This book offers the beginning undergraduate student some of the vista of modern mathematics by developing and presenting the tools needed to gain an understanding of the arithmetic of elliptic curves over finite fields and their applications to modern cryptography. This gradual introduction also makes a significant effort to teach students how to produce or discover a proof by presenting mathematics as an exploration, and at the same time, it provides the necessary mathematical underpinnings to investigate the practical and implementation side of elliptic curve cryptography (ECC). Elements of abstract algebra, number theory, and affine and projective geometry are introduced and developed, and their interplay is exploited. Algebra and geometry combine to characterize congruent numbers via rational points on the unit circle, and group law for the set of points on an elliptic curve arises from geometric intuition provided by Bézout's theorem as well as the construction of projective space. The structure of the unit group of the integers modulo a prime explains RSA encryption, Pollard's method of factorization, Diffie–Hellman key exchange, and ElGamal encryption, while the group of points of an elliptic curve over a finite field motivates Lenstra's elliptic curve factorization method and ECC. The only real prerequisite for this book is a course on one-variable calculus; other necessary mathematical topics are introduced on-the-fly. Numerous exercises further guide the exploration.
With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and application
The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography.
Anchored by a comprehensive treatment of the practical aspects of elliptic curve cryptography (ECC), this guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for ...
This open access book systematically explores the statistical characteristics of cryptographic systems, the computational complexity theory of cryptographic algorithms and the mathematical principles behind various encryption and decryption ...
This practical guide to modern encryption breaks down the fundamental mathematical concepts at the heart of cryptography without shying away from meaty discussions of how they work.
This book is meant for those without a strong mathematics background with only just enough math to understand the algorithms given. The book contains a slide presentation, questions and answers, and exercises throughout.
This book provides a compact course in modern cryptography.
Even though the elegant mathematical theory underlying cryptosystems is considerably more involved than for other systems, this text requires the reader to have only an elementary knowledge of basic algebra.
... H.O. CORDES 203 Hochschild cohomology of von Neumann algebras , A. SINCLAIR & R. SMITH 204 Combinatorial and geometric group theory , A.J. DUNCAN , N.D. GILBERT & J. HOWIE ( eds ) 205 Ergodic theory and its connections with harmonic ...
... 2003] , Linear complexity and related complexity measures for sequences, Progress in [730] Cryptology – Indocrypt 2003, Lecture Notes in Comput. Sci., vol. 2904, Springer-Verlag, 2003, 1–17. [NISH 1999] H. NIEDERREITER & I. E. ...