Technical Books

This note explains the following topics, Arithmetic of Elliptic Curves, Classical EllipticCurve Cryptography, Efficient Implementation, Introduction to Pairing, PairingBased Cryptography, Sample ApplicationECDSA Batch Verification.

Elliptic curves belong to the most fundamental objects in mathematics and connect many different research areas such as number theory, algebraic geometry and complex analysis. Their definition and basic properties can be stated in an elementary way, Roughly speaking, an elliptic curve is the set of solutions to a cubic equation in two variables over a field. This PDF covers the following topics related to Elliptic Curves, Analytic theory of elliptic curves, Elliptic integrals, The topology of elliptic curves, Elliptic curves as complex tori, Complex tori as elliptic curves, Geometric form of the group law, Abels theorem, The jinvariant, The valence formula, Geometry of elliptic curves, Affine and projective varieties, Smoothness and tangent lines, Intersection theory for plane curves, The group law on elliptic curves, Abels theorem and RiemannRoch, Weierstrass normal forms, The jinvariant, Arithmetic of elliptic curves, Rational points on elliptic curves, Reduction modulo primes and torsion points, An intermezzo on group cohomology, The weak MordellWeil theorem, Heights and the MordellWeil theorem.

This note covers fermats method of infinite descent, Some remarks on algebraic curves, Weierstrass equations, The group law, Isogeny, The invariant differential, Elliptic curves over finite fields, Formal groups, Elliptic curves over local fields, Elliptic curves over number fields, Kummer theory, Mordell weil theorem, Heights, Dual isogenies and weil pairing, Galois cohomology, Descent by cyclic isogeny.

This note describes the following topics, Solving cubic equations in two variables, Group law on the cubic curve, Theta functions, Rank two vector bundles on elliptic curves.

This note contains below topics, Background on curves and function fields, Elliptic curves over function fields, Surfaces and the Tate conjecture.

The goal of these notes is to introduce orbifolds and stacks through the example of elliptic curves. This note contains below topics, Elliptic Curves and the Moduli Problem, Families of Elliptic Curves and the Universal Curve, The Orbifold and Modular Forms.

This note covers the following topics, Sets, Functions and Relations, Proofs and Induction, Number Theory, Counting, Probability, Logic, Graphs, Finite Automata.

This note covers the following topics, Logic, Asymptotic Notation, Convex Functions and Jensens Inequality, Basic Number Theory, Counting, Binomial coefficients, Graphs and Digraphs, Finite Probability Space, Finite Markov Chains.

This lecture note describes the following topics, Sets and Notation, Induction, Proof Techniques, Divisibility, Prime Numbers, Modular Arithmetic, Relations and Functions, Mathematical Logic, Counting, Binomial Coefficients, The InclusionExclusion Principle, The Pigeonhole Principle, Asymptotic Notation, Graphs, Trees, Planar Graphs.

The aim of this note is to introduce fundamental concepts and techniques in set theory in preparation for its many applications in computer science. Topics covered includes, Mathematical argument, Sets and Logic, Relations and functions, Constructions on sets, Wellfounded induction.

The contents include, Mathematical Logic, Relations, Algebraic structures, Recurrence Relation, Graph Theory.

This page covers the following topics related to Discrete Mathematics, Logic and Sets, Relations and Functions, the Natural Numbers, Division and Factorization, Languages, Finite State Machines, Finite State Automata, Turing Machines, Groups and Modulo Arithmetic, Introduction to Coding Theory, Group Codes, Public Key Cryptography, Principle of Inclusionexclusion, Generating Functions, Number of Solutions of a Linear Equation, Recurrence Relations, Graphs, Weighted Graphs, Search Algorithms, Digraphs.

This PDF covers the following topics related to Discrete Mathematics, Introduction, Sets, Functions, Counting, Relations, Sequences, Modular Arithmetic, Asymptotic Notation, Orders.

This note explains the following topics related to Discrete Mathematics, Mathematical Logic, Relations, Algebraic structures, Elementary Combinatorics, Recurrence Relation, Graph Theory.

This note explains the following topics, introduction to preliminaries, Counting, Sequences, Symbolic logic and proofs, Graph theory, Additional topics.