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UNIVERSITI PUTRA MALAYSIA

ON THE IMPROVEMENT OF ADDITION CHAIN IN APPLICATIONS TO ELLIPTIC CURVE CRYPTOSYSTEM

MOHAMAD AFENDEE MOHAMED

IPM 2011 9

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ON THE IMPROVEMENT OF ADDITION CHAIN IN

APPLICATIONS TO ELLIPTIC CURVE CRYPTOSYSTEM

By

MOHAMAD AFENDEE MOHAMED

Thesis Submitted to the School of Graduate Studies, Universiti

Putra Malaysia in Fulfilment of the Requirements for the Degree of

Doctor of Philosophy

November 2011

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DEDICATION

My

MA DRE and PA DRE Whom without there’ll be no I

FA MI L I A Whom for I have my blood flowing

MU J E R Whom with I spend my ups and downs

N I C K O and N I C K Y Whom with I grew my childhood

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Abstract of thesis presented to the Senate of Universiti Putra Malaysia in

fulfillment of the requirement for the degree of Philosophy of Doctor

ON THE IMPROVEMENT OF ADDITION CHAIN IN

APPLICATIONS TO ELLIPTIC CURVE CRYPTOSYSTEM

By

MOHAMAD AFENDEE MOHAMED

November 2011

Chair: Mohamad Rushdan Md Said, PhD

Faculty: Institute for Mathematical Research

A hard problem most of the time can be broken into a sequence of simple tasks

from which a solution to the original problem is obtainable. Originally, elliptic

curve cryptography is based on a non-singular algebraic curve of genus 1. The

process of encrypting message involves modular operation of huge integer n

acting on points on elliptic curve. This operation namely scalar multiplication

is formularized as Q = nP . By restricting to only addition and doubling

operations of two previous terms, it can be transformed into an equivalent

iteration of Q = (2 . . . (2(2(P ) + br−1P ) + br−2P ) + . . .) + b0P ). The resulting

ascending sequence is called an addition chain. Finding an optimal chain was

proven to be an NP-complete problem. Notwithstanding, this gives way to the

emergence of many heuristics methods offering near optimal solution. All in

all, the study of efficient point arithmetic on elliptic curves can be reduced to

the study of optimizing an addition chain.

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This thesis centers around an investigation into a new method to improve scalar

multiplication operation. Ultimately, the objective is to minimize the execution

time of EC point arithmetic. One of the ways to achieve this is through shorter

addition chain. The proposed method will be developed from scratch, and are

subjected to some theoretical analysis. For the purpose of empirical test, some

parameters are defined in the course to validating the findings.

Existing methods exploit the binary(m-ary) representation of an integer n,

whilst the new method to be proposed opens up a new window of research into

the problem. An integer n is decomposed into its prime factor pe11 pe22 . . . pess .

A classical one-layered approach is transformed into a two-layered approach.

Efficiency can be improved at prime pi layer, prime power peii layer and the

combination of prime power layer that make up an n.

Initially, a Decomposition Method (DM) is developed based on prime power

factorization of an integer n. Each factor pi is assigned a unique rule from

which an addition chain will be generated. An ei multiple of each rule pi gener-

ates an addition chain for peii of which be further combined altogether to build

up n. Mathematical analysis shows that the chain generated by this method

is confined to the similar boundary as that of an optimal chain studied by A.

Brauer. Experiment shows a significant advantage over existing methods under

appropriate working conditions. An improved version, Signed Decomposition

Method (SDM) introduces a subtraction operation into the sequence to gener-

ate an addition subtraction chain. P. Erdos stated that addition subtraction

chain is always at most as lengthy as an addition chain. It follows that SDM

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outperforms its predecessor by 8 percents. Moreover, the generated chains are

shown to outclass existing methods with significant improvement.

Additionally, we developed a Composition Method (CM) which bears the idea

of computing addition chain directly from the respective rule for n. Unlike

decomposition based methods, CM brings back the approach to one-layered

with most of its properties are inherited from DM at prime layer. An im-

proved version called Signed Composition Method (SCM) is also proposed as

an implication of introducing a subtraction operation into CM. The generated

chain by SCM has recorded an improvement of 10 percent over its predecessor.

