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Number Theoretic Methods in Cryptography : Complexity lower bounds / by Igor Shparlinski
(Progress in Computer Science and Applied Logic ; 17)

データ種別 電子ブック
出版者 Basel : Birkhäuser Basel : Imprint: Birkhäuser
出版年 1999
本文言語 英語
大きさ IX, 182 p : online resource

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EB0089058

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内容注記 I Preliminaries
1 Introduction
2 Basic Notation and Definitions
3 Auxiliary Results
II Approximation and Complexity of the Discrete Logarithm
4 Approximation of the Discrete Logarithm Modulo p
5 Approximation of the Discrete Logarithm Modulo p — 1
6 Approximation of the Discrete Logarithm by Boolean Functions
7 Approximation of the Discrete Logarithm by Real and Complex Polynomials
III Complexity of Breaking the Diffie-Hellman Cryptosystem
8 Polynomial Approximation and Arithmetic Complexity of the Diffie-Hellman Key
9 Boolean Complexity of the Diffie-Hellman Key
IV Other Applications
10 Trade-off between the Boolean and Arithmetic Depths of Modulo p Functions
11 Special Polynomials and Boolean Functions
12 RSA and Blum-Blum-Shub Generators of Pseudo-Random Numbers
V Concluding Remarks
13 Generalizations and Open Questions
14 Further Directions
一般注記 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
著者標目 *Shparlinski, Igor author
SpringerLink (Online service)
件 名 LCSH:Mathematics
LCSH:Data structures (Computer science)
LCSH:Data encryption (Computer science)
LCSH:Computers
LCSH:Number theory
FREE:Mathematics
FREE:Number Theory
FREE:Data Encryption
FREE:Theory of Computation
FREE:Data Structures, Cryptology and Information Theory
分 類 DC23:512.7
巻冊次 ISBN:9783034886642 REFWLINK
ISBN 9783034886642
URL http://dx.doi.org/10.1007/978-3-0348-8664-2
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