A Literature Review on Scalar Recoding Algorithms

ISSN 2039 - 5086 Vol. 5 N. 4 August 2015

International Journal on

Communications Antenna and Propagation

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(IRECAP)

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Contents:

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A Literature Review on Scalar Recoding Algorithms in Elliptic Curve Cryptography by Mohsen Bafandehkar, Sharifah Md Yasin, Ramlan Mahmod

183

Statistical Peak to Average Power Ratio Bound by M. Y. Bendimerad

190

Geographic Routing Protocols for Wireless Sensor Networks: Design and Security Perspectives by Ali I. Adnan, Zurina M. Hanapi

197

Comparison of Rectangular and Circular Microstrip Antenna’s Array for a WIFI and RFID Applications at 2.45 GHZ and 5.8 GHZ by Loubna Berrich, Mohammed Lahsaini, Lahbib Zenkouar

212

A Measurement Study of the Quality of Zero Line Programming Technique for Smartphones’ Applications by Mohammad Masoud, Yousef Jaradat, Ismael Jannoud, Omar Alheyasat

222

Design of UWB Multilayer Patch Antenna Using T-Probe for Breast Tumor Detection by Soufian Lakrit, Hassan Ammor

228

Reflection Phase Characteristics of EBG Structures and WLAN Band Notched Circular Monopole Antenna Design by Naveen Jaglan, Samir Dev Gupta

233

Radio Link Design for Unmanned Aerial Vehicles (UAVs) with SQAM/TQAM Configuration and Alamouti/STBC Codes by Amirhossein Fereidountabar, Gian Carlo Cardarilli

241

Design of a Low Noise Amplifier Using the Quarter Wave Transformers Matching Technique in the Frequency Band [9-13] GHz by I. Toulali, M. Lahsaini, L. Zenkouar

248

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Compact Printed Dipole Antenna for L- Band Radar Applications by R. Poonkuzhali, D. Thiripurasundari, Zachariah C. Alex, T. Balakrishnan

International Journal on Communications Antenna and Propagation (IRECAP) Rongxing Lu Division of Communication Engineering, School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore

Editorial Board:

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Dalhousie University - Department of Eng. Mathematics and Internetworking SUPELEC University NEC Laboratories Europe - Network Research Division Florida International University - School of Computing and Information Sciences Dalhousie University - Department of Electrical and Computer Eng. Universidad Politécnica de Madrid - Dep. de Electromagnet. y Teoría de Circuitos TU Dresden - Institut für Nachrichtentechnik Johannes Kepler University Linz - Institute of Telecooperation TU Darmstadt - Computer Science Department IIT Delhi - Centre for Applied Research in Electronics Technical University of Crete – Dep. of Electronic and Computer Engineering IMST GmbH - Department of Antennas & EM Modelling University of Mississippi - Center for Wireless Communication Jackson State University The University of Texas at Dallas - Department of Computer Science Federal University of Ceara - Computer Science Department University of Limerick Department of Computer Science - National Tsing Hua University Manhattan College Universidad Complutense de Madrid - Dep. de Ingeniería del Software Kyushu University – Dep. of Computer Science and Communication Engineering Institute for Infocomm Research

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(Canada) (France) (Germany) (U.S.A.) (Canada) (Spain) (Germany) (Austria) (Germany) (India) (Greece) (Germany) (U.S.A.) (U.S.A.) (U.S.A.) (Brazil) (Ireland) (Taiwan) (U.S.A.) (Spain) (Japan) (Singapore)

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Nauman Aslam C. Faouzi Bader Marcus Brunner Shu-Ching Chen Zhizhang (David) Chen José A. Encinar Adolf Finger Ismail Khalil Abdelmajid Khelil Shiban Koul Polychronis Koutsakis Marta Martínez Vázquez Mustafa M. Matalgah Natarajan Meghanatan Mittal Neeraj José Neuman De Souza Mairtin O’Droma Hung-Min Sun Mehmet Ulema Luis Javier García Villalba Kiyotoshi Yasumoto Chen Zhi Ning

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International Journal on Communications Antenna and Propagation (I.Re.C.A.P.), Vol. 5, N. 4 ISSN 2039 – 5086 August 2015

