementary number theory methods, such as congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 19x + 46 has no integer point.

A τ function formalism for Sakai's elliptic Painleve equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta–Ramani–Grammaticos. We also give a simple geometric description of the elliptic Painleve equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of the elliptic Painleve equation expressed in terms of the elliptic hypergeometric function 10E9.

This thesis details the concepts behind several bilinear pairings, its implementations and importante. Unlike more tradicional presentations, we focus on the more general case, using the Picard Group defined over hyperelliptic curves, not focusing on particular cases, such as elliptic curves, genus 2 curves, or particular classes of curves.

A generalized bc-system associated to a Hermitian vector bundle over a Riemann surface is introduced in close analogy to the usual rank one case. Some of the geometric analogies to the well-known case are studied. In particular, if there are no zero-modes, the “non-Abelian” theta divisor appears. In the general case where zero-modes exist, it seems to be more difficult to find a natural description. It is discussed in detail why no such system exists in low genus that is on the Riemann sphere and on elliptic curves.

Let X be a smooth connected projective manifold defined over a number field k. We give below a conjectural qualitative description of X(k’) for k’ a sufficiently large number field containing k according to the algebro-geometric positivity/negativity of the cotangent bundle of X. When X is ‘of general type’, our conjecture coincides with Lang’s conjecture claiming the ‘Mordellicity’ of X . In the general case, we decompose X by means of a new fibration, its ‘core map’, the fibres of which are ‘special’, its ‘orbifold base’ being of ‘general type’. The ‘special’ manifolds generalise rational and elliptic curves, and we conjecture them to be ‘potentially dense’. Extending Lang’s (and Vojta’s) conjecture to the orbifold setting, we get the above conjectural qualitative description of X(k’). Even for orbifold curves, our conjecture is open, but is implied by the abc-conjecture.KeywordsLogarithmic pairsOrbifold pairsCampana conjecturesHyperbolicityFaltings theoremKodaira dimensionMSC codes11A45111J9711G3211G3514D1014E0514E2214E9914G0514H2514H3014J2014J9914M2232A2232J2532Q45

We deal with an ant colony based routing model for wireless multi‐hop networks. Our model adopts an elliptic curve equation, which is beneficial to design pheromone dynamics for load balancing and packet delivery robustness. Due to the attribute of an elliptic curve equation, our model prevents the over‐utilization of a specific node, distinctively from conventional ant colony based schemes. Numerical simulations exhibit the characteristics of our model with respect to various parameters.

We study solvability of convolution equations for functions with discrete support in , a special case being functions with support in the integer points. The more general case is of interest for several grids in Euclidean space, like the body-centred and face-centred tessellations of 3-space, as well as for the non-periodic grids that appear in the study of quasicrystals. The theorem of existence of fundamental solutions by de Boor et al is generalized to general discrete supports, using only elementary methods. We also study the asymptotic growth of sequences and arrays using the Fenchel transformation.

Optimizing communication is a key issue in generating efficient SPMD codes in compiling distributed arrays on data parallel languages, such as High Performance Fortran. In HPF, the array distribution may involve alignment and cyclic(k)-distribution such that the enumeration of the local set and the enumeration of the communication set exhibit regular patterns which can be modeled as integer lattices. In the special case of unit-strided alignment, many techniques of the communication set enumeration have been proposed, while in the general case of the non-unit-strided alignment, inspector-like run-time codes are needed to build repeating pattern table or to scan over local elements such that the communication set can be constructed. Unlike other works on this problem of the general alignment and cyclic(k) distribution, our approach derives an algebraic solution for such an integer lattice that models the communication set by using the Smith-Normal-Form analysis, therefore, efficient enumeration of the communication set can be generated. Based on the integer lattice, we also present our algorithm for the SPMD code generation. In our approach, when the parameters are known, the SPMD program can be efficiently constructed without any inspector-like run-time codes.

The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms—including Gallant–Lambert–Vanstone (GLV) and Galbraith–Lin–Scott (GLS) multiplication, as well as higher-dimensional and higher-genus constructions—is to produce a short basis of a certain integer lattice involving the eigenvalues of the endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar coefficients, and the faster the resulting scalar multiplication. Typically, we use the eigenvalues to write down a long basis, which we then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more specialized algorithm. In this work, we use elementary facts about quadratic rings to immediately write down a short basis for the elliptic curve GLV, GLS, GLV+GLS, and Q \mathbb {Q} -curve constructions, and for genus 2 real multiplication constructions. We do not pretend that this represents a significant optimization in scalar multiplication, since the lattice reduction step is always an offline precomputation—but it does give a better insight into the structure of scalar decompositions. In any case, it is always more convenient to use a ready-made short basis than it is to compute a new one.