Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

https://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

The Arithmetic of Elliptic Curves

We discuss the use of elliptic curves in cryptography. In particular, we propose an analogue of the Diffie-Hellmann key exchange protocol which appears to be immune from attacks of the style of Western, Miller, and Adleman. With the current bounds for infeasible attack, it appears to be about 20% faster than the Diffie-Hellmann scheme over GF(p). As computational power grows, this disparity should get rapidly bigger.

/pdf/use-of-elliptic-curves-in-cryptography-credwc2ehh.pdf

Use of Elliptic Curves in Cryptography

In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

/pdf/factoring-polynomials-with-rational-coefficients-4jcm2kwak4.pdf

Factoring Polynomials with Rational Coefficients

/pdf/factoring-polynomials-with-rational-coeficients-3cerm8jvrh.pdf

Factoring polynomials with rational coeficients

Controlling chaos

Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction 2. The basic process 3. Definite quadratic forms 4. Indefinite quadratic forms 5. Binary cubic forms 6. Other forms Chapter III. Theorems of Blichfeldt and Minkowski 1. Introduction 2. Blichfeldt's and Mnowski's theorems 3. Generalisations to non-negative functions 4. Characterisation of lattices 5. Lattice constants 6. A method of Mordell 7. Representation of integers by quadratic forms Chapter IV. Distance functions 1. Introduction 2. General distance-functions 3. Convex sets 4. Distance functions and lattices Chapter V. Mahler's compactness theorem 1. Introduction 2. Linear transformations 3. Convergence of lattices 4. Compactness for lattices 5. Critical lattices 6. Bounded star-bodies 7. Reducibility 8. Convex bodies 9. Speres 10. Applications to diophantine approximation Chapter VI. The theorem of Minkowski-Hlawka 1. Introduction 2. Sublattices of prime index 3. The Minkowski-Hlawka theorem 4. Schmidt's theorems 5. A conjecture of Rogers 6. Unbounded star-bodies Chapter VII. The quotient space 1. Introduction 2. General properties 3. The sum theorem Chapter VIII. Successive minima 1. Introduction 2. Spheres 3. General distance-functions Chapter IX. Packings 1. Introduction 2. Sets with V(/varphi) =n^2/Delta(/varphi) 3. Voronoi's results 4. Preparatory lemmas 5. Fejes Toth's theorem 6. Cylinders 7. Packing of spheres 8. The proudctio of n linear forms Chapter X. Automorphs 1. Introduction 2. Special forms 3. A method of Mordell 4. Existence of automorphs 5. Isolation theorems 6. Applications of isolation 7. An infinity of solutions 8. Local methods Chapter XI. Ihomogeneous problems 1. Introduction 2. Convex sets 3. Transference theorems for convex sets 4. The producti of n linear forms Appendix References Index quotient space. successive minima. Packings. Automorphs. Inhomogeneous problems.

An Introduction to the Geometry of Numbers

An Introduction to Diophantine Approximation

Publisher Summary This chapter presents local-global theorems and their limitations—that is, the information that can be obtained from p-adic considerations. Some of the results extend to algebraic number-fields or in other ways. There is a very close relation between the behavior of rational quadratic forms over Q and over the Qp. The chapter presents theorems on “weak Hasse principle” and “strong Hasse principle.” Strong Hasse principle theorem shows the advantage of the p-adic language both for enunciations and proofs. The chapter also introduces a new classification of lattices—namely, into spinor genera

Rational Quadratic Forms

ABELIAN l -ADIC REPRESENTATIONS AND ELLIPTIC CURVES

Introduction 1 Curves of genus: introduction 2 p-adic numbers 3 The local-global principle for conics 4 Geometry of numbers 5 Local-global principle: conclusion of proof 6 Cubic curves 7 Non-singular cubics: the group law 8 Elliptic curves: canonical form 9 Degenerate laws 10 Reduction 11 The p-adic case 12 Global torsion 13 Finite basis theorem: strategy and comments 14 A 2-isogeny 15 The weak finite basis theorem 16 Remedial mathematics: resultants 17 Heights: finite basis theorem 18 Local-global for genus principle 19 Elements of Galois cohomology 20 Construction of the jacobian 21 Some abstract nonsense 22 Principle homogeneous spaces and Galois cohomology 23 The Tate-Shafarevich group 24 The endomorphism ring 25 Points over finite fields 26 Factorizing using elliptic curves Formulary Further reading Index

J. W. S. Cassels

Papers

An Introduction to the Geometry of Numbers

An Introduction to Diophantine Approximation

Rational Quadratic Forms

ABELIAN l -ADIC REPRESENTATIONS AND ELLIPTIC CURVES

Lectures on elliptic curves