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Algebraic geometry for coding theory and cryptography : IPAM, Los Angeles, CA, February 2016

Author: Everett W Howe; Kristin E Lauter; Judy L Walker
Publisher: Cham : Springer, 2017.
Series: Association for Women in Mathematics Series, v. 9.
Edition/Format:   eBook : Document : Conference publication : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Conference papers and proceedings
Congresses
Additional Physical Format: Print version:
Howe, Everett W.
Algebraic Geometry for Coding Theory and Cryptography : IPAM, Los Angeles, CA, February 2016.
Cham : Springer International Publishing, ©2017
Material Type: Conference publication, Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Everett W Howe; Kristin E Lauter; Judy L Walker
ISBN: 9783319639314 3319639315
OCLC Number: 1012883090
Description: 1 online resource (160 pages).
Contents: Preface; Contents; Contributors; 1 Representations of the Multicast Network Problem; 1.1 Introduction; 1.1.1 Achieving Multicast Network Requirements; 1.2 Coding Points and Reduced Multicast Networks; 1.3 Code Graphs; 1.4 Fq-Vector Labelings and Matrices; 1.4.1 Fq-Vector Labelings and Rational Points; 1.4.2 Fq-Vector Labelings and Grassmannians; 1.4.3 An Open Question on Fq-Vector Labelings of Code Graphs; References; 2 Hypersurfaces in Weighted Projective Spaces Over Finite Fields with Applications to Coding Theory; 2.1 Introduction; 2.2 Polynomials with Many Zeros. 2.3 Hypersurfaces in Weighted Projective Planes P(1,a1,a2)2.4 Weighted Projective Reedâ#x80;#x93;Muller Codes; 2.4.1 Generalized Reedâ#x80;#x93;Muller Codes, Projective Reedâ#x80;#x93;Muller Codes, and Projective Nested Cartesian Codes; 2.4.2 Weighted Projective Reedâ#x80;#x93;Muller Codes; 2.4.2.1 Length and Dimension; 2.4.2.2 Minimum Distance; 2.4.2.3 A Particular Case; 2.4.2.4 Another Particular Case; 2.4.2.5 Relative Parameters; References; 2.A Appendix: Weighted Projective Spaces; 2.A.1 Definitions of Weighted Projective Spaces; 2.A.1.1 WPS as a Proj Functor; 2.A.1.2 Quotients. 2.A.1.3 WPS as a Quotient of the Punctured Affine Space2.A.1.4 WPS as a Finite Quotient of the Projective Space; 2.A.2 The Singular Locus; 2.A.3 Affine Parts; 2.A.3.1 Quotient of the Affine Space by a Cyclic Group; 2.A.3.2 Affine Parts; 2.A.3.3 A Special Case; 2.A.3.4 Action of Gm; 2.A.4 Rationality; 2.A.5 Weighted Forms; 2.A.5.1 Definition; 2.A.5.2 Weighted Binary Forms; 2.A.5.3 Weighted Ternary Forms; References; 3 Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication; 3.1 Introduction; 3.1.1 The State of the Art. 3.1.2 Our Contributions, and Beyond3.1.3 Vanilla Abelian Varieties; 3.2 Genus One Curves: Elliptic Curve Point Counting; 3.2.1 Schoof's Algorithm; 3.2.2 Frobenius Eigenvalues and Subgroups; 3.2.3 Modular Polynomials and Isogenies; 3.2.4 Elkies, Atkin, and Volcanic Primes; 3.2.5 Computing the Type of a Prime; 3.2.6 Atkin's Improvement; 3.2.7 Elkies' Improvement; 3.3 The Genus-2 Setting; 3.3.1 The Jacobian; 3.3.2 Frobenius and Endomorphisms of JC; 3.3.3 Real Multiplication; 3.3.4 From Schoof to Pila; 3.3.5 The Gaudryâ#x80;#x93;Schost Approach; 3.3.6 Point Counting with Efficiently Computable RM. 3.3.7 Generalizing Elkies' and Atkin's Improvements to Genus-23.3.8 mu-Isogenies; 3.4 Invariants; 3.4.1 Invariants for RM Abelian Surfaces; 3.4.2 Hilbert Modular Polynomials for RM Abelian Surfaces; 3.4.3 Invariants for Curves and Abelian Surfaces; 3.4.4 Pulling Back Curve Invariants to RM Invariants; 3.5 Atkin Theorems in Genus 2; 3.5.1 Roots of Gmu and the Order of Frobenius; 3.5.2 The Factorization of Gmu; 3.5.3 The Characteristic Polynomial of Frobenius; 3.5.4 Prime Types for Real Multiplication by OF; 3.5.5 The Parity of the Number of Factors of Gmu. 3.6 The Case F = Q(sqrt5): Gundlachâ#x80;#x93;MÃơller Invariants.
Series Title: Association for Women in Mathematics Series, v. 9.
Responsibility: Everett W. Howe, Kristin E. Lauter, Judy L. Walker, editors.

Abstract:

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  Read more...

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