Encryption and Decryption Algorithms for Plain Text and Images using Fractional Calculus

Description

This book offers an alternative for encrypting and decrypting messages using objects called integer and fractional-order estimators or observers, by means of security codes. The authors first establish the class of observers capable of carrying out this work. Then, the type of observers to treat either the integer or fractional order type and their main characteristics is mentioned. The book also presents an essential property of some systems such as Liouville, which is vital for the encryption and decryption of messages in integer and fractional order nonlinear systems by using the synchronization property of chaotic systems. Finally, it addresses some logistic maps such as Mandelbrot sets including Julia and fractal sets, taking advantage of their characteristics to encrypt or recover messages. Preface Contents Notations and Abbreviations List of Figures 1 Introduction 1.1 Chaotic System Synchronization and Encryption Algorithms 1.1.1 Encryption Through Chaotic Systems 1.2 Key or Security Code 1.3 Security Analysis 1.3.1 Cryptographic Attacks (Cryptanalysis) 1.3.2 Differential Cryptanalysis 1.3.3 Linear Cryptanalysis 1.4 Specific Attacks for Stream Cipher-Type Chaotic Cryptosystems 1.4.1 Message Extraction 1.4.2 Parametric Estimation 1.4.3 Brute Force Attacks References 2 Synchronization of Chaotic Systems 2.1 Chaotic Systems 2.1.1 Lyapunov Exponents 2.2 Stability 2.2.1 Nonlinear Systems 2.2.2 Stability and Linearization 2.2.3 Lyapunov's Direct Method 2.3 State Observers 2.3.1 Luenberger Observer 2.4 Fractals and Synchronization References 3 Stream Cyphers and Block Cyphers 3.1 Message and Data Carrier Signals 3.1.1 Decimal and Binary Numbers 3.1.2 Binary to Decimal and Decimal to Binary Conversions 3.1.3 Representation of Plaintext with 8 Integers 3.1.4 Representation of Plain Images with 8 Bit Integers 3.1.5 Data Carrier Signal 3.2 Stream Ciphers and State Observers 3.2.1 Pseudorandom Number Generator 3.2.2 The Luenberger Observer in a Stream Cipher 3.3 Block Ciphers and Observers 3.3.1 Block Cipher References 4 Liouvillian Systems and Cryptography 4.1 Introduction 4.2 Transmitter 4.3 Receiver 4.3.1 Super-Twisting Based Receiver 4.3.2 Proof of Stability 4.3.3 Reconstruction of the States Based Receiver 4.4 Numerical Simulation 4.5 Vulnerability to Cryptanalysis 4.6 Concluding Remarks References 5 State Observers and Cryptography 5.1 Introduction 5.2 Encryption 5.2.1 Generating Pseudo-Random Numbers 5.2.2 Encryption Algorithm 5.3 Data Recovery 5.3.1 Exponential Polynomial Receiver 5.3.2 Stability 5.3.3 Liouvillian System Properties Based Receiver 5.4 Numerical Simulation 5.5 Concluding Remarks References 6 Fractional Systems 6.1 Gamma Function 6.1.1 Some Properties of the Gamma Function 6.2 Beta Function 6.3 Euler's Number and Its Relation to the Gamma Function 6.4 Miscellaneous Examples 6.5 Fractional-Order Differential Equations 6.5.1 Laplace Transform of Fractional-Order Functions 6.5.2 Solution of FODE by Means of the Laplace Transform 6.6 Fractional Dynamical System 6.6.1 Commensurate Fractional-Order Systems 6.6.2 Incommensurate Fractional-Order Systems References 7 Fractional-Order Liouvillian Systems and Encryption 7.1 Introduction 7.2 Preliminaries 7.3 Fractional Derivative Numerical Estimation 7.4 Encryption Algorithm 7.5 Decryption 7.6 Numerical Results 7.7 Security Analysis 7.8 Concluding Remarks References 8 Fractional-Order Robust State Observers and Encryption 8.1 Introduction 8.2 Preliminaries 8.3 Encryption Algorithm 8.4 Receiver and Decryption 8.5 Numerical Results 8.5.1 Situations that Lead to Decryption Failure 8.6 Security Analysis 8.7 Concluding Remarks References 9 Secure Communications by Using Atangana-Baleanu Fractional Derivative 9.1 Introduction 9.2 Preliminaries 9.3 Encryption Algorithm 9.4 Receiver and Decryption 9.5 Numerical Results 9.6 Security Analysis 9.7 Concluding Remarks References Index