Character Sums with Exponential Functions and their Applications [Hardcover]

A treatment of a lively topic from number theory for graduate students and researchers.The theme of this book is the study of the distribution of integer powers modulo a prime number. It provides numerous new links between number theory and computer science as well as other areas of mathematics. Possible applications include (but are not limited to) complexity theory, random number generation, cryptography, and coding theory. The main method discussed is based on bounds of exponential sums and the book contains many estimates of these, including new estimates of classical Gaussian sums. It also contains many open questions and proposals for further research.The theme of this book is the study of the distribution of integer powers modulo a prime number. It provides numerous new links between number theory and computer science as well as other areas of mathematics. Possible applications include (but are not limited to) complexity theory, random number generation, cryptography, and coding theory. The main method discussed is based on bounds of exponential sums and the book contains many estimates of these, including new estimates of classical Gaussian sums. It also contains many open questions and proposals for further research.The theme of this book is the study of the distribution of integer powers modulo a prime number. It provides numerous new, sometimes quite unexpected, links between number theory and computer science as well as to other areas of mathematics. Possible applications include (but are not limited to) complexity theory, random number generation, cryptography, and coding theory. The main method discussed is based on bounds of exponential sums. Accordingly, the book contains many estimates of such sums, including new estimates of classical Gaussian sums. It also contains many open questions and proposals for further research.Part I. Preliminaries: 1. Introduction; 2. Notation and auxiliary results; Part II. Bounds of Character Sums: 3. Bounds of long chl