Algebraic Geometry for Coding Theory and Cryptography

1. Front Matter

   Pages i-xv

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2. Representations of the Multicast Network Problem

   • Sarah E. Anderson, Wael Halbawi, Nathan Kaplan, Hiram H. López, Felice Manganiello, Emina Soljanin et al.

   Pages 1-23

3. Hypersurfaces in Weighted Projective Spaces Over Finite Fields with Applications to Coding Theory

   • Yves Aubry, Wouter Castryck, Sudhir R. Ghorpade, Gilles Lachaud, Michael E. O’Sullivan, Samrith Ram

   Pages 25-61

4. Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

   • Sean Ballentine, Aurore Guillevic, Elisa Lorenzo García, Chloe Martindale, Maike Massierer, Benjamin Smith et al.

   Pages 63-94

5. Locally Recoverable Codes from Algebraic Curves and Surfaces

   • Alexander Barg, Kathryn Haymaker, Everett W. Howe, Gretchen L. Matthews, Anthony Várilly-Alvarado

   Pages 95-127

6. Variations of the McEliece Cryptosystem

   • Jessalyn Bolkema, Heide Gluesing-Luerssen, Christine A. Kelley, Kristin E. Lauter, Beth Malmskog, Joachim Rosenthal

   Pages 129-150

Covering topics in algebraic geometry, coding theory, and cryptography, this volume presents interdisciplinary group research completed for the February 2016 conference at the Institute for Pure and Applied Mathematics (IPAM) in cooperation with the Association for Women in Mathematics (AWM). The conference gathered research communities across disciplines to share ideas and problems in their fields and formed small research groups made up of graduate students, postdoctoral researchers, junior faculty, and group leaders who designed and led the projects. Peer reviewed and revised, each of this volume's five papers achieves the conference’s goal of using algebraic geometry to address a problem in either coding theory or cryptography. Proposed variants of the McEliece cryptosystem based on different constructions of codes, constructions of locally recoverable codes from algebraic curves and surfaces, and algebraic approaches to the multicast network coding problem are only some of the topics covered in this volume. Researchers and graduate-level students interested in the interactions between algebraic geometry and both coding theory and cryptography will find this volume valuable.

• Algebraic geometry
• network-constrained generator matrices
• isogenies for genus-2 curves
• pairing-based cryptography
• coding theory
• trace-zero subgroups
• codes with locality constraints
• affine and projective algebraic sets
• Frobenius conjugates
• elliptic and hyperelliptic curves
• public key cryptography
• post-quantum cryptography
• modular polynomials

• Center for Communications Research, San Diego, USA

  Everett W. Howe

• Microsoft Research, One Microsoft Way, Redmond, USA

  Kristin E. Lauter

• Department of Mathematics, University of Nebraska-Lincoln, Lincoln, USA

  Judy L. Walker

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