ssca-protected point arithmetic using side-channel

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u Ottawa
L’UiiivorsUb canadicnnc
Canada’s university
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FACULTE DES ETUDES SUPERIEURES FACULTY OF GRADUATE AND
ET POSTOCTORALES u Ottawa POSDOCTORAL STUDIES
L’Universitd canadienne
Canada’s university
Patrick Longa
M.A.Sc. (Electrical Engineering)
School of Information Technology and Engineering
'TACUlfOCOLETDlPARTM^
Accelerating the Scalar Multiplication on Elliptic Curve Cryptosystems over Prime Fields
TITRE DE LA THESE I TITLE OF THESIS
A. Miri
DIRECtEUifpR^^
CCLDIrECTeURTC^
EXAMINATEURS (EXAMINATRICES) DE LA THESE / THESIS EXAMINERS
A. Adler
M. Bolic
Gary W. Slater
Le Doyen de la Facuitedes etudes supdrieures et postdoctoraies / Dean of the Faculty of Graduate and Postdoctoral Studies
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Accelerating the Scalar Multiplication on
Elliptic Curve Cryptosystems
over Prime Fields
by
Patrick Longa
Thesis submitted to
The Faculty of Graduate and Postdoctoral Studies
in partial fulfillment of the requirements
for the degree of
Master of Applied Science
in
Electrical and Computer Engineering
Ottawa-Carleton Institute for Electrical and Computer Engineering
School of Information Technology and Engineering
University of Ottawa
© Patrick Longa
Ottawa, Canada, 2007
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Abstract
Elliptic curve cryptography (ECC), independently introduced by Koblitz and Miller in the
80’s, has attracted increasing attention in recent years due to its shorter key length
requirement in comparison with other public-key cryptosystems such as RSA. Shorter key
length means reduced power consumption and computing effort, and less storage
requirement, factors that are fundamental in ubiquitous portable devices such as PDAs,
cellphones, smartcards, and many others. To that end, a lot of research has been carried out
to speed-up and improve ECC implementations, mainly focusing on the most important and
time-consuming ECC operation: scalar multiplication.
In this thesis, we focus in optimizing such ECC operation at the point and scalar
arithmetic levels, specifically targeting standard curves over prime fields. At the point
arithmetic level, we introduce two innovative methodologies to accelerate ECC formulae:
the use of new composite operations, which are built on top of basic point doubling and
addition operations; and the substitution of field multiplications by squarings and other
cheaper operations. These techniques are efficiently exploited, individually or jointly, in
several contexts: to accelerate computation of scalar multiplications, and the computation of
pre-computed points for window-based scalar multiplications (up to 30% improvement in
comparison with previous best method); to speed-up computations of simple side-channel
attack (SSCA)-protected implementations using innovative atomic structures (up to 22%
improvement in comparison with scalar multiplication using original atomic structures); and
to develop parallel formulae for SIMD-based applications, which are able to execute three
and four operations simultaneously (up to 72% of improvement in comparison with a
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sequential scalar multiplication).
At the scalar arithmetic level, we develop new sublinear (in terms of Hamming weight)
multibase scalar multiplications based on NAF-like conversion algorithms that are shown to
be faster than any previous scalar multiplication method. For instance, proposed multibase
scalar multiplications reduce computing times in 10.9% and 25.3% in comparison with
traditional NAF for unprotected and SSCA-protected scenarios, respectively. Moreover, our
conversion algorithms overcome the problem of converting any integer to multibase
representation, solving an open problem that was defined as hard. Thus, our algorithms
make the use of multiple bases practical for applications as ECC scalar multiplication for
first time.
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Acknowledgements
I would like to thank my wife and family for their support and understanding through all
these years. Their love and confidence in me were the inspiration to give the best of myself
to this work.
Special gratitude is to Dr. Miri. His faith in my work and capacity, and his constant
support and guidance at all levels were priceless factors that helped me to achieve more than
I thought possible.
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To my wife, Veronica
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Table of Contents
ABSTRACT II
ACKNOWLEDGEMENTS IV
TABLE OF CONTENTS VI
LIST OF TABLES X
LIST OF FIGURES XIII
LIST OF ALGORITHMS XIV
LIST OF ACRONYMS XV
CHAPTER 1: INTRODUCTION 1
1.1 Motivation.............................................................................................................................1
1.2 Significance of this Work....................................................................................................3
1.3 Thesis Outline.......................................................................................................................5
CHAPTER 2: BACKGROUND 7
2.1 Preliminaries.........................................................................................................................7
2.2 Introduction to Elliptic Curves............................................................................................9
2.2.1 Elliptic Curve Discrete Logarithm Problem (ECDLP).................................................10
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2.3 ECC Arithmetic.................................................................................................................12
2.3.1 Level 3: Finite Field Arithmetic....................................................................................13
2.3.2 Level 2: Point Arithmetic..............................................................................................15
2.3.3 Level 1: Scalar Arithmetic.............................................................................................22
CHAPTER 3: EFFICIENT POINT ARITHMETIC OVER PRIME FIELDS VIA
COMPOSITE OPERATIONS 31
3.1 Previous Work...................................................................................................................32
3.2 Composite Operations dP + Q..........................................................................................34
3.2.1 Improved Doubling-Addition (DA) operation.............................................................35
3.2.2 Improved Tripling-Addition (TA) operation................................................................37
3.2.3 Generalization to Composite Operations dP + Q........................................................40
3.2.4 Performance comparison...............................................................................................42
3.3 Composite Operations dP.................................................................................................43
3.3.1 Improved Tripling (T) operation..................................................................................43
3.3.2 Generalization to Composite Operations dP................................................................44
3.3.3 Performance comparison............................... 48
3.4 Applications.......................................................................................................................49
3.4.1 Computation of pre-computed points...........................................................................50
3.4.2 Speeding-up existent scalar multiplications.................................................................54
CHAPTER 4: FAST AND FLEXIBLE POINT ARITHMETIC OVER PRIME FIELDS 56
4.1 Flexible Methodology for Replacing Multiplications by Cheaper Operations.............57
4.1.1 Fast Point Formulae for Traditional Operations...........................................................59
4.1.2 Fast Point Formulae for Composite Operations...........................................................62
4.2 Performance Comparison.................................................................................................65
CHAPTER 5: SSCA-PROTECTED POINT ARITHMETIC USING SIDE-CHANNEL
ATOMICITY 68
5.1 Side-Channel Attacks.......................................................................................................69
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