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UNIVERSITI PUTRA MALAYSIA
NEW SIGNED-DIGIT {0,1,3}-NAF SCALAR MULTIPLICATION
ALGORITHM FOR ELLIPTIC CURVE OVER BINARY FIELD
SHARIFAH MD YASIN
FSKTM 2011 31
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NEW SIGNED-DIGIT {0,1,3}-NAF SCALAR MULTIPLICATION ALGORITHM
FOR ELLIPTIC CURVE OVER BINARY FIELD
SHARIFAH MD YASIN
DOCTOR OF PHILOSOPHY
UNIVERSITI PUTRA MALAYSIA
2011
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NEW SIGNED-DIGIT {0,1,3}-NAF SCALAR MULTIPLICATION ALGORITHM
FOR ELLIPTIC CURVE OVER BINARY FIELD
By
SHARIFAH MD YASIN
Thesis Submitted to the School of Graduate Studies, University Putra Malaysia, in
Fulfilment of the Requirements for the Degree of Doctor of Philosophy
February 2011
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Abstract of thesis presented to the Senate of University Putra Malaysia in fulfillment of
the requirement for the degree of Doctor of Philosophy
NEW SIGNED-DIGIT {0,1,3}-NAF SCALAR MULTIPLICATION ALGORITHM
FOR ELLIPTIC CURVE OVER BINARY FIELD
By
SHARIFAH MD YASIN
February 2011
Chair: Associate Professor Ramlan Mahmod, PhD
Faculty: Computer Science and Information Technology
Elliptic curve cryptography introduced by Neil Koblitz and Victor Miller in the
80’s. Elliptic curve cryptosystem is very popular recently in comparison with the
other public-key cryptosystem such as RSA. This is because elliptic curve
cryptosystem uses smaller key size than RSA but it provides equivalent security
level with the RSA. Smaller key size means less storage requirement, low power
consumption and computing cost. These features are suitable for portable devices
such as PDAs, mobile phone, and smartcards.
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Scalar multiplication is a major operation in elliptic curve cryptosystem. It is the
most time consuming and costly operation. It involves with three levels of
computations: Scalar arithmetic (Level 1), Point arithmetic (Level 2) and Field
arithmetic (Level 3). In the literature, improving scalar representation and
efficient point operation can reduce the scalar multiplication cost.
In this research, a new left-to-right recoding algorithm is proposed to convert a
binary expansion into a new {0,1,3}-NAF representation. The new scalar
representation is in base 2 using digit 0, 1 and 3. The special NAF property is
adopted in the new {0,1,3}-NAF scalar. Also, a new point tripling formula and
algorithm are introduced for elliptic curve over binary field using Lopez-Dahab
projective coordinates. The tripling operation is done by computing (2P+P) using
one doubling followed by one mixed addition. The tripling cost is (12M+7S)
which is cheaper than the cost of computing (2P+P) using traditional method. The
new tripling formula saved (2M+2S) when compared with the traditional method.
Finally, a new signed-digit {0,1,3}-NAF scalar multiplication algorithm is
proposed to utilize the new scalar representation and the new tripling operation.
Mathematical analysis is carried out to identify important features, advantages
and disadvantages of the new signed-digit {0,1,3}-NAF scalar multiplication
algorithm. Cost measurement is based on number of point operations per scalar
throughout the execution of the scalar multiplication algorithm. The cost of three
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scalar multiplication algorithms are compared: double-and-add, addition-
subtraction and new signed-digit {0,1,3}-NAF scalar multiplication. By
comparison with the double-and-add algorithm, the new signed-digit {0,1,3}-NAF
scalar multiplication algorithm has better performance when the value of p is
greater than or equal to 2, where p is the number of digit 3 in the new {0,1,3}-
NAF representation. By comparison with the addition-subtraction algorithm, the
new signed-digit {0,1,3}-NAF scalar multiplication algorithm has better
performance when (h2 – h1) > 0 where h1 and h2 are the Hamming weight of the
new {0,1,3}-NAF and {-1,0,1}-NAF scalars respectively. Also, from observation,
it is effective to use the new signed-digit {0,1,3}-NAF scalar multiplication
algorithm when the percentage of the nonzero digit in the binary expansion is less
than 75%.
