The Geometry of Elliptic Curves

Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences generalize a theorem of Olson [9] ( K = Q) and give divisibility conditions for the discriminant and the coefficients of E related with the paper of Zimmer [13] as well as diophantine equations related with Fermat's equation that are discussed for K Q and K a function field. We are interested in elliptic curves over global fields K (i.e. : K is a finite number field or K is a function field of one variable over a finite field) and especially in the torsion group of E(K), where E(K) is the group of K-rational points of E. It is well known that E(K) is finitely generated, it is conjectured that if K is a number field then the order of the torsion group of E(K) is bounded by some number depending only on K (cf. Demjanenko [1]). In any case in order to handle with E(K) the first step is to determine the torsion group. In principle this is not so difficult; if one uses the results of Lutz [6] and Zimmer [13], one sees immediately that for every E there exist points of q-power-order only for a finite number of primes q, as the equations for points of order q are known (in principle) one has only t ~ test what orders really occur. But as the computational work grows very rapidly with q it is usefull to look for sharper necessary conditions, and this shall be done in this paper.

The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve – which we refer to as "inter-universal Teichmüller theory" – by applying the theory of semi-graphs of anabelioids , Frobenioids , the étale theta function , and log-shells developed in earlier papers by the author. We begin by fixing what we call "initial \Theta -data" , which consists of an elliptic curve E_F over a number field F , and a prime number l\ge 5 , as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite étale coverings to the once-punctured elliptic curve X_F determined by E_F . These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring \mathbb{F}_l=\mathbb{Z}/l\mathbb{Z} acting on the l -torsion points of the elliptic curve. We then construct " \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters" associated to the given \Theta -data. These \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field – which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field – are, in some sense, "dismantled" or "disentangled" from one another. All \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters are isomorphic to one another, but may also be related to one another by means of a " \Theta -link" , which relates certain Frobenioid-theoretic portions of one \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater to another in a fashion that is not compatible with the respective conventional ring/scheme theory structures . In particular, it is a highly nontrivial problem to relate the ring structures on either side of the \Theta -link to one another. This will be achieved, up to certain "relatively mild indeterminacies" , in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an "alien ring structure" [associated, say, to the domain of the \Theta -link] in terms of a given ring structure [associated, say, to the codomain of the \Theta -link] will be applied in the final paper of the series to obtain results in diophantine geometry . Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a p -adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

Let${\mathbb M}$be an affine variety equipped with a foliation, both defined over a number field${\mathbb K}$. For an algebraic$V\subset {\mathbb M}$over${\mathbb K}$, write$\delta _{V}$for the maximum of the degree and log-height ofV. Write$\Sigma _{V}$for the points where the leaves intersectVimproperly. Fix a compact subset${\mathcal B}$of a leaf${\mathcal L}$. We prove effective bounds on the geometry of the intersection${\mathcal B}\cap V$. In particular, when$\operatorname {codim} V=\dim {\mathcal L}$we prove that$\#({\mathcal B}\cap V)$is bounded by a polynomial in$\delta _{V}$and$\log \operatorname {dist}^{-1}({\mathcal B},\Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of${\mathcal B}\cap V$by an algebraic map$\Phi $. For instance, under suitable conditions we show that$\Phi ({\mathcal B}\cap V)$contains at most$\operatorname {poly}(g,h)$algebraic points of log-heighthand degreeg.We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections$P,Q$of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever$P,Q$are simultaneously torsion their order of torsion is bounded effectively by a polynomial in$\delta _{P},\delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given$V\subset {\mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in$\delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.

We show that finding an efficiently computable injective homomorphism from the XTR subgroup into the group of points over GF(p2) of a particular type of supersingular elliptic curve is at least as hard as solving the Diffie–Hellman problem in the XTR subgroup. This provides strong evidence for a negative answer to the question posed by Vanstone and Menezes at the Crypto 2000 Rump Session on the possibility of efficiently inverting the MOV embedding into the XTR subgroup. As a side result we show that the Decision Diffie–Hellman problem in the group of points on this type of supersingular elliptic curves is efficiently computable, which provides an example of a group where the Decision Diffie–Hellman problem is simple, while the Diffie–Hellman and discrete logarithm problems are presumably not. So-called distortion maps on groups of points on elliptic curves that play an important role in our cryptanalysis also lead to cryptographic applications of independent interest. These applications are an improvement of Joux’s one round protocol for tripartite Diffie–Hellman key exchange and a non-refutable digital signature scheme that supports escrowable encryption. We also discuss the applicability of our methods to general elliptic curves defined over finite fields which includes a classification of elliptic curve groups where distortion maps exist.

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.

Elliptic curves have been intensively studied in algebraic geometry and number theory. Recently, they have been used in devising efficient algorithms for factoring integers [8, 9], for primality proving [2, 11, 12] and for the construction of pseudorAndom bit generators [4]. In Chapter 8 we study how elliptic curves can be used in constructing efficient and secure cryptosystems, while in Chapter 10 we will discuss how good error-correcting codes may be obtained by using elliptic curves.

We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a sextic CM-field $K$ containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.

It is somnewhat surprising that the systematic evaluation of the heights of rational points on a curve and on its jacobian variety and particularly of their relation to each other should yield any new information. Nonetheless this appears to be the case and the result is described in this article. Although the main theorem is not even a special case of the very fascinating conjecture of Mordell, still it is an estimate that already reveals that rational points on curves of genus at least 2 are much harder to come by than on curves of genus 0 or 1. It is a quantitative limitation on the heights of such points which is well-known to be false in the case of genus 0 or 1. Incidentally, there is a good explanation why an estimate of this type can be obtained so cheaply, whereas Mordell's conjecture itself could not: namely, results obtained by our methods will more or less automatically apply to the analogous funetion case [where the ground field is a function field in one variable over a finite field, rather than an algebraic number field]. And in this case, unless further restrictions are imposed, there are curves of any genus with an infinite number of rational poilnts whose heights increase exactly at the rate which we will find. Let k be an algebraic number field of finite degree over Q. Let C be a non-singular projective curve over k of genus g at least 2. Mordell's conjecture asserts that the set of k-rational points on C is finite. Now suppose

1: Some Topics in Elementary Number Theory. 2: Finite Fields and Quadratic Residues. 3: Cryptography. 4: Public Key. 5: Primality and Factoring. 6: Elliptic Curves.

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end.

A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.