Progress in Computer Science and Applied Logic

# Number Theoretic Methods in Cryptography

## Complexity lower bounds

Authors: Shparlinski, Igor

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The book introduces new techniques which imply rigorous lower bounds on the complexity of some number theoretic and cryptographic problems. These methods and techniques are based on bounds of character sums and numbers of solutions of some polynomial equations over finite fields and residue rings. It also contains a number of open problems and proposals for further research. We obtain several lower bounds, exponential in terms of logp, on the de­ grees and orders of • polynomials; • algebraic functions; • Boolean functions; • linear recurring sequences; coinciding with values of the discrete logarithm modulo a prime p at suf­ ficiently many points (the number of points can be as small as pI/He). These functions are considered over the residue ring modulo p and over the residue ring modulo an arbitrary divisor d of p - 1. The case of d = 2 is of special interest since it corresponds to the representation of the right­ most bit of the discrete logarithm and defines whether the argument is a quadratic residue. We also obtain non-trivial upper bounds on the de­ gree, sensitivity and Fourier coefficients of Boolean functions on bits of x deciding whether x is a quadratic residue. These results are used to obtain lower bounds on the parallel arithmetic and Boolean complexity of computing the discrete logarithm. For example, we prove that any unbounded fan-in Boolean circuit. of sublogarithmic depth computing the discrete logarithm modulo p must be of superpolynomial size.

Reviews

"This volume gives a thorough treatment of the complexity of the discrete logarithm problem in a prime field, as well as related problems. The final chapter on further directions gives an interesting selection of problems."

--Zentralblatt Math

• Introduction

Shparlinski, Igor

Pages 3-12

30,19 €
• Basic Notation and Definitions

Shparlinski, Igor

Pages 13-18

30,19 €
• Auxiliary Results

Shparlinski, Igor

Pages 19-36

30,19 €
• Approximation of the Discrete Logarithm Modulo p

Shparlinski, Igor

Pages 39-47

30,19 €
• Approximation of the Discrete Logarithm Modulo p - 1

Shparlinski, Igor

Pages 49-52

30,19 €

eBook 74,96 €
price for Spain (gross)
• ISBN 978-3-0348-8664-2
• Digitally watermarked, DRM-free
• Included format: PDF
• ebooks can be used on all reading devices
Hardcover 135,19 €
price for Spain (gross)
valid through March 31, 2018
• ISBN 978-3-7643-5888-4
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.
• The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 93,55 €
price for Spain (gross)
• ISBN 978-3-0348-9723-5
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.
• The final prices may differ from the prices shown due to specifics of VAT rules

## Bibliographic Information

Bibliographic Information
Book Title
Number Theoretic Methods in Cryptography
Book Subtitle
Complexity lower bounds
Authors
Series Title
Progress in Computer Science and Applied Logic
Series Volume
17
1999
Publisher
Birkhäuser Basel
Springer Basel AG
eBook ISBN
978-3-0348-8664-2
DOI
10.1007/978-3-0348-8664-2
Hardcover ISBN
978-3-7643-5888-4
Softcover ISBN
978-3-0348-9723-5
Series ISSN
2297-0576
Edition Number
1
Number of Pages
IX, 182
Topics