In conclusion, performance of a scalar multiplication algorithm depends on the
digits used in the scalar representation, and the efficiency of the point operations
involved in the scalar multiplication algorithm. Efficiently managed point
operations can give optimal number of point operations involved throughout the
execution of the scalar multiplication algorithm and can reduced cost of the scalar
multiplication.
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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai
memenuhi keperluan untuk ijazah Doktor Falsafah
ALGORITMA BARU DIGIT BERTANDA {0,1,3}-NAF PENDARABAN
SKALAR UNTUK KELUK ELIPTIK DALAM MEDAN BINARI
Oleh
SHARIFAH MD YASIN
Februari 2011
Pengerusi: Profesor Madya Ramlan Mahmod, PhD
Fakulti: Sains Komputer dan Teknologi Maklumat
Kriptografi keluk eliptik diperkenalkan oleh Neil Koblitz dan Victor Miller dalam 80an.
Sistem kripto keluk eliptik adalah terkenal di masa kini berbanding dengan sistem kripto
kunci awam seperti RSA. Ini adalah kerana sistem kripto keluk elliptik menggunakan
saiz kunci yang lebih kecil dari RSA tetapi menyediakan tahap sekuriti yang serupa
dengan RSA. Saiz kunci yang kecil bermakna keperluan storannya kecil, penggunaan
kuasa dan kos komputeran yang rendah. Fitur tersebut adalah sesuai untuk alat mudah
alih seperti PDA, telefon mudah alih dan kad pintar.
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Pendaraban skalar adalah operasi utama dalam sistem kripto keluk eliptik. Ia adalah
operasi yang menggunakan masa yang lama dan mahal. Ia melibatkan tiga tahap
komputeran: Aritmetik skalar (Tahap 1), Aritmetik titik (Tahap 2) dan Aritmetik medan
(Tahap 3). Dalam literatur, pembaikan pewakilan skalar dan operasi titik yang efisien
boleh mengurangkan kos pendaraban skalar.
Dalam penyelidikan ini, satu algoritma pengkodan kiri-ke-kanan yang baru di cadangkan
untuk menukar pewakilan binari kepada pewakilan {0,1,3}-NAF yang baru. Pewakilan
skalar yang baru ini adalah dalam asas dua menggunakan digit 0, 1 dan 3. Ciri istimewa
NAF di adaptasikan di dalam skalar {0,1,3}-NAF yang baru. Formula titik gandaan tiga
yang baru dan algoritmanya diperkenalkan kepada keluk eliptik medan binari
menggunakan kordinat projektif Lopez-Dahab. Operasi gandaan tiga ini dilakukan
dengan pengkomputan 2P+P menggunakan satu gandaan dua diikuti dengan satu
tambahan campuran. Kos gandaan tiga adalah 12M+7S, di mana ianya lebih murah
daripada kos pengkomputan 2P+P menggunakan kaedah tradisional. Formula gandaan
tiga yang baru menjimatkan (2M+2S) bila di bandingkan dengan kos kaedah tradisional.
Akhir sekali, algoritma digit bertanda {0,1,3}-NAF pendaraban skalar yang baru di
cadangkan untuk menggunakan sepenuhnya pewakilan skalar dan operasi gandaan tiga
yang baru.
