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Many Rational Points - Coding Theory and Algebraic Geometry (Hardcover, 2003 ed.)
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Many Rational Points - Coding Theory and Algebraic Geometry (Hardcover, 2003 ed.)
Series: Mathematics and Its Applications, 564
Expected to ship within 10 - 15 working days
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2 Triangle Groups: An Introduction 279 3 Elementary Shimura Curves
281 4 Examples of Shimura Curves 282 5 Congruence Zeta Functions
283 6 Diophantine Properties of Shimura Curves 284 7 Klein Quartic
285 8 Supersingular Points 289 Towers of Elkies 9 289 7.
CRYPTOGRAPHY AND APPLICATIONS 291 1 Introduction 291 Discrete
Logarithm Problem 2 291 Curves for Public-Key Cryptosystems 3 295
Hyperelliptic Curve Cryptosystems 4 297 CM-Method 5 299 6
Cryptographic Exponent 300 7 Constructive Descent 302 8 Gaudry and
Harley Algorithm 306 9 Picard Jacobians 307 Drinfeld Module Based
Public Key Cryptosystems 10 308 11 Drinfeld Modules and One Way
Functions 308 12 Shimura's Map 309 13 Modular Jacobians of Genus 2
Curves 310 Modular Jacobian Surfaces 14 312 15 Modular Curves of
Genus Two 313 16 Hecke Operators 314 8. REFERENCES 317 345 Index
Xll Preface The history of counting points on curves over finite
fields is very ex- tensive, starting with the work of Gauss in 1801
and continuing with the work of Artin, Schmidt, Hasse and Weil in
their study of curves and the related zeta functions Zx(t), where m
Zx(t) = exp (2: N t ) m m 2': 1 m with N = #X(F qm). If X is a
curve of genus g, Weil's conjectures m state that L(t) Zx(t) = (1 -
t)(l - qt) where L(t) = rr~!l (1 - O'.
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