A Literature Review on Scalar Recoding Algorithms in Elliptic Curve Cryptography Mohsen Bafandehkar, Sharifah Md Yasin, Ramlan Mahmod

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Abstract – Elliptic Curve Cryptosystem (ECC) is a type of public key cryptography (PKC) based on the algebraic structure of elliptic curve over finite fields. In mid 80s Neal Koblitz and Victor Miller independently proposed the use of elliptic curves in cryptography. For a smaller key size, ECC is able to provide same level of security with RSA. This feature made ECC one of the most popular PKC algorithms today. Scalar multiplication is known as the fundamental operation in ECC algorithm and protocols. The efficiency of ECC is critically depends on efficiency of scalar multiplication operation. Scalar multiplication involves with three levels of computations: scalar arithmetic, point arithmetic and field arithmetic. Improving the first two levels will lead to significant increment in efficiency of scalar multiplication. Scalar arithmetic level can improve by employing an enhanced scalar recoding algorithm that can reduce the Hamming weight or decrease the number of operations in the scalar representation process. This paper reviews some of the recoding algorithms and techniques. Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Elliptic Curve Cryptosystem, Public Key Cryptography, Scalar Multiplication, Scalar

I.

Introduction

Scalar multiplication is known as the fundamental operation in ECC algorithm and protocols. The efficiency of ECC is censoriously depends on efficiency of scalar multiplication operation [3]. Scalar multiplication involves with three levels of computations: scalar arithmetic, point arithmetic and field arithmetic. Improving the first two levels will lead to significant increment in efficiency of scalar multiplication [4]. Utilizing an enhanced scalar recoding algorithm will lead to scalar arithmetic level improvement [4]. This objective can achieved either by reducing the Hamming weight or decreasing the number of operations in the scalar representation process[5]. This work has reviewed six scalar recoding algorithms. Some of the traditional scalar recoding algorithms such as binary method and Traditional NAF (non-adjacent form) method has been studied. And some of the current algorithms including Modified Booth’s algorithm and NAF-Block Recoding Method has been reviewed. The novelty of this work is to review and compare {0, 1, 3}-NAF algorithm which is considerably recent with traditional recoding algorithms .This review will be based on analyzing the data obtained from literature research and extracted from technical reports. The result of this study will be beneficial for further researches and future works. It will be helpful in finding a most effective recoding algorithm for future enhancement to overcome the existing restrictions. This will lead to performance enhancement in same level of security in resource constrained devices.

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Recoding, Non-Adjacent Form

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Over recent years, human lives have been massively computerized and depend not only on the usage of digital devices but reliability of technologies and trustworthy of the services they provide. The reality is, digital devices have become necessary tools of nowadays highly mobile lifestyle. These small, multi-purpose and fairly low-cost devices have several functionalities such as sending and receiving electronic mail and messages, storing documents and sensitive data, remotely accessing to servers and data. Although, these devices are giving us numerous benefits, they also raise new risks to organizations and individuals [1]-[28]. There is a trade in between features and security of digital devices. The more feature available in digital devices the higher level of security is required. A digital environment would not be secure without implementing the security techniques such as encryption technique. Encrypting the sensitive data is one of the biggest defenses to preserve their secrecy and confidentiality from unpermitted users. The concept of encryption has been around since the Ancient Babylonia, yet only recently has been applied to the digital field. Elliptic curves in cryptography has been eccentrically proposed in mid 80s by Neal Koblitz and Victor Miller [1]. It’s experimentally proven that for a smaller key size, ECC is able to afford same level of security with RSA [2]. This property has made ECC one of the widely popular PKC algorithms.

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183

Mohsen Bafandehkar, Sharifah Md Yasin, Ramlan Mahmod

The expected average number of additions is the average Hamming weight of the binary representation and is ( ).