Furthermore, SCM has shown an advantage over existing methods for selected

integers.

Earlier comparison between DM and CM favours DM for most n. However,

experimental result for SDM against SCM shows that for small integers SDM is

in favour over SCM, but as n grows, SCM gradually start to outperform SDM.

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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia

sebagai memenuhi keperluan untuk ijazah Doktor Falsafah

TENTANG PENAMBAHBAIKAN RANTAIAN PENAMBAHAN

DALAM APLIKASI KRIPTOGRAFI KELUK ELIPTIK

Oleh

MOHAMAD AFENDEE MOHAMED

November 2011

Pengerusi: Mohamad Rushdan Md Said, PhD

Fakulti: Institut Penyelidikan Matematik

Kebiasaanya, sesuatu permasalahan yang genting dapat dijelma menjadi satu

siri permasalahan yang sederhana kesukarannya di mana penyelesaian kepada

masalah asal mudah didapati. Pada awalnya, kriptografi keluk eliptik di-

dasarkan pada keluk aljabar tak-tunggal dengan kerabat 1. Proses pengkrip-

tanan mesej melibatkan operasi nombor bulat besar modulo n yang bertindak

ke atas titik-titik di atas keluk eliptik. Operasi yang dinamakan hasil darab

skala ini boleh dirumuskan sebagai Q = nP . Dengan membataskan operasi

hanya kepada hasil tambah dan ganda-dua oleh dua sebutan terdahulu, ianya

boleh ditransformasikan menjadi iterasi setara Q = (2 . . . (2(2(P ) + br−1P ) +

br−2P )+ . . .)+b0P ). Turutan menaik yang dihasilkan dikenali sebagai rantaian

tambahan. Menemukan sesuatu rantai optimum telah terbukti menjadi per-

masalahan NP-lengkap. Perihal ini memberi ruang kepada kemunculan kaedah-

kaedah heuristik yang menyediakan penyelesaian hampir optimum. Dengan

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ini, kajian tentang aritmetik titik yang cekap pada keluk eliptik boleh diper-

mudahkan menjadi kajian cara pengoptimuman rantaian tambahan.

Tesis ini berkisar tentang penyelidikan suatu kaedah baru untuk mempercekap-

kan hasil darab skala. Ianya bertujuan untuk meningkatkan kelajuan pelak-

sanaan aritmetik titik EC. Hal ini boleh dicapai melalui pemendekan rantaian

tambahan. Kaedah baru ini akan dibangunkan dari awal, dan dianalisis se-

cara matematik. Untuk tujuan ujian makmal, beberapa pembolehubah akan

digunakan untuk mengesahkan penemuan-penemuan ini.

Kaedah-kaedah sedia ada mengeksploitasi perwakilan dedua(de-m) untuk sesu-

atu nombor bulat n, sementara kaedah baru yang akan dicadangkan pula mem-

buka lembaran baru untuk mengkaji permasalahan ini. Nombor bulat n dil-

eraikan kepada hasil darab unsur perdana berbentuk pe11 pe22 . . . pess . Pendekatan

klasik satu-lapisan digubah menjadi pendekatan dua-lapisan. Kecekapan boleh

dipertingkatkan di lapisan perdana pi, lapisan kuasa perdana peii dan juga

lapisan penggabungan sesama kuasa perdana dalam membentuk sesuatu n.

Pada awalnya, Kaedah Terurai (DM) telah dibangunkan berasaskan kepada

peleraian kuasa perdana dari sesuatu nombor bulat n. Setiap perdana pi,

diberikan satu peraturan yang unik bagi menghasilkan rantaian tambahan.

Suatu ei penggandaan setiap peraturan pi menghasilkan rantaian tambahan

untuk peii yang jika digabungkan sesama sendiri akan menghasilkan n. Analisis

bermatematik menunjukkan bahawa rantaian yang dihasilkan oleh kaedah ini

terbatas pada lingkungan rantai optimum yang dikaji oleh A. Brauer. Ujian

menunjukkan penambahbaikan yang besar telah terhasil melalui kaedah ini

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berbanding dengan kaedah-kaedah yang terdahulu. Suatu versi yang lebih baik

dinamakan Kaedah Terurai Bertanda (SDM), memperkenalkan operasi pengu-

rangan ke dalam turutan bagi menghasilkan rantaian tambahan kurangan. P.

