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Number Theoretic Methods in Cryptography : Complexity lower bounds - Igor Shparlinski

Number Theoretic Methods in Cryptography

Complexity lower bounds

Hardcover

Published: 15th February 1999
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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.

"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

Preface
Acknowledgments
Preliminaries
Introduction
Basic Notation and Definitions
Auxiliary Results
Approximation and Complexity of the Discrete Logarithm
Approximation of the Discrete Logarithm Modulo p
Approximation of the Discrete Logarithm Modulo p - 1
Approximation of the Discrete Logarithm by Boolean Functions
Approximation of the Discrete Logarithm by Real and Complex Polynomials
Complexity of Breaking the Diffie-Hellman Cryptosystem
Polynomial Approximation and Arithmetic Complexity of the Diffie-Hellman Key
Boolean Complexity of the Diffie-Hellman Key
Other Applications
Trade-off between the Boolean and Arithmetic Depths of Modulo p Functions
Special Polynomials and Boolean Functions
RSA and Blum-Blum-Shub Generators of Pseudo-Random Numbers
Concluding Remarks
Generalizations and Open Questions
Further Directions
Bibliography
Index
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783764358884
ISBN-10: 3764358882
Series: Progress in Computer Science and Applied Logic
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 182
Published: 15th February 1999
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 1.0