Literature Review

Algorithm II: Right-to-Left Binary Scalar Multiplication Input: An element D of a group ℊ and a non-negative integer = ( … ) Output: The element , ∈ ℊ 1. ′ = 1, ′′ = , = 0 2. ℎ ≪ −1 ′ 3. if =1 ℎ = ′ + ′′ ′′ ′′ 4. =2 ; = +1 5. end while 6. return ′

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III.2. Modified Booth Recoding Modified Booth Recoding has been a very popular technique for reducing the computational cost associated with multiplication. It is especially useful when the size of input is unknown because it guarantees a maximum of or one half of the total computations compared to 2's complement [10]. When converting to modified booth recoding form, using Table I, every other bit is considered the i’th bit starting with zero. For example converting (01100101)2 to modified booth recoding means:

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Elliptic curves are a highly important and pervasive concept of number theory. They have been under study and analysis for decades. They are algebraic/geometric entities which obviously are concerns with several mathematical branches. They have for instance, been used to solve Fermat's last theorem. The utilization of elliptic curves in cryptography has been proposed for the first time by Koblitz and Victor Miller individually in mid-1980’s [1] based on the ECDLP. ECC is known as a sort of PKC which is built upon algebraic structure of elliptic curve over finite fields [6]. Difficulty of the ECDLP plays a major role in the security of ECC, and this problem can be resolved in exponential time [1]. Meanwhile it has to be added that performance of this algorithm is mainly intertwined with the efficiency of its scalar multiplication algorithm [7]. This operation is the most costly and time consuming operation in ECC [8]. Accordingly, efficiency of ECC is critically depends on efficiency of scalar multiplication operation [3]. In addition, efficiency of the scalar multiplication operation is directly depends and can be optimized by improving the performance of scalar recoding [4]. Therefore according to literatures, optimization of recoding algorithm can result in speed up the whole ECC operations [9].

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III. Review on Recoding Techniques and Algorithms

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In this section some of the recoding algorithm are reviewed and analyzed. Suitable scalar algorithm for each algorithm are represented as well.

′ = (010) = (1) = (101) = (−1) = (100) = (−2) = (011) = (2) ′ = (000) = (0)

The advantage of modified booth recoding is that it has a maximum hamming weight of approximately , where n is the bit length of the scalar.

III.1. Binary Representation

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Binary representation and its usage for scalar multiplication has been known for more than 2000 years. Left-to-right and right-to-left binary scalar multiplication algorithms are presented as Algorithms I and II accordingly. The number of doubling operations is approximately log ( )for both algorithms.

0 0 0 0 1 1 1 1

Algorithm I:Left-to-Right Binary Scalar Multiplication Input: An element D of a group ℊ and a non-negative integer = ( … ) Output: The element ∈ ℊ 1. = 1, = − 1 2. ℎ ≥0 3. =2 4. if =1 ℎ = + ′ 5. = − 1 6. end while 7. return

TABLE I MODIFIED BOOTH RECORDING Inputs Result Meaning X 0 0 0 String of Zeros 0 1 X End of String of 1's 1 0 X Single 1 1 1 2X End of String of 1's 0 0 -2X Beginning of 1's 0 1 -X Single Zero 1 0 -X Beginning of 1's 1 1 0 String of 1's

III.3. Non-Adjacent Form (NAF) [11] proposed a right-to-left NAF recoding which converts a binary number into NAF with digit {-1,0,1}. In NAF recoding, a binary number = ( ,…, ) with ∈ {0,1}, converted into a canonical form ′ = ( ′ , ′ ,…, ′ ) with ′ ∈ {1, 0,1}.

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Mohsen Bafandehkar, Sharifah Md Yasin, Ramlan Mahmod

The Algorithm III has the nature to scanning in a right to left fashion. It scan the digits in sequence, starting from to . In order to manifest the execution or working of this algorithm the look-up table presented in Table II can be seen for reference. Based on the work flow of the algorithm, first the Algorithm III has to complete its execution than the output is stored before the initialization of scalar multiplication algorithm. For the purpose of storing the output an extra memory is required. Addition and subtraction algorithm will perform the operations for scalar multiplication.