Erdos menyatakan bahawa paling panjang rantaian tambahan kurangan adalah

sepanjang rantaian tambahan. Hasilnya, SDM berjaya menambahbaik DM se-

banyak 8 peratus. Lebih lagi, ia berjaya mengatasi kebolehan kaedah-kaedah

yang sebelumnya dengan penambahbaikan yang besar dan sejajar.

Sebagai tambahan, kami telah bangunkan Kaedah Komposisi (CM) yang be-

rasaskan pengiraan rantaian tambahan secara langsung dari peraturan bagi n.

Berbeza dari kaedah penguraian, CM kembali kepada pendekatan satu lapisan

dengan kebanyakan sifatnya diwarisi dari DM pada lapisan perdana. Suatu

versi penambahbaikan yang dinamakan Kaedah Komposisi Bertanda (SCM)

juga dicadangkan sebagai implikasi memperkenalkan operasi pengurangan ke

dalam CM. Rantaian yang dihasilkan oleh SCM telah mencatatkan peningkatan

sebanyak 10 peratus berbanding dengan CM. Tambahan pula, SCM telah me-

nunjukkan kelebihan berbanding kaedah sedia ada bagi kebanyakan n.

Perbandingan awal antara DM dan CM memihak ke arah DM untuk semua

n. Walau bagaimanapun, uji kaji antara SDM dan SCM menunjukkan bahawa

untuk n yang kecil, SDM adalah lebih baik dari SCM, tetapi untuk n yang

besar, SCM sebaliknya mengatasi SDM.

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ACKNOWLEDGEMENTS

This thesis describes the months of research works I conducted during my PhD

candidature at INSPEM, Universiti Putra Malaysia. The studies centered on

elliptic curve cryptography. This thesis shows the result of contribution of many

people whom expressed their ideas, experiences, knowledge and competence

and like. For this reason, here is my short but sole and sincere appreciation.

First and foremost, all praise to the Almighty Allah for His blessings and mer-

cifuls that enable me to learn.

I am priviledged to have known Dr. Rushdan who had offered progressive

support, ingeneous critics and brilliant suggestions throughout the tenure. I

am deeply indebted to Prof. Kamel for his invaluable insights and thorough

inductiveness on theoretical foundation of my studies. I am grateful to Dr.

Zuriati for her encouragement and continous support from the first day of my

PhD.

My sincere appreciation to Prof. Kamel for his lectures on number theory

and algebraic number theory and for being with so much patient with my

overtimely manner questions, Dr Isamiddin for his lectures on abstract algebra

and for being able to answer if not all, most of my non-stopped questions, and

Dr. Ali for his lectures on computational algorithm, of which altogether, they

have had (almost) completed the foundation I need to conduct this research.

My special thanks to INSPEM secretaries for the documentation and admin-

istrative issues, anonymous reviewers for which have contributed to the im-

provement of the quality of my articles, thesis examiners for their reviews,

constructive feedbacks and corrections which have put up my thesis for the

better.

Last but never least, my everything, my mother for her unlimited love, my wife

and my family for whom I am inspired to do this study.

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I certify that a Thesis Examination Committee has met on 15 November 2011

to conduct the final examination of Mohamad Afendee Mohamed on his thesis

entitled “On the Improvement of Addition Chain in Applications to Elliptic

Curve Cryptosystem” in accordance with the Universities and University Col-

leges Act 1971 and the Constitution of the Universiti Putra Malaysia [P.U.(A)

106] 15 March 1998. The Committee recommends that the student be awarded

the Doctor of Philosophy.