Analisa matematik dijalankan untuk mengenalpasti fitur penting, kebaikan dan
kelemahan algoritma digit bertanda {0,1,3}-NAF pekalian scalar yang baru. Ukuran kos
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adalah berasaskan bilangan operasi titik per skalar di sepanjang larian algoritma
pendaraban skalar. Kos untuk tiga algoritma pendaraban skalar dibandingkan: double-
and-add, addition-subtraction dan digit bertanda {0,1,3}-NAF pendaraban skalar yang
baru. Melalui perbandingan dengan algoritma double-and-add, algoritma digit bertanda
{0,1,3}-NAF pekalian scalar yang baru mempunyai prestasi yang lebih baik apabila nilai
p adalah lebih besar atau sama dengan 2, di mana p adalah bilangan digit 3 dalam
pewakilan scalar {0,1,3}-NAF yang baru. Melalui perbandingan dengan algoritma
addition-subtraction, algoritma digit bertanda {0,1,3}-NAF pendaraban skalar yang baru
mempunyai prestasi yang lebih baik apabila (h2 – h1) > 0, di mana h1 dan h2 adalah berat
Hamming untuk scalar-skalar {0,1,3}-NAF dan {-1,0,1}-NAF mengikut urutan. Melalui
pemerhatian juga, algoritma digit bertanda {0,1,3}-NAF pendaraban skalar yang baru
adalah efektif di gunakan apabila peratus digit bukan kosong dalam pewakilan binari
adalah kurang dari 75%.
Kesimpulannya, prestasi algoritma pendaraban skalar bergantung kepada digit yang
digunakan dalam pewakilan skalar, dan keberkesanan operasi titik yang terlibat dalam
algoritma pendaraban skalar. Operasi titik yang disusun secara efisien boleh memberikan
bilangan operasi titik yang optima di sepanjang larian algoritma pendaraban skalar dan
boleh mengurangkan kos pendaraban skalar.
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ACKNOWLEDGEMENTS
I would like to thank my supervisor, Associate Professor Dr Ramlan Mahmod, for giving
me helpful comments, guidance and continuous support to complete this thesis. Also, I
would like to thank the committee members, Associate Professor Dr Azmi Jaafar and
Associate Professor Dr Muhammad Rushdan, for their valuable comments. Lastly, I
would like to express my gratitude to my family for their support and inspiration.
Sharifah Md Yasin
February 2011
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I certify that a Thesis Examination Committee has met on 24th February 2011 to conduct
the final examination of Sharifah Md Yasin on her thesis entitled “New Signed-Digit
{0,1,3}-NAF Scalar Multiplication Algorithm for Elliptic Curve over Binary Field” in
accordance with the Universities and Universities Colleges 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 degree of Doctor of Philosophy.
Members of the Thesis Examination Committee were as follows:
Ali Mamat, PhD
Associate Professor
Faculty of Computer Science and Information Technology
Universiti Putra Malaysia
(Chairman)
Zuriati Ahmad Zukarnain, PhD
Associate Professor
Faculty of Computer Science and Information Technology
Universiti Putra Malaysia
(Internal Examiner)
Kamel ariffin Mohd Atan, PhD
Associate Professor
Faculty of Science
Universiti Putra Malaysia
(Internal Examiner)
Tsuyoshi Takagi, PhD
Professor
Faculty of Mathematics
Kyushu University, Japan
(External Examiner)
________________________
BUJANG KIM HUAT, PhD
Professor and Deputy Dean
School of Graduate Studies
Universiti Putra Malaysia
Date:
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This thesis was submitted to the Senate of Universiti Putra Malaysia and has been
accepted as the fulfillment of the requirement for the degree of Doctor of Philosophy.
The members of the Supervisory Committee were as follows:
Ramlan Mahmod, PhD
Associate Professor
Faculty of Computer Science and Information Technology
Universiti Putra Malaysia
(Chairman)
Azmi Jaafar, PhD
Associate Professor
Faculty of Computer Science and Information Technology
Universiti Putra Malaysia
(Member)
Muhammad Rushdan Md Said, PhD
Associate Professor
Faculty of Science
Universiti Putra Malaysia
(Member)
_______________________________
HASANAH MOHD GHAZALI, 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 has not been previously, and is not
concurrently, submitted for any other degree at Universiti Putra Malaysia or at
any other institution.
