Cryptography for secure encryption (Q2139905)

This book aims to provide strong encryption using cryptography. The book also relates to how mathematical concepts are used to develop the essential components of cryptography. As we all know, mathematical foundations are as crucial as cryptography. This textbook assumes that the reader has prior knowledge of calculus and linear algebra. This book can be used for a course in cryptography in a master's degree program in mathematics, computer science, or cyber security. It can also be used in a senior-level undergraduate course in cryptography for mathematics or computer science majors. This book is divided into two parts. In the first part of the book (Chapters 2--7), the author introduces concepts from probability theory, information theory, complexity theory, modern algebra, and number theory that will be used later in the study of cryptography. On the other hand, in the second part of the book (Chapters 8--14), the author develops the basic notions of cryptography. The author outlines the contents of the book's second part in detail. Chapter 8 presents the major symmetric key cryptosystems, including the simple substitution cryptosystem, the affine cipher, the Hill \(2\times 2\) cipher, the Vigenere cipher, the Vernam cipher, and the stream cipher. He discusses Feistel-type block ciphers such as DES and AES. Cryptanalysis methods such as frequency analysis and the Kasiski method are also given. In Chapter 9, the author introduces public key cryptography, including the two most important public-key cryptosystems: RSA and ElGamal. He discusses standard attacks on these cryptosystems. In Chapter 10, he investigates digital signature schemes and hash functions. In Chapter 11, he considers the construction of bit generators for use in stream ciphers. The author gives a practical definition of the ``randomness'' of a sequence of bits (the next-bit test) and shows that under the discrete logarithm assumption, the Blum-Micali bit generator is pseudorandom. Moreover, under the quadratic residue assumption, he proves that the Blum-Blum-Shub bit generator is pseudorandom. In Chapter 12, the author addresses the problem of the distribution of keys and introduces the Diffie-Hellman key exchange protocol (DHKEP), and discusses standard attacks on the DHKEP, including the man-in-the-middle attack, baby-step/giant-step, and index calculus. Index calculus is an attack tailored to the choice of group \(G = U (Z_p)\), \(p\) prime, in the DHKEP. In Chapter 13, the author introduces elliptic curves and the elliptic curve group. An elliptic curve over a field \(K\) is the set of points satisfying an equation of the form \(y^2 = x^3 + ax + b\), where the curve is smooth, that is, the cubic has non-zero elliptic discriminant. The set of points on an elliptic curve, together with the point at infinity, is endowed with a binary operation (point addition) to yield the elliptic curve group \(E(K)\). A cyclic subgroup of \(E(K)\) is used in the Diffie-Hellman key exchange protocol in place of \(U (Z_p)\) to define the elliptic curve key exchange protocol (ECKEP). The ECKEP is more secure than the ordinary Diffie-Hellman protocol since the index calculus attack cannot be applied to an elliptic curve group. Chapter 14 considers the case where the curve \(y^2 = x^3 + ax + b\) is not smooth. It explores the connection between the group of points on such curves (the non-singular points \(E_{ns}(K)\)) and another group of points \(G_c(K)\), which generalizes the circle group. This is an active area of research that may provide insight into the nature of point addition in the smooth case. Each chapter contains a set of exercises of various degrees of difficulty that help summarize and review the main ideas of the chapter. The author finds that cryptographic algorithms and protocols provide good programming problems. In most of the places in the text, the author includes the GAP code for a better understanding of the subject in practical use. This book is self-contained and valuable to all audiences interested in cyber security subject.