Number Theoretic Methods in Cryptography: Complexity lower bounds (Progress in Computer Science and Applied Logic)

Shparlinski, Igor

ISBN 10: 3764358882 / ISBN 13: 9783764358884

Published by Birkh?user, 1999

About this Item

Square Tight Binding.Clean interior, save for small p/o signature to top of front paste down, else new.

Seller Inventory # 12907

Bibliographic Details

• Title: Number Theoretic Methods in Cryptography: Complexity lower bounds (Progress in Computer Science and Applied Logic)
• Publisher: Birkh?user
• Publication Date: 1999
• Language: English
• ISBN 10: 3764358882
• ISBN 13: 9783764358884
• Binding: Hardcover
• Condition: Like New
• Seller catalogs:
  • Science

About this title

Synopsis

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.

Review

"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