Konyagin S.V., Shparlinski I.E. Character Sums with Exponential Functions and their Applications

New York: Cambridge University Press, 2004. — viii+163 p. — (Cambridge Tracts in Mathematics, 136). — ISBN: 0-521-64263-9, 0-511-04036-9.The book "Character sums with exponential functions and their applications" by S. Konyagin and I. Shparlinski is a welcome addition to the theory of exponential and character sums, a classical topic from analyic number theory. It includes a number of timely applications in algebraic number theory, function fields, complexity theory, pseudo-random number generation, cryptography, and coding theory. This is an advanced monograph which will likely be of interest mostly to specialists and advanced students. In fact, a number of open research questions are outlined in the course of presentation.Preface
AcknowledgementPreliminariesNotation and Auxiliary ResultsBounds of Character Sums
Bounds of Long Character Sums
Bounds of Short Character Sums
Bounds of Character Sums for Almost All Moduli
Bounds of Gaussian SumsMultiplicative Translations of Sets
Multiplicative Translations of Subgroups of F*_p
Multiplicative Translations of Arbitrary Sets Modulo pApplications to Algebraic Number Fields
Representatives of Residue Classes
Cyclotomic Fields and Gaussian PeriodsApplications to Pseudo-Random Number Generators
Prediction of Pseudo-Random Number Generators
Congruential Pseudo-Random Number GeneratorsApplications to Finite Fields
Small mth Roots Modulo p
Supersingular Hyperelliptic Curves
Distribution of Powers of Primitive RootsApplications to Coding Theory and Combinatorics
Difference Sets in V_p
Dimension of BCH Codes
An Enumeration Problem in Finite FieldsBibliography
Index