Algorithmic Number Theory

1. Front Matter

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2. Invited Papers

   1. Running Time Predictions for Factoring Algorithms

      • Ernie Croot, Andrew Granville, Robin Pemantle, Prasad Tetali

      Pages 1-36

   2. A New Look at an Old Equation

      • R. E. Sawilla, A. K. Silvester, H. C. Williams

      Pages 37-59

3. Elliptic Curves Cryptology and Generalizations

   1. Abelian Varieties with Prescribed Embedding Degree

      • David Freeman, Peter Stevenhagen, Marco Streng

      Pages 60-73

   2. Almost Prime Orders of CM Elliptic Curves Modulo p

      • Jorge Jiménez Urroz

      Pages 74-87

   3. Efficiently Computable Distortion Maps for Supersingular Curves

      • Katsuyuki Takashima

      Pages 88-101

   4. On Prime-Order Elliptic Curves with Embedding Degrees k = 3, 4, and 6

      • Koray Karabina, Edlyn Teske

      Pages 102-117

4. Arithmetic of Elliptic Curves

   1. Computing in Component Groups of Elliptic Curves

      • J. E. Cremona

      Pages 118-124

   2. Some Improvements to 4-Descent on an Elliptic Curve

      • Tom Fisher

      Pages 125-138

   3. Computing a Lower Bound for the Canonical Height on Elliptic Curves over Totally Real Number Fields

      • Thotsaphon Thongjunthug

      Pages 139-152

   4. Faster Multiplication in GF(2)[x]

      • Richard P. Brent, Pierrick Gaudry, Emmanuel Thomé, Paul Zimmermann

      Pages 153-166

5. Integer Factorization

   1. Predicting the Sieving Effort for the Number Field Sieve

      • Willemien Ekkelkamp

      Pages 167-179

   2. Improved Stage 2 to P ± 1 Factoring Algorithms

      • Peter L. Montgomery, Alexander Kruppa

      Pages 180-195

6. K3 Surfaces

   1. Shimura Curve Computations Via K3 Surfaces of Néron–Severi Rank at Least 19

      • Noam D. Elkies

      Pages 196-211

   2. K3 Surfaces of Picard Rank One and Degree Two

      • Andreas-Stephan Elsenhans, Jörg Jahnel

      Pages 212-225

7. Number Fields

   1. Number Fields Ramified at One Prime

      • John W. Jones, David P. Roberts

      Pages 226-239

   2. An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields

      • Alar Leibak

      Pages 240-252

   3. Functorial Properties of Stark Units in Multiquadratic Extensions

      • Jonathan W. Sands, Brett A. Tangedal

      Pages 253-267

   4. Enumeration of Totally Real Number Fields of Bounded Root Discriminant

      • John Voight

      Pages 268-281

8. Point Counting

   1. Computing Hilbert Class Polynomials

      • Juliana Belding, Reinier Bröker, Andreas Enge, Kristin Lauter

      Pages 282-295

This book constitutes the refereed proceedings of the 8th International Algorithmic Number Theory Symposium, ANTS 2008, held in Banff, Canada, in May 2008. The 28 revised full papers presented together with 2 invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections on elliptic curves cryptology and generalizations, arithmetic of elliptic curves, integer factorization, K3 surfaces, number fields, point counting, arithmetic of function fields, modular forms, cryptography, and number theory.

• Prime
• algebraic geometry
• algebraic number fields
• algebraic number theory
• algorithm
• algorithmic number theory
• algorithms
• computational number theory
• cryptography
• cryptology
• diophantine equation
• discrete Fourier transform
• elliptic curve computations
• factoring
• number theory
• algorithm analysis and problem complexity

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