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Gröbner Bases, Coding, and Cryptography

  • Book
  • © 2009

Overview

Editors:
  1. Massimiliano Sala

    1. Boole Centre for Research in Informatics, University College Cork, Cork, Ireland

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  2. Shojiro Sakata

    1. Faculty of Engineering Dept. Production Systems Engineering, Toyohashi University of Technology, Tempaku-cho, Japan

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  3. Teo Mora

    1. Dipto. Matematica, Università Genova, Genova, Italy

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  4. Carlo Traverso

    1. Dipto. Matematica, Università Pisa, Pisa, Italy

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  5. Ludovic Perret

    1. LIP6, University of Paris VI, Paris, France

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  • Covers exciting new research directions on Gröbner bases and their applications
  • Includes supplementary material: sn.pub/extras

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Table of contents (33 chapters)

  1. Front Matter

    Pages 1-16
    Download chapter PDF

  2. Gröbner Bases, Coding, and Cryptography: a Guide to the State-of-Art

    1. Gröbner Bases, Coding, and Cryptography: a Guide to the State-of-Art

      • Massimiliano Sala
      Pages 1-8

  3. Invited Papers

    1. Front Matter

      Pages 9-9
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    2. Gröbner Technology

      • Teo Mora
      Pages 11-25

    3. The FGLM Problem and Möller’s Algorithm on Zero-dimensional Ideals

      • Teo Mora
      Pages 27-45

    4. An Introduction to Linear and Cyclic Codes

      • Daniel Augot, Emanuele Betti, Emmanuela Orsini
      Pages 47-68

    5. Decoding Cyclic Codes: the Cooper Philosophy

      • Teo Mora, Emmanuela Orsini
      Pages 69-91

    6. A Tutorial on AG Code Construction from a Gröbner Basis Perspective

      • Douglas A. Leonard
      Pages 93-106

    7. Automorphisms and Encoding of AG and Order Domain Codes

      • John B. Little
      Pages 107-120

    8. Algebraic Geometry Codes from Order Domains

      • Olav Geil
      Pages 121-141

    9. The BMS Algorithm

      • Shojiro Sakata
      Pages 143-163

    10. The BMS Algorithm and Decoding of AG Codes

      • Shojiro Sakata
      Pages 165-185

    11. A Tutorial on AG Code Decoding from a Gröbner Basis Perspective

      • Douglas A. Leonard
      Pages 187-196

    12. FGLM-Like Decoding: from Fitzpatrick’s Approach to Recent Developments

      • Eleonora Guerrini, Anna Rimoldi
      Pages 197-218

    13. An Introduction to Ring-Linear Coding Theory

      • Marcus Greferath
      Pages 219-238

    14. Gröbner Bases over Commutative Rings and Applications to Coding Theory

      • Eimear Byrne, Teo Mora
      Pages 239-261

    15. Overview of Cryptanalysis Techniques in Multivariate Public Key Cryptography

      • Olivier Billet, Jintai Ding
      Pages 263-283

    16. A Survey on Polly Cracker Systems

      • Françoise Levy-dit-Vehel, Maria Grazia Marinari, Ludovic Perret, Carlo Traverso
      Pages 285-305

    17. Block Ciphers: Algebraic Cryptanalysis and Gröbner Bases

      • Carlos Cid, Ralf-Philipp Weinmann
      Pages 307-327

    18. Algebraic Attacks on Stream Ciphers with Gröbner Bases

      • Frederik Armknecht, Gwenolé Ars
      Pages 329-348
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Keywords

About this book

Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Gröbner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations.

This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will findthat all chapters fit nicely within a unified notation.

Editors and Affiliations

  • Boole Centre for Research in Informatics, University College Cork, Cork, Ireland

    Massimiliano Sala

  • Faculty of Engineering Dept. Production Systems Engineering, Toyohashi University of Technology, Tempaku-cho, Japan

    Shojiro Sakata

  • Dipto. Matematica, Università Genova, Genova, Italy

    Teo Mora

  • Dipto. Matematica, Università Pisa, Pisa, Italy

    Carlo Traverso

  • LIP6, University of Paris VI, Paris, France

    Ludovic Perret

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