Free Delivery Over $100
|
|
|||
|
||||
OverviewThe 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. Full Product DetailsAuthor: Igor ShparlinskiPublisher: Birkhauser Verlag AG Imprint: Birkhauser Verlag AG Edition: 1999 ed. Volume: 17 Dimensions: Width: 15.50cm , Height: 1.20cm , Length: 23.50cm Weight: 1.000kg ISBN: 9783764358884ISBN 10: 3764358882 Pages: 182 Publication Date: 15 February 1999 Audience: College/higher education , Professional and scholarly , Undergraduate , Postgraduate, Research & Scholarly Format: Hardback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsI 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.ReviewsThis 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. <p>--Zentralblatt Math 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 Author InformationTab Content 6Author Website:Countries AvailableAll regions |
||||