According to [13] the NAF has the advantage on average case with the hamming weight of approximately where l is the bit length of the scalar. In Algorithm V, there are approximately three times as many doublings as additions. Algorithm V: Left-to-Right Scalar multiplication of NAF Input: Affine point , positive integer with NAF(k) = ( ,…, ) Output: Affine point 1. ←2 2. for = − 3 0 3. if =1 4. ←2 + 5. else if = −1 6. ←2 − 7. else 8. ←2 9. return

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AlgorithmIII: Right-To-Left NAF Recoding [9] Input: A binary number = ( ,…, ) Output: ′ = ( , ′ ,…, ′ ) 1. ⟵ 0; ⟵ 0; ⟵ 0; 2. for i from 0 to m do ( ) 3. ⟵ 4. ′ ⟵ + − 2 5. return ( , ′ ,…, ′ )

Since in computation of scalar multiplication, addition operation is more expensive than doubling operation this algorithm is less expensive to perform scalar multiplication. The [12] adopted [9] idea in Algorithm VI. This algorithm outputs number with the NAF property but it cannot use look-up table.

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(Note: X = 0

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TABLE II LOOK-UP TABLE OF ALGORITHM III [12] + + ′ 0 0 X 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 X 1 0

Algorithm VI: Right-To-Left NAF Recoding[12] Input: = ( ,…, ) Output: ( ) 1. = , = 0, =0 2. for i from m-1 to 0 do 3. if ( = ) then 4. ′ = − 5. while ( > + 1) do 6. = − 1, ′ = 1 − − , = 1 − 7. = , = −1 8. ′ = − 9. while ( > 0) do 10. = − 1, ′ = 1 − , = 1 − 11. return ( ′ , … , ′ )

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The actual idea behind using NAF is to reduce the computational cost associated with multiplication. The NAF is best known as to efficiently minimize the number of 1s in the scalar number which effectively reduces the total computationalcost [12]. Whenever a string of 1s is encountered in the scalar number it is replaced by a (1) and a (-1). The following is an example for converting a binary number into a NAF number. Example: ×

=

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7 = (00111) × = (01001) =8 −1 =7

Algorithm IV: Computing the NAF of a Positive Integer Input: Positive integer Output:NAF(k) as a signed binary representation ( ,…, ) 1. ← 0 2. ℎ ≥1 3. if 4. ←2−( 4) 5. ← − 6. else 7. ←0 8. ← , ← +1 9. return( ,…, )

Later, [12] proposed an optimal left-to-right minimal weight signed-digit recoding algorithm (Algorithm VII). This algorithm use look-up table as shown in Table III. The employed recording method does not possess NAF property. However, it retains a minimal Hamming weight which is as efficient as the Reitweisner’s algorithm. III.4. Complementary Recoding A complementary recoding technique has been introduced by [13]. Algorithm VIII is presenting this method. It can converts a binary to a complementary number.

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Mohsen Bafandehkar, Sharifah Md Yasin, Ramlan Mahmod

TABLE IV HAMMING WEIGHT OF BINARY AND COMPLEMENTARY ALGORITHM Algorithm Hamming weight Binary, w(r) 7 Complementary, w(r) 5

Algorithm VII: Left-to-Right Minimal weight signedDigit Rcoding [12] Input: ( ,…, ) Output: ( , ′ ,…, ′ ) 1. ⟵ 0; ⟵ 0; ⟵ 0; ⟵ 0; 2. for i from 0 to m do ( ) 3. ⟵ 4. ⟵ −2 + + 5. return ( , ′ ,…, ′ )

III.5. {0, 1, 3}-NAF Recoding

TABLE III LOOK-UP TABLE OF ALGORITHM VII ′ 0 0 0 x 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 x 0 1 1 0 1 x 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 x 1 0 (Note:X = 0 1)

TABLE V LOOK-UP TABLE FOR {0, 1, 3}-NAF RECODING Input Special Case Output ′

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1 0 0 0 X 2 0 0 1 0 3 0 0 1 1 4 0 0 1 1 5 0 1 0 X 6 0 1 1 1 7 1 0 1 0 8 1 0 1 1 9 1 1 0 0 10 1 1 0 1 11 1 1 0 1 12 1 1 1 0 13 1 1 1 0 14 1 1 1 1 15 1 1 1 1 *(Count the number of consecutive '1' == 0) then ′ = 1

