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Arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. This book presents an introduction to the discipline of arithmetic dynamics.
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This book is designed to provide a path for the reader into an amalgamation oftwo venerable areas ofmathematics, Dynamical Systems and Number Theory. Many of the motivating theorems and conjectures in the new subject of Arithmetic Dynamics may be viewed as the transposition ofclassical results in the theory ofDiophantine equations to the setting of discrete dynamical systems, especially to the iteration theory ofmaps on the projective line and other algebraic varieties. Although there is no precise dictionary connecting the two areas, the reader will gain a flavor of the correspondence from the following associations: Diophantine Equations Dynamical Systems rational and integral rational and integral points on varieties points in orbits torsion points on periodic and preperiodic abelian varieties points ofrational maps There are a variety of topics covered in this volume, but inevitably the choice reflects the author's tastes and interests. Many related areas that also fall under the heading ofarithmetic or algebraic dynamics have been omitted in order to keep the book to a manageable length. A brief list of some of these omitted topics may be found in the introduction. Online Resources The reader will find additonal material, references and errata at http://www. math. brown. ectu/-jhs/ADSHome. html Acknowledgments The author has consulted a great many sources in writing this book. Every attempt has been made to give proper attribution for all but the most standard results.
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ThecreationofpublickeycryptographybyDi?eandHellmanin1976andthe subsequent invention of the RSA public key cryptosystem by Rivest, Shamir, and Adleman in 1978 are watershed events in the long history of secret c- munications. It is hard to overestimate the importance of public key cr- tosystems and their associated digital signature schemes in the modern world of computers and the Internet. This book provides an introduction to the theory of public key cryptography and to the mathematical ideas underlying that theory. Public key cryptography draws on many areas of mathematics, including number theory, abstract algebra, probability, and information theory. Each of these topics is introduced and developed in su?cient detail so that this book provides a self-contained course for the beginning student. The only prerequisite is a ?rst course in linear algebra. On the other hand, students with stronger mathematical backgrounds can move directly to cryptographic applications and still have time for advanced topics such as elliptic curve pairings and lattice-reduction algorithms. Amongthemanyfacetsofmoderncryptography,thisbookchoosestoc- centrate primarily on public key cryptosystems and digital signature schemes. This allows for an in-depth development of the necessary mathematics - quired for both the construction of these schemes and an analysis of their security. The reader who masters the material in this book will not only be well prepared for further study in cryptography, but will have acquired a real understanding of the underlying mathematical principles on which modern cryptography is based.