Number Theory with Applications to Cryptography

✍ Scribed by Stefano Spezia (editor)

• Publisher: Arcler Press
• Year: 2019
• Tongue: English
• Leaves: 310
• Category: Library

✦ Synopsis

Number Theory with Applications to Cryptography takes into account the application of number theory in the field of cryptography. It comprises elementary methods of Diophantine equations, the basic theorem of arithmetic and the Riemann Zeta function. This book also discusses about Congruences and their use in mock theta functions, Method of Iterative Sliding Window for Shorter Number of Operations in case of Modular Exponentiation and Scalar Multiplication, Discrete log problem, elliptic curves, matrices and public-key cryptography and Implementation of Pollard Rho over binary fields using Brent Cycle Detection Algorithm. It also provides the reader with the significant insights of number theory to the practice of cryptography in order to understand discrete log problem, matrices, elliptic curves and public-key cryptography and the applications of Fibonacci sequence on continued fractions.

✦ Table of Contents

Cover
Title Page
Copyright
DECLARATION
ABOUT THE EDITOR
TABLE OF CONTENTS
List of Contributors
List of Abbreviations
Preface
SECTION I: DIOPHANTINE EQUATIONS
Chapter 1 A Disaggregation Approach for Solving Linear Diophantine Equations
Abstract
Introduction
Lattice and Basis Reduction
Equivalent Modular Equations and Their Lattice Representation
Disaggregation of a System of Equations With Basis Reduction
Conclusion
References
Chapter 2 Diophantine Equations. Elementary Methods
Abstract
Introduction and Main Results
Acknowledgements
References
Chapter 3 Diophantine Equations. Elementary Methods II
Abstract
Introduction and Main Results
Acknowledgements
References
Chapter 4 Almost and Nearly Isosceles Pythagorean Triples
Abstract
Introduction
Almost and Nearly Pythagorean Triples
Almost Isosceles Pythagorean Triple
Acknowledgments
References
Chapter 5 A Public Key Cryptosystem based on Diophantine Equations of Degree Increasing Type
Abstract
Introduction
Review of ASC
Our Cryptosystem
Security Analysis
Sizes of Keys and Cipher Polynomials
Conclusion
Acknowledgements
References
SECTION II: THE RIEMANN ZETA FUNCTION AND THE FUNDAMENTAL THEOREM OF ARITHMETIC
Chapter 6 Hamiltonian for the Zeros of the Riemann Zeta Function
Abstract
Acknowledgements
References
Chapter 7 Fractional Parts and Their Relations to the Values of the Riemann Zeta Function
Abstract
Background
Notation
The Fractional Transform
Main Results
Conclusion
References
SECTION III: CONGRUENCES
Chapter 8 11-Dissection and Modulo 11 Congruences Properties for Partition Generating Function
Abstract
Introduction
Preliminaries
Components and Congruences For M = 11
References
Chapter 9 Effective Congruences for Mock Theta Functions
Abstract
Introduction and Statement of The Results
Nuts and Bolts
Statement of The General Theorem and Its Proof
Acknowledgments
References
Chapter 10 On Integer Solutions of the Cubic Equations Over Certain Fields Zn
Abstract
Introduction
Main Results
References
Chapter 11 Iterative Sliding Window Method for Shorter Number of Operations in Modular Exponentiation and Scalar Multiplication
Abstract
Introduction
Iterative Sliding Window Method (ISWM)
Iterative Recoded Swm (IRSWM)
Conclusion And Future Works
References
SECTION IV: DISCRETE LOG PROBLEM, ELLIPTIC CURVES, MATRICES AND PUBLIC-KEY CRYPTOGRAPHY
Chapter 12 Implementation of Pollard Rho over binary fields using Brent Cycle Detection Algorithm
Abstract
Introduction
Basic Definition
Pollard Rho Algorithm
Modified Pollard Rho
Experimental Results
Conclusion and Further Research
Acknowledgment
References
Chapter 13 Cryptanalysis of a Proposal Based on the Discrete Logarithm Problem Inside Sn
Abstract
Introduction
The Scheme of Doliskani Et al.
Finding Discrete Logarithms In Cyclic Subgroups of SN
Experimental Validation
Conclusions
Author Contributions
References
Chapter 14 Research on Attacking a Special Elliptic Curve Discrete Logarithm Problem
Abstract
Introduction
Preliminary
Partitions of Group Elements
A Group Represented by Disjoint Orbits
A Special Polynomial Construction
Experimental Results
Conclusion
Acknowledgments
References
Chapter 15 Are Matrices Useful in Public-Key Cryptography?
Abstract
Introduction
Circulant Matrices
Security of The Proposed Elgamal Cryptosystem
Is The Elgamal Cryptosystem Over SC(D, Q) Really Useful?
An Algorithm
References
SECTION V: CONTINUED FRACTIONS
Chapter 16 An Application of Fibonacci Sequence on Continued Fractions
Abstract
Introduction
Basic Lemma
Proof of Theorem 1.1
Acknowledgments
References
Chapter 17 On The Quantitative Metric Theory of Continued Fractions in Positive Characteristic
Abstract
Introduction
Quantitative Metrical Theorems
Proofs
References
Chapter 18 Some New Continued Fraction Sequence Convergent to the Somos QuadraticRecurrence Constant
Abstract
Introduction
Estimating γ(1/2)
Estimating γ(1/3)
Acknowledgements
References
Chapter 19 Continued Fractions for Some Transcendental Numbers
Abstract
Introduction
The Main Result
Acknowledgements
References
Index
Back Cover