Efficient arithmetic on hyperelliptic curves:
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| 1. Verfasser: | |
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| Format: | Abschlussarbeit Buch |
| Sprache: | English |
| Veröffentlicht: |
Essen
Inst. für Experimentelle Mathematik, Univ.
2002
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| Schriftenreihe: | Preprint / Institut für Experimentelle Mathematik, Universität Essen
2002,4 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | VI, 112 S. |
Internformat
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Datensatz im Suchindex
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| adam_text | Titel: Efficient arithmetic on hyperelliptic curves
Autor: Lange, Tanja
Jahr: 2002
Contents
1 Introduction 1
2 Mathematical Background 7
2.1 Notation and Definitions............................................................7
2.2 Algorithms for the Ideal Class Group..............................................16
2.3 Cardinality of Pic°(X/F9n)..................................19
2.4 The Frobenius Endomorphism......................................................22
3 Computing in the Divisor Class Group for Genus 2 25
3.1 Different Cases......................................................................26
3.2 Addition in Most Common Case ..................................................27
3.3 Addition in Case degui = 1, degu2 = 2..........................................30
3.4 Doubling ............................................................................30
4 Efficient Determination of the Class Number for Koblitz Curves 33
4.1 Computation of P(T) ..............................................................33
4.2 Recurrence Formulae for the Class Number......................................34
4.3 Examples............................................................................38
4.3.1 Binary Koblitz Curves......................................................38
4.3.2 Curves over F3..............................................................44
4.3.3 Curves over F4..............................................................46
4.3.4 Curves over F5..............................................................47
5 Speeding Up the Computation of m-folds for Koblitz Curves 49
5.1 Standard ways of computing m-folds..............................................49
5.2 Representing Integers to the Base of r............................................50
5.3 On the Finiteness of the Representation..........................................52
5.4 Reducing the length of the representation........................................64
5.4.1 Representing (rn — l)/(r — 1) in Z[r] ....................................67
5.4.2 Inversion of Elements e0 + eir 4-----b e2g_ir29~l in Q[r]..............68
5.4.3 Computing r-adic Expansions of Reduced Length........................68
5.5 Density of the Expansion..........................................................70
5.6 Experimental results................................................................72
5.6.1 Curves of genus 2 over F2..................................................72
5.6.2 Curves of genus 3 over F2 ................................................73
5.6.3 Curves of genus 4 over F2..................................................75
5.7 Comparison......................................76
v
5.7.1 Complexity compared to binary double-and-add ........................76
5.7.2 Complexities taking into account the storage............................77
5.7.3 Timings......................................................................79
5.8 Alternatives..........................................................................79
5.9 Koblitz Curve Cryptosystems Revisited..........................................81
5.9.1 Protocols....................................................................81
5.9.2 Collisions....................................................................82
5.9.3 Attacks.................................................84
6 Trace-Zero Variety 87
6.1 Different Kinds of Divisor Classes..................................................88
6.2 Describing Equations................................................................89
6.3 Computing in the Trace Zero Variety..............................................92
6.4 Security and Comparison..........................................................95
