Character Sums with Exponential Functions and their Applications

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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.

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Editorial Reviews

From the Publisher
"It is perhaps surprising that such a diverse collections of problems may be unified under one theme. This book should serve as a useful reference for mathematicians interested in such problems as well as a valuable source of open questions for research projects." Bulletin of the American Mathematical Society

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

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Product Details

  • ISBN-13: 9780521642637
  • Publisher: Cambridge University Press
  • Publication date: 9/28/1999
  • Series: Cambridge Tracts in Mathematics Series , #136
  • Pages: 172
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.67 (d)

Table of Contents

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.

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