Members of the Thesis Examination Committee were as follows:

Hishamuddin Zainuddin, PhD

Associate Professor

Faculty of Science

Universiti Putra Malaysia

(Chairman)

Azmi Jaafar, PhD

Associate Professor

Faculty of Computer Science and Information Technology

Universiti Putra Malaysia

(Internal Examiner)

Siti Hasana Sapar, PhD

Senior Lecturer

Faculty of Science

Universiti Putra Malaysia

(Internal Examiner)

Raphael Phan, PhD

Lecturer

Loughborough University

England

(External Examiner)

SEOW HENG FONG, PhD

Professor and Deputy Dean

School of Graduate Studies

Universiti Putra Malaysia

Date: 20 December 2011

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This thesis was submitted to the Senate of Universiti Putra Malaysia and has

been accepted as fulfilment of the requirement for the degree of Doctor of

Philosophy. The members of Supervisory Committee were as follows:

Mohamad Rushdan Md Said, PhD

Associate Professor

Faculty of Science

Universiti Putra Malaysia

(Chairman)

Kamel Ariffin Mohd Atan, PhD

Professor

Faculty of Science

Universiti Putra Malaysia

(Member)

Zuriati Ahmad Zukarnain, PhD

Associate Professor

Faculty of Computer Science and Information Technology

Universiti Putra Malaysia

(Member)

BUJANG KIM HUAT, PhD

Professor and Dean

School of Graduate Studies

Universiti Putra Malaysia

Date:

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DECLARATION

I declare that the thesis is my original work except for quotations and citations

which have been duly acknowledged. I also declare that it has not been pre-

viously, and is not concurrently, submitted for any other degree at Universiti

Putra Malaysia or at any other institution.

MOHAMAD AFENDEE MOHAMED

Date: 15 November 2011

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TABLE OF CONTENTS

Page

DEDICATIONS ii

ABSTRACT iii

ABSTRAK vi

ACKNOWLEDGMENTS ix

APPROVAL x

DECLARATION xii

LIST OF TABLES xvi

LIST OF FIGURES xvii

LIST OF ABBREVIATIONS xviii

LIST OF NOTATIONS xix

CHAPTER

1 INTRODUCTION 1

1.1 Cryptology 11.1.1. Foundation of Cryptography 21.1.2. Keyed Cryptography 41.1.3. Public Key Cryptography 5

1.2 Problem Statement 91.3 Objectives 111.4 Contribution 111.5 Thesis Organisation 13

2 ELLIPTIC CURVE 16

2.1 Introduction 162.2 Mathematical Background 172.3 Group Structure 242.4 Elliptic Curve Over Finite Fields 312.5 Endomorphism 362.6 Summary 39

3 SCALAR MULTIPLICATION 40

3.1 Introduction 403.2 Addition Chain 413.3 Integer representation 47

3.3.1. Binary representation 483.3.2. m-ary representation 603.3.3. ψ-ary representation 64

3.4 Addition Chain Methods 683.4.1. Binary Method 68

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3.4.2. m-ary Method 713.4.3. ψ-ary Method 73

3.5 Window Size Against Key Size 743.5.1. Algorithm Development 743.5.2. Results 76

3.6 Summary 79

4 DECOMPOSITION METHOD 80

4.1 Introduction 804.2 Decomposition Method 814.3 Algorithm Development 90

4.3.1. Building Up Rules 924.3.2. Constructing Chain 95

4.4 Analysis 964.5 Result 1014.6 Summary 103

5 SIGNED DECOMPOSITION METHOD 105

5.1 Introduction 1055.2 Decomposition Method Revisited 1075.3 Signed Decomposition Method 1095.4 Algorithm Development 1195.5 Analysis 1205.6 Results 1265.7 Summary 129

6 COMPOSITION BASED METHODS AND COMPARA-

TIVE STUDIES 130

6.1 Introduction 1306.2 Composition Method 1316.3 Signed Composition Method 1336.4 Algorithm Development 1356.5 Analysis 1376.6 Results 139

6.6.1. DM versus CM 1416.6.2. SCM versus CM 1456.6.3. SDM versus DM 1496.6.4. SDM versus SCM 153

6.7 Summary 157

7 CONCLUSION 158

7.1 Work Done 1587.2 Open Questions 160

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BIBLIOGRAPHY 162

BIODATA OF STUDENT 172

LIST OF PUBLICATIONS 173

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