6.5 Example..............................................................................99
7 Conclusion 101
7.1 Generalizations and Practical Considerations....................................101
7.2 Side-Channel Attacks ..............................................................101
7.3 Outlook..............................................................................105
8 Bibliography 107
vi
|
| adam_txt |
Titel: Efficient arithmetic on hyperelliptic curves
Autor: Lange, Tanja
Jahr: 2002
Contents
1 Introduction 1
2 Mathematical Background 7
2.1 Notation and Definitions.7
2.2 Algorithms for the Ideal Class Group.16
2.3 Cardinality of Pic°(X/F9n).19
2.4 The Frobenius Endomorphism.22
3 Computing in the Divisor Class Group for Genus 2 25
3.1 Different Cases.26
3.2 Addition in Most Common Case .27
3.3 Addition in Case degui = 1, degu2 = 2.30
3.4 Doubling .30
4 Efficient Determination of the Class Number for Koblitz Curves 33
4.1 Computation of P(T) .33
4.2 Recurrence Formulae for the Class Number.34
4.3 Examples.38
4.3.1 Binary Koblitz Curves.38
4.3.2 Curves over F3.44
4.3.3 Curves over F4.46
4.3.4 Curves over F5.47
5 Speeding Up the Computation of m-folds for Koblitz Curves 49
5.1 Standard ways of computing m-folds.49
5.2 Representing Integers to the Base of r.50
5.3 On the Finiteness of the Representation.52
5.4 Reducing the length of the representation.64
5.4.1 Representing (rn — l)/(r — 1) in Z[r] .67
5.4.2 Inversion of Elements e0 + eir 4-----b e2g_ir29~l in Q[r].68
5.4.3 Computing r-adic Expansions of Reduced Length.68
5.5 Density of the Expansion.70
5.6 Experimental results.72
5.6.1 Curves of genus 2 over F2.72
5.6.2 Curves of genus 3 over F2 .73
5.6.3 Curves of genus 4 over F2.75
5.7 Comparison.76
v
5.7.1 Complexity compared to binary double-and-add .76
5.7.2 Complexities taking into account the storage.77
5.7.3 Timings.79
5.8 Alternatives.79
5.9 Koblitz Curve Cryptosystems Revisited.81
5.9.1 Protocols.81
5.9.2 Collisions.82
5.9.3 Attacks.84
6 Trace-Zero Variety 87
6.1 Different Kinds of Divisor Classes.88
6.2 Describing Equations.89
6.3 Computing in the Trace Zero Variety.92
6.4 Security and Comparison.95
6.5 Example.99
7 Conclusion 101
7.1 Generalizations and Practical Considerations.101
7.2 Side-Channel Attacks .101
7.3 Outlook.105
8 Bibliography 107
vi |
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| spelling | Lange, Tanja Verfasser (DE-588)142444731 aut Efficient arithmetic on hyperelliptic curves vorgelegt von Tanja Lange Essen Inst. für Experimentelle Mathematik, Univ. 2002 VI, 112 S. txt rdacontent n rdamedia nc rdacarrier Preprint / Institut für Experimentelle Mathematik, Universität Essen 2002,4 Zugl.: Essen, Univ., Diss., 2002 Hochschulschrift gtt Hyperelliptische Kurve (DE-588)4282659-7 gnd rswk-swf Divisorenklasse (DE-588)4150325-9 gnd rswk-swf Arithmetik (DE-588)4002919-0 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Arithmetik (DE-588)4002919-0 s DE-604 Divisorenklasse (DE-588)4150325-9 s Hyperelliptische Kurve (DE-588)4282659-7 s Institut für Experimentelle Mathematik, Universität Essen Preprint 2002,4 (DE-604)BV006667081 2002,4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015380389&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lange, Tanja Efficient arithmetic on hyperelliptic curves Hochschulschrift gtt Hyperelliptische Kurve (DE-588)4282659-7 gnd Divisorenklasse (DE-588)4150325-9 gnd Arithmetik (DE-588)4002919-0 gnd |
| subject_GND | (DE-588)4282659-7 (DE-588)4150325-9 (DE-588)4002919-0 (DE-588)4113937-9 |
| title | Efficient arithmetic on hyperelliptic curves |
| title_auth | Efficient arithmetic on hyperelliptic curves |
| title_exact_search | Efficient arithmetic on hyperelliptic curves |
| title_exact_search_txtP | Efficient arithmetic on hyperelliptic curves |
| title_full | Efficient arithmetic on hyperelliptic curves vorgelegt von Tanja Lange |
| title_fullStr | Efficient arithmetic on hyperelliptic curves vorgelegt von Tanja Lange |
| title_full_unstemmed | Efficient arithmetic on hyperelliptic curves vorgelegt von Tanja Lange |
| title_short | Efficient arithmetic on hyperelliptic curves |
| title_sort | efficient arithmetic on hyperelliptic curves |
| topic | Hochschulschrift gtt Hyperelliptische Kurve (DE-588)4282659-7 gnd Divisorenklasse (DE-588)4150325-9 gnd Arithmetik (DE-588)4002919-0 gnd |
| topic_facet | Hochschulschrift Hyperelliptische Kurve Divisorenklasse Arithmetik |
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