We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over GF(2'). We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.

https://www.ams.org/mcom/1987-48-177/S0025-5718-1987-0866109-5/S0025-5718-1987-0866109-5.pdf

Elliptic curve cryptosystems

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

We introduce a short signature scheme based on the Computational Diffie-Hellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a low-bandwidth channel.

/pdf/short-signatures-from-the-weil-pairing-5ajupg0cpi.pdf

Short Signatures from the Weil Pairing

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.

Algebraic Number Theory

1 Absolute Values.- 2 Proper Sets of Absolute Values. Divisors and Units.- 3 Heights.- 4 Geometric Properties of Heights.- 5 Heights on Abelian Varieties.- 6 The Mordell-Weil Theorem.- 7 The Thue-Siegel-Roth Theorem.- 8 Siegel's Theorem and Integral Points.- 9 Hilbert's Irreducibility Theorem.- 10 Weil Functions and Neron Divisors.- 11 Neron Functions on Abelian Varieties.- 12 Algebraic Families of Neron Functions.- 13 Neron Functions Over the Complex Numbers.- Review of S. Lang's Diophantine Geometry, by L. J. Mordell.- Review of L. J. Mordell's Diophantine Equations, by S. Lang.

Fundamentals of Diophantine Geometry

This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations. Although the book grew out of the author's earlier book "Differential and Riemannian Manifolds", the focus has now changed from the general theory of manifolds to general differential geometry, and includes new chapters on Jacobi lifts, tensorial splitting of the double tangent bundle, curvature and the variation formula, a generalization of the Cartan-Hadamard theorem, the semiparallelogram law of Bruhat-Tits and its equivalence with seminegative curvature and the exponential map distance increasing property, a major example of seminegative curvature (the space of positive definite symmetric real matrices), automorphisms and symmetries, and immersions and submersions. These are all covered for infinite-dimensional manifolds, modeled on Banach and Hilbert Spaces, at no cost in complications, and some gain in the elegance of the proofs. In the finite-dimensional case, differential forms of top degree are discussed, leading to Stokes' theorem (even for manifolds with singular boundary), and several of its applications to the differential or Riemannian case. Basic formulas concerning the Laplacian are given, exhibiting several of its features in immersions and submersions.

Fundamentals of differential geometry

This book is meant as a text for a first-year graduate course in analysis. In a sense, the subject matter covers the same topics as elementary calculus - linear algebra, differentiation, integration - but treated in a manner suitable for people who will be using it in further mathematical investigations. The book begins with point-set topology, essential for all analysis. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces. The book then turns to the subject of integration and measure. After a general introduction, it covers duality and representation theorems, some applications (such as Dirac sequences and Fourier transforms), integration and measures on locally compact spaces, the Riemann-Stjeltes integral, distributions, and integration on locally compact groups. Part four deals with differential calculus (with values in a Banach space). The next part deals with functional analysis. It includes several major spectral theorems of analysis, showing how one can extend to infinite dimensions certain results from finite-dimensional linear algebra; a discussion of compact and Fredholm operators; and spectral theorems for Hermitian operators. The final part, on global analysis, provides an introduction to differentiable manifolds. The text includes worked examples and numerous exercises, which should be viewed as an integral part of the book. The organization of the book avoids long chains of logical interdependence, so that chapters are as independent as possible. This allows a course using the book to omit material from some chapters without compromising the exposition of material from later chapters.

Serge Lang

Papers

Algebraic Number Theory

Fundamentals of Diophantine Geometry

Fundamentals of differential geometry

Real and Functional Analysis

Number of points of varieties in finite fields.