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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.
Of all the alternatives for encryption and decryption of messages shown here, a vulnerability analysis to cryptographic attacks (cryptoanalysis) is made, this is a security analysis, an important topic on the subject of secure communications. The book is self-contained, that is to say, the necessary tools to address the issues such as fractional calculus are given in the same book and several examples are presented. Moreover, this book includes exercises that are left to the reader. The book is directed to an audience such as professionals in the areas of mathematics, physics and engineering and researchers in general and related areas with a minimum of knowledge in higher mathematics. However, it also contains advanced research topics for people interested in encryption and decryption, observers, synchronization and secure communications areas.
The book is organized as follows. In Chap. 1, a brief overview of the main topics covered is presented giving an introduction to the state of the art on encryption and decryption algorithms, synchronization of chaotic systems, security keys or codes, security analysis such as cryptographic attacks, linear and differential cryptanalysis, in addition to specific attacks for chaotic cryptosystems of type stream cipher. In Chap. 2, some definitions are given about the Lyapunov exponents, stability, and state observers also fractals and synchronization are briefly introduced. Chapter 3 shows the stream and block ciphers and observers, binary representations as well as some conversions from binary to decimal and vice versa, representations of plain text and images in integer bits and ciphers with generalized synchronization. Chapter 4 deals with the study of Liouville systems and cryptography, and a supertwisting observer is addressed as a receiver as well as its vulnerability to cryptanalysis. Chapter 5 presents some basic concepts of state observers, the exponential polynomial observer is used as a receptor, and the receptors are based on properties related to Liouville systems. Chapter 6 shows some basic elements of fractional calculus and some observers. Chapter 7 deals with the implementation of systems with the property of Liouville and fractional order systems used for the encryption and decryption of plain-text and image messages. In Chap. 8, we present robust fractional order state observers as means of encryption and decryption, presenting a security analysis and situations that lead to decryption failures. Finally, in Chap. 9, a new topic is described in secure communications, and we present encryption and decryption algorithms by using state observers that are represented by means of fractional-order chaotic systems with the Atangana-Baleunu fractional derivative. Additionally, the reader will find throughout this material some exercises to strengthen the knowledge acquired.
Introduction
Synchronization of Chaotic Systems
Stream Cyphers and Block Cyphers
Liouvillian Systems and Cryptography
State Observers and Cryptography
Fractional Systems
Fractional-Order Liouvillian Systems and Encryption
Fractional-Order Robust State Observers and Encryption
Secure Communications by Using Atangana-Baleanu Fractional Derivative