### このページのリンク ## 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 Basel : Birkhäuser Basel : Imprint: Birkhäuser 1999 英語 IX, 182 p : online resource

EB0089058

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内容注記 I Preliminaries1 Introduction2 Basic Notation and Definitions3 Auxiliary ResultsII Approximation and Complexity of the Discrete Logarithm4 Approximation of the Discrete Logarithm Modulo p5 Approximation of the Discrete Logarithm Modulo p — 16 Approximation of the Discrete Logarithm by Boolean Functions7 Approximation of the Discrete Logarithm by Real and Complex PolynomialsIII Complexity of Breaking the Diffie-Hellman Cryptosystem8 Polynomial Approximation and Arithmetic Complexity of the Diffie-Hellman Key9 Boolean Complexity of the Diffie-Hellman KeyIV Other Applications10 Trade-off between the Boolean and Arithmetic Depths of Modulo p Functions11 Special Polynomials and Boolean Functions12 RSA and Blum-Blum-Shub Generators of Pseudo-Random NumbersV Concluding Remarks13 Generalizations and Open Questions14 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 DC23:512.7 ISBN:9783034886642 9783034886642