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Mathematics and Cryptography

Authored by: David O. Novick

Circuits and Systems for Security and Privacy

Print publication date:  May  2016
Online publication date:  May  2016

Print ISBN: 9781482236880
eBook ISBN: 9781482236897
Adobe ISBN:

10.1201/b19499-3

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Abstract

2.1 Introduction

2.1.1 Mathematics: The Toolchest of Modern Cryptography

2.1.1.1 History: From old tools to new

2.1.1.2 A treasure trove of techniques

2.1.1.3 Purpose of this chapter

2.1.2 Notation Summary

2.2 Probability Theory

2.2.1 Introduction

2.2.2 Basic Concepts

2.2.3 The Birthday Problem

2.3 Combinatorics

2.3.1 Introduction

2.3.2 Permutations

2.3.2.1 Representations

2.3.2.2 Operations on permutations

2.3.2.3 Properties of permutations

2.3.2.4 Subset permutations

2.3.3 Binomial Coefficients

2.3.4 Recurrence Relations

2.4 Number Theory

2.4.1 Introduction

2.4.2 Greatest Common Divisor

2.4.2.1 Euclid’s algorithm

2.4.3 Modular Arithmetic

2.4.4 The Chinese Remainder Theorem

2.4.5 Quadratic Residues and Quadratic Reciprocity

2.4.6 The Prime Numbers

2.4.6.1 Distribution of the primes

2.4.6.2 Primality testing

2.4.7 Euler’s Totient Function

2.4.8 Fermat’s Little Theorem

2.4.9 Euler’s Theorem

2.5 Efficient Integer Arithmetic

2.5.1 Introduction

2.5.2 Montgomery Multiplication

2.5.3 Barrett Reduction

2.5.4 Karatsuba Multiplication

2.6 Abstract Algebra

2.6.1 Introduction

2.6.2 Algebraic Structures: An Overview

2.6.3 Groups

2.6.3.1 Group structure and operations

2.6.3.2 Common properties of groups

2.6.3.3 Examples of groups

2.6.3.4 The discrete logarithm problem

2.6.4 Rings

2.6.4.1 Ring structure and operations

2.6.4.2 Examples of rings

2.6.4.3 Ideals

2.6.5 Vector Spaces

2.6.5.1 Vector space structure and operations

2.6.5.2 Examples of vector spaces

2.6.6 Fields

2.6.6.1 Field structure and operations

2.6.6.2 Examples of fields

2.6.6.3 Finite (Galois) fields

2.6.6.4 Finite field elements when degree d > 1

2.6.6.5 Finite field use in AES

2.7 Elliptic Curves

2.7.1 Introduction

2.7.2 Elliptic Curves

2.7.3 Point Representation

2.7.3.1 Projective coordinates

2.7.3.2 Jacobian coordinates

2.7.3.3 Other coordinate systems

2.7.4 Point Addition and Doubling

2.8 Conclusion

2.9 Appendix A: Symbols and Notation

2.10 Appendix B: References

2.10.1 General

2.10.2 Probability Theory

2.10.3 Information Theory

2.10.4 Combinatorics

2.10.5 Number Theory

2.10.6 Efficient Modular Integer Arithmetic

2.10.7 Abstract Algebra

2.10.8 Group Theory

2.10.9 Algebraic Geometry

2.10.10 Elliptic Curves

2.10.11 Lattice Theory

2.10.12 Computational Complexity

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