Character Sums with Exponential Functions and their Applications

A treatment of a lively topic from number theory for graduate students and researchers.

Sergei Konyagin (Author), Igor Shparlinski (Author)

9780521642637, Cambridge University Press

Hardback, published 2 September 1999

172 pages
22.9 x 15.2 x 1.1 cm, 0.4 kg

"...this is a work that should be present in any decent mathematical library." Mathematical Reviews

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 character sums
4. Bounds of short character sums
5. Bounds of character sums for almost all moduli
6. Bounds of Gaussian sums
Part III. Multiplicative Translations of Sets: 7. Multiplicative translations of subgroups of F*p
8. Multiplicative translations of arbitrary sets modulo p
Part IV. Applications to Algebraic Number Fields: 9 Representatives of residue classes
10. Cyclotomic fields and Gaussian periods
Part V. Applications to Pseudo-random Number Generators: 11. Prediction of pseudo-random number generators
12. Congruential pseudo-random number generators
Part VI. Applications to Finite Fields: 13. Small mth roots modulo p
14. Supersingular hyperelliptic curves
15. Distribution of powers of primitive roots
16. Difference sets in Vp
17. Dimension of BCH codes
18. An enumeration problem in finite fields.

Subject Areas: Combinatorics & graph theory [PBV]