نظریه اعداد با کاربردهایی در رمزنگاری: تاریخ و پژوهش

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. 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 Discusses the application of number theory in the field of cryptography. The book covers elementary methods of Diophantine equations, the basic theorem of arithmetic, the Riemann Zeta function, congruences and their use in mock theta functions, and the method of iterative sliding window for shorter number of operations.