Combinatorial Arithmetic on Elliptic Curves

214 Pages·2017·1.53 MB·English

Combinatorial Arithmetic
on Elliptic Curves
by
Gabriel Gauthier-Shalom
A thesis
presented to the University of Waterloo
in fulfillment of the
thesis requirement for the degree of
Doctor of Philosophy
in
Combinatorics and Optimization
Waterloo, Ontario, Canada, 2017
c Gabriel Gauthier-Shalom 2017
(cid:13)
Examining Committee Membership
The following served on the Examining Committee for this thesis. The decision of the
Examining Committee is by majority vote.
External Examiner: Paulo Barreto
Professor
Supervisor: David Jao
Associate Professor
Internal Member: Alfred Menezes
Professor
Internal Member: Kevin Purbhoo
Associate Professor
Internal-external Member: David McKinnon
Professor
ii
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,
including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.
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Abstract
We propose a scalar multiplication technique on an elliptic curve, which operates on
triples of collinear points. The computation of this operation requires a new approach
to operation chains, with similarities to Montgomery ladders for x-only scalar multipli-
cation. We develop a diagrammatic calculus with a combinatorial flavor for the purpose
of developing these operation chains. Some interesting algebra arises when studying this
diagrammatic calculus, which leads us to improvements to our algorithms. We propose
some cryptographic applications for our scalar multiplication technique.
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Acknowledgements
Thank you Professor David Jao for your infinite patience.
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Dedication
This is dedicated to C´ecile.
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Table of Contents
1 Introduction 1
1.1 Elliptic Curve Point Multiplication . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Montgomery’s x-only Point Multiplication . . . . . . . . . . . . . . . . . . 2
1.3 Line Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Diagrammatic Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.1 Improved Line Multiplication . . . . . . . . . . . . . . . . . . . . . 6
1.4.2 Three Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.3 Trilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Applications and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Elliptic Curve Cryptography 9
2.1 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The Discrete Logarithm Problem . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Introduction to Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Elliptic Curve Point Addition . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Elliptic Curve Point Multiplication . . . . . . . . . . . . . . . . . . 14
2.5 Elliptic Curves for Cryptography . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Efficient Elliptic Curve Arithmetic . . . . . . . . . . . . . . . . . . . . . . 17
2.7 x-only point multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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3 Line Multiplication 21
3.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Line Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Obstacles to Line Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Cyclically Oriented Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Cyclic Line Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Generic Line Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Formula for Linear Sum Function . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 Generic Line Multiplication Operation Chain . . . . . . . . . . . . . . . . . 39
3.8.1 Line Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.8.2 Line Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8.3 Line Multiplication Ladder . . . . . . . . . . . . . . . . . . . . . . . 41
3.9 Improving on Generic Algorithm . . . . . . . . . . . . . . . . . . . . . . . 44
3.10 Nine Point Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.11 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Generalized Line Multiplication 50
4.1 Generalized Elliptic Curve Line Multiplication . . . . . . . . . . . . . . . . 50
4.1.1 Generalizing to Abelian Groups . . . . . . . . . . . . . . . . . . . . 52
4.2 Generic Linear 2-Set Multiplication . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Generic Linear Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 Generic Line Multiplication . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Linear Sets over a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.1 Generic Line Multiplication Over a Field . . . . . . . . . . . . . . . 59
4.4.2 Improved Line Multiplication Over a Field . . . . . . . . . . . . . . 61
4.5 Application: Cipolla’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.1 Cipolla Using Line Multiplication . . . . . . . . . . . . . . . . . . . 66
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5 Diagrammatic Algebra 68
5.1 Label Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1 Isomorphisms of Incidence Structures . . . . . . . . . . . . . . . . . 70
5.1.2 Nine Point Diagram Structure . . . . . . . . . . . . . . . . . . . . . 71
5.2 Labeled Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.1 Automorphisms on Labeled Diagrams . . . . . . . . . . . . . . . . . 74
5.2.2 Labeled Diagram Arithmetic . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Diagrams With Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.1 Automorphisms on Diagrams with Symmetry . . . . . . . . . . . . 78
5.3.2 Unlabeled Line Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.3 Cyclic Line Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.4 Nine Point Diagrams with Symmetry . . . . . . . . . . . . . . . . . 79
5.4 Diagrammatic Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4.1 Isomorphic Diagrammatic Sum . . . . . . . . . . . . . . . . . . . . 82
5.4.2 Diagrammatic Sum Symmetries . . . . . . . . . . . . . . . . . . . . 86
5.4.3 Diagrammatic Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Homomorphisms of Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5.1 Algebraic Definition of Diagrammatic Homomorphisms . . . . . . . 92
5.5.2 Forgetful Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 94
5.5.3 Line Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6 Linear Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6.1 Cyclic Line Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6.2 Forward Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7 Nine Point Diagram Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . 102
5.7.1 Completion Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.7.2 Diagrams Within Diagrams . . . . . . . . . . . . . . . . . . . . . . 107
5.7.3 Line Addition and Completion Diagrams . . . . . . . . . . . . . . . 109
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6 Diagrammatic Calculus 112
6.1 Diagram Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Nine Point Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2.1 Nine Point Diagram Function . . . . . . . . . . . . . . . . . . . . . 116
6.2.2 Algebraic Relations Between Line Coordinates . . . . . . . . . . . . 117
6.2.3 Nine Point Diagram Automorphisms . . . . . . . . . . . . . . . . . 118
6.3 Nine Point Diagram Orientation . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3.1 Formulas for Nine Point Diagram Orientation . . . . . . . . . . . . 120
6.3.2 Orientation in Terms of Two Lines . . . . . . . . . . . . . . . . . . 122
6.3.3 Relation to Cyclic Orientation . . . . . . . . . . . . . . . . . . . . . 125
6.4 Forward Differences in Nine Point Diagrams . . . . . . . . . . . . . . . . . 126
6.4.1 Alternate Representations . . . . . . . . . . . . . . . . . . . . . . . 130
6.4.2 Proof of Theorem 6.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.4.3 Cyclic Line Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.5 Completion Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.5.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.5.2 Relations from Line Sum Function . . . . . . . . . . . . . . . . . . 141
6.5.3 Pairing Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.5.4 Linear Sum Diagram Orientation . . . . . . . . . . . . . . . . . . . 144
6.6 Diagrammatic Line Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.7 Cyclic Line Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7 Three Torsion Algebra 151
7.1 Elliptic Curve Three Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1.1 Action of Three Torsion . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2 Trilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2.1 Cyclic Orientation from Determinant Forms . . . . . . . . . . . . . 159
7.2.2 Forward Difference from Trilinear Forms . . . . . . . . . . . . . . . 163
7.2.3 Point Addition and Trilinear Forms . . . . . . . . . . . . . . . . . . 165
7.3 Trilinear Forms on Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.3.1 Trilinear Form Relations . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3.2 Proof of Theorem 7.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 168
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