© 2003

Cryptographic Applications of Analytic Number Theory

Complexity Lower Bounds and Pseudorandomness

  • Igor Shparlinski

Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 22)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Introduction

    1. Igor Shparlinski
      Pages 1-14
  3. Preliminaries

    1. Front Matter
      Pages 15-15
    2. Igor Shparlinski
      Pages 17-26
    3. Igor Shparlinski
      Pages 27-36
    4. Igor Shparlinski
      Pages 37-60
    5. Igor Shparlinski
      Pages 61-65
    6. Igor Shparlinski
      Pages 67-81
    7. Igor Shparlinski
      Pages 83-102
    8. Igor Shparlinski
      Pages 103-106
  4. Approximation and Complexity of the Discrete Logarithm

    1. Front Matter
      Pages 107-107
    2. Igor Shparlinski
      Pages 109-122
  5. Approximation and Complexity of the Diffie—Hellman Secret Key

  6. Other Cryptographic Constructions

    1. Front Matter
      Pages 195-195

About this book


The book introduces new ways of using analytic number theory in cryptography and related areas, such as complexity theory and pseudorandom number generation.

Key topics and features:

- various lower bounds on the complexity of some number theoretic and cryptographic problems, associated with classical schemes such as RSA, Diffie-Hellman, DSA as well as with relatively new schemes like XTR and NTRU

- a series of very recent results about certain important characteristics (period, distribution, linear complexity) of several commonly used pseudorandom number generators, such as the RSA generator, Blum-Blum-Shub generator, Naor-Reingold generator, inversive generator, and others

- one of the principal tools is bounds of exponential sums, which are combined with other number theoretic methods such as lattice reduction and sieving

- a number of open problems of different level of difficulty and proposals for further research

- an extensive and up-to-date bibliography

Cryptographers and number theorists will find this book useful. The former can learn about new number theoretic techniques which have proved to be invaluable cryptographic tools, the latter about new challenging areas of applications of their skills.


Cryptography Digital Signature Algorithm Nonce Sage complexity complexity theory computer science finite field number theory

Editors and affiliations

  • Igor Shparlinski
    • 1
  1. 1.Department of ComputingMacquarie UniversityAustralia

Bibliographic information

  • Book Title Cryptographic Applications of Analytic Number Theory
  • Book Subtitle Complexity Lower Bounds and Pseudorandomness
  • Authors Igor Shparlinski
  • Series Title Progress in Computer Science and Applied Logic
  • DOI
  • Copyright Information Birkhäuser Basel 2003
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-7643-6654-4
  • Softcover ISBN 978-3-0348-9415-9
  • eBook ISBN 978-3-0348-8037-4
  • Edition Number 1
  • Number of Pages IX, 414
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
    Applications of Mathematics
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


From the reviews:

“Igor Shparlinski is a very prolific mathematician and computer scientist … . The book is written at a very high level, suitable for graduate students and researchers in computer science and mathematics. … book has a unique perspective, and is not really comparable to other books in the area. … book contains many deep results, and the mathematically-sophisticated reader can find much that is novel. … this is an impressive work that will be of significant interest to researchers in cryptography and algorithmic number theory.” (Jeffrey Shallit, SIGACT News, Vol. 41 (3), September, 2010)