______________________
SHARIFAH MD YASIN
Date: 24 February 2011
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TABLE OF CONTENTS
Page
ABSTRACT ii
ABSTRAK v
ACKNOWLEDGEMENTS viii
APPROVAL ix
DECLARATION xi
LIST OF TABLES 16
LIST OF FIGURES 18
LIST OF ABBREVIATIONS 21
CHAPTER
1 INTRODUCTION 1
1.1 Introduction 1
1.2 Problem Statement 10
1.3 Research Objective 12
1.4 Contribution of Research 14
1.5 Scope of Research 15
1.6 Research Methodology 15
1.7 Thesis outline 17
2 ELLIPTIC CURVE OVER BINARY FIELD 19
2.1 Introduction 19
2.2 Groups 19
2.3 Finite Fields 20
2.4 Polynomials Basis 23
2.5 Elliptic Curve Cryptosystem (ECC) 32
2.5.1 Elliptic Curve Group Operations 32
2.5.2 Elliptic Curve Over Binary Field 34
2.6 Coordinate Representation 36
2.6.1 Affine Coordinates 36
2.6.2 Projective Coordinates 38
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2.7 Lopez Dahab (LD) Projective Coordinate 39
2.7.1 Lopez Dahab (LD) Point Addition 40
2.7.2 Lopez Dahab (LD) Point Doubling 42
2.8 Elliptic Curve Cryptography Simulating ElGamal 44
3 ELLIPTIC CURVE SCALAR MULTIPLICATION 46
3.1 Introduction 46
3.2 Scalar Multiplication 46
3.3 Level 1: Scalar Arithmetic 47
3.3.1 Signed-Digit (SD) Recoding 52
3.3.2 Scalar Multiplication Algorithm 59
3.4 Level 2: Curve/Point Arithmetic 78
3.5 Level 3: Finite Field Arithmetic 80
3.6 Summary 81
4 NEW SIGNED-DIGIT {0,1,3}-NAF SCALAR MULTIPLICATION 83
4.1 Introduction 83
4.2 Level 1: Improved Scalar Arithmetic 83
4.2.1 New Signed-Digit {0,1,3}-NAF Scalar Representation 84
4.2.2 New Signed-Digit {0,1,3}-NAF Scalar Multiplication 90
Algorithm
4.3 Level 2: Improved Point Arithmetic 93
4.3.1 New Point Tripling (3P) Using Mixed Addition 95
4.4 Summary 101
5 RESULT AND DISCUSSION 102
5.1 Introduction 102
5.2 Mathematical Analysis 102
5.3 Analysis of Scalar Recoding 104
5.3.1 Hamming Weight 105
5.3.2 Finding from Scalar Recoding Analysis 114
5.4 Analysis of Scalar Multiplication Algorithms 115
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5.4.1 Double-and-add vs. Signed-Digit {0,1,3}-NAF Scalar 116
Multiplication Algorithm
5.4.2 Addition-Subtraction vs. Signed-Digit {0,1,3}-NAF 130
Scalar Multiplication Algorithm
5.4.3 Finding from Analysis of Scalar Multiplication Algorithm 149
5.5 Summary 151
6 CONCLUSIONS 152
6.1 Introduction 152
6.2 Concluding Remarks 152
6.2.1 Improved Scalar Representation 153
6.2.2 Improved Point Operation 153
6.2.3 Proposed a New Scalar Multiplication Algorithm 154
6.2.4 Result Analysis 154
6.3 Discussion 155
6.3.1 Proposed Signed-Digit {0,1,3}-NAF Method 156
and Left-to-Right Unsigned Width-2 Sliding
Window Method
6.4 Future Research 158
REFERENCES 159
APPENDICES 169
BIODATA OF STUDENT 224
