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＜電子ブック＞
Number Theoretic Methods in Cryptography : Complexity lower bounds / by Igor Shparlinski
(Progress in Computer Science and Applied Logic. ISSN:22970584 ; 17)

版	1st ed. 1999.
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
著者標目	*Shparlinski, Igor author SpringerLink (Online service)
件　名	LCSH:Number theory LCSH:Cryptography LCSH:Data encryption (Computer science) LCSH:Computer science LCSH:Data structures (Computer science) LCSH:Information theory FREE:Number Theory FREE:Cryptology FREE:Theory of Computation FREE:Data Structures and Information Theory
一般注記	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 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 HTTP:URL=https://doi.org/10.1007/978-3-0348-8664-2

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