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Public key cryptography is a major interdisciplinary subject with many real-world applications, such as digital signatures. A strong background in the mathematics underlying public key cryptography is essential for a deep understanding of the subject, and this book provides exactly that for students and researchers in mathematics, computer science and electrical engineering. Carefully written to communicate the major ideas and techniques of public key cryptography to a wide readership, this text is enlivened throughout with historical remarks and insightful perspectives on the development of the subject. Numerous examples, proofs and exercises make it suitable as a textbook for an advanced course, as well as for self-study. For more experienced researchers it serves as a convenient reference for many important topics: the Pollard algorithms, Maurer reduction, isogenies, algebraic tori, hyperelliptic curves and many more.

The Mathematics of Secrets takes readers on a tour of the mathematics behind cryptography--the science of sending secret messages. Joshua Holden shows how mathematical principles underpin the ways that different codes and ciphers operate, as he focuses on both code making and code breaking. He discusses the majority of ancient and modern ciphers currently known, beginning by looking at substitution ciphers, built by substituting one letter or block of letters for another. Explaining one of the simplest and historically well-known ciphers, the Caesar cipher, Holden establishes the key mathematical idea behind the cipher and discusses how to introduce flexibility and additional notation. He explores polyalphabetic substitution ciphers, transposition ciphers, including one developed by the Spartans, connections between ciphers and computer encryption, stream ciphers, ciphers involving exponentiation, and public-key ciphers, where the methods used to encrypt messages are public knowledge, and yet, intended recipients are still the only ones who are able to read the message. Only basic mathematics up to high school algebra is needed to understand and enjoy the book.

This text illustrates algorithms and cryptosystems using examples and the open-source computer algebra system of Sage. It enables students to run their own programs and develop a deep and solid understanding of the mechanics of cryptography. The author, a noted educator in the field, covers the methods, algorithms, and applications of modern cryptographic systems. He provides a highly practical learning experience by progressing at a gentle pace, keeping mathematics at a manageable level, and including numerous end-of-chapter exercises.

\"Codes are a part of everyday life, from the ubiquitous Universal Price Code (UPC) to postal zip codes. They need not be intended for secrecy. They generally use groups of letters (sometimes pronounceable code words) or numbers to represent other words or phrases. There is typically no mathematical rule to pair an item with its representation in code. A few more examples will serve to illustrate the range of codes\"-- Provided by publisher.

Electronic communication and financial transactions have assumed massive proportions today. But they come with high risks. Achieving cyber security has become a top priority, and has become one of the most crucial areas of study and research in IT. This book introduces readers to perhaps the most effective tool in achieving a secure environment, i.e. cryptography. This book offers more solved examples than most books on the subject, it includes state of the art topics and discusses the scope of future research. Preface Overview of Cryptography Introduction Goals of Cryptography Classification of Cryptosystem Practically Useful Cryptosystem Cryptanalysis Basic Algebra Group Ring Field Exercise Number Theory Introduction Prime Numbers Cardinality of Primes Extended Euclidean Algorithm Primality Testing Factorization and Algorithms for it Congruences Quadratic Congruence Exponentiation and Logarithm Discrete Logarithm Problem and Algorithms for it Exercise Probability and Perfect Secrecy Basic Concept of Probability Birthday Paradox Perfect Secrecy Vernam One Time Pad Random Number Generation Pseudo-random Number Generator Exercise Complexity Theory Running Time and Size of Input Big-O Notation Types of algorithm Complexity Classes Exercise Classical Cryptosystems Classification of Classical Cryptosystem Block Cipher Stream Cipher Cryptanalysis of Cryptosystems Exercise 　 Block Ciphers Introduction Modes of Operation Padding Design Considerations Data Encryption Standard Advanced Encryption Standard Exercise Hash Function Compression and Hash Functions Hash function for cryptography Random Oracle Model Cryptographic Hash Functions Exercise Public Key Cryptosystem Introduction Diffie-Hellman Key Exchange Protocol RSA Cryptosystem Rabin Cryptosystem ElGamal Cryptosystem Elliptic Curve Cryptosystem Exercises Digital Signature Formal Definitions Attack Goals for Digital Signature Digital Signature in Practice Some Popular Digital Signatures Exercises Research Directions in Cryptography Pairing-Based Cryptography Zero-knowledge Proof System Authenticated Group Key Exchange Attribute-Based Cryptography Homomorphic Encryption Secure Multi-party Computation Secret Sharing Post-Quantum Cryptography Side-Channel Analysis References Index Sahadeo Padhye has a doctorate in Cryptography, and currently working as Associate Professor at Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, India. His research interests include Public key Cryptography, Elliptic Curve Cryptography, Digital Signatures, Lattice Based Cryptography. He has published many research papers in reputed international journals and conferences in Cryptography. Rajeev A Sahu has a doctorate in Cryptography and currently working as a Post-Doctoral Researcher at Université Libre de Bruxelles, Belgium. He has also worked as an Assistant Professor at C.R. Rao Advanced Institute of Mathematics Statistics & Computer Science, Hyderabad, India. His research interests are Identity-Based Cryptography, Elliptic Curve Cryptography, Digital Signature, Searchable Encryption, Post-Quantum Cryptography on which he has published over two dozen research papers in reputed international journals and conferences in Computer Science and Cryptography. Vishal Saraswat received his Ph.D. in Cryptography from University of Minnesota, Minneapolis, USA and has held regular and visiting positions at a variety of institutions, including IIT Jammu, IIT Hyderabad, ISI Kolkata, Univ. of Hyderabad and AIMSCS, Hyderabad. His research interests include anonymity and privacy, searchable encryption, postquantum crypto, and active and passive cryptanalysis, topics on which he has published several papers in reputed international journals and conferences.

This book offers a unique survey of the objects of discrete mathematics known as Boolean bent functions. As these maximal, nonlinear Boolean functions and their generalizations have many theoretical and practical applications in combinatorics, coding theory, and cryptography, the text provides a detailed survey of their main results, presenting a systematic overview of their generalizations and applications, and considering open problems in classification and systematization of bent functions. It provides a detailed survey of bent functions and their main results, presenting a systematic overview of their generalizations and applications; presents a systematic and detailed survey of hundreds of results in the area of highly nonlinear Boolean functions in cryptography. --