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Algorithm VIII: Scalar multiplication using complementary recoding Input: An integer K and P is a point on an elliptic curve Output: = 1. = (100.0)( ) 2. = 3. = − − 1 4. =0 5. For = − 1 down to 0 6. =2 7. If ( = 1) 8. = + 9. Else If ( = −1) 10. = − 11. End If 12. return Q

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An innovative {0, 1, 3}-NAF scalar representation using digit set {0, 1, 3} was proposed based on traditional NAF technique by [14]. The proposed scalar representation adopts the non-adjacency property of NAF. It uses base 2 with digits 0, 1 and 3. The new recoding uses a left-to-right mode and converts any binary that has adjacent nonzero digits into the new {0, 1, 3}-NAF representation. In the recoding process, Algorithm IX is used together with a look-up table as shown in Table V for conversion from a binary to the new {0,1,3}-NAF representation.

0 0 1 1 0 1 1 1 0 1 1 1 1 1 1 in the

0 1 * 0 1 0 ′ =1 3 0 1 0 0 0 ′ =1 3 3 ′ =0 3 ′ =0 0 ′ =1 3 ′ =0 0 ′ =1 3 s-block) If(((no.of '1')%2)

Algorithm IX: {0, 1, 3}-NAF Recoding Input: = ( ,…, ) Output: ′ = ( ′ , ′ , … , ′ ){ , , } 1. ← 0; ← 0; ← 0; ← 0; ′ ← 0; 2. −1 0 ( ) 3. ←⌊ ⌋

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The only operation required for recoding is bitwise subtraction. The procedure of this method is given below: Assume a scalar is ( ,…, , ) − −1 then = ∑ 2 = (100 … 0)( ) where = …. and if = 1, = 0; if = 1 = 0,1 … − 1.

(

4. (

,

, ,

, )≡

( 3) ( 5) ( 6) 8) ( 9) ( 10) ( 13) ( 15) ℎ ′ =0 [( 5. , , , , )≡ ′ {( 2) ( 4) ( 7)}] ℎ =1 ) [( ≡ 6. , , , , ′ {( 11) ( 12) ( 14)}] ℎ =3 7. ′ , ′ ,…, ′

Example: = 687 = (1010101111) − −1 = (100. .0)( ) = (100000000) − (0101010000) − 1 = (10 − 10 − 10 − 1000 − 1) = 1024 − 256 − 64 − 16 − 1 = 687 According to Table IV, the complementary method can reduce the Hamming weight of a binary. However, this method is only efficient if the total number of 0 is less than the total number of 1 in the binary number.

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1) (

Int. Journal on Communications Antenna and Propagation, Vol. 5, N. 4

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Mohsen Bafandehkar, Sharifah Md Yasin, Ramlan Mahmod

Consider a binary number, =( ,…, ) where ∈ {0, 1}, and is the bit length of . Let ℎ denote the Hamming weight of r and be the number of occurrences of ≠ 0, for all ∈ , where 0 < < and 0 < ≤ . Algorithm IX together with Table V can convert a binary r into ′ = ∑ ′ 2 where ′ ∈ {0, 1, 3} and ℎ the Hamming weight of ’. Also, . ’ can be written as ′ = ( ′ , … , ′ ){ , , }

The process of recoding the scalar k into signed binary representation is as follows: a) Firstly partition the given input into n blocks of binary with equal size. b) Sufficient padding bits of 0 will be appended to the left of the most left block if there is a block with different size. In order to represent all blocks with equal size. c) Then assign index to each block of binary to extract the NAF equivalent value from look up table for each block. d) The last step is to combine the blocks and get the final result in NAF. In this method, look up table is used to store NAF values for the blocks. The block size used = 8 bits and NAF values are stored using 9 bytes. Look up table contains 256 values of NAF for the present case with 8 bits block size as 2 = 256. Few such NAF values are shown in Table VI.

Example: For the {0, 1, 3}-NAF recoding, let k = 12632 and P a point on the elliptic curve E. Given the binary expansion of k: 12632 = 2 + 2 + 2 + 2 + 2 + 2 = = (11000101011000) = (300010103000){ , , }

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Algorithm X: {0, 1, 3}-NAF Scalar Multiplication Input: is recoded as{0,1,3} − , =∑ 2 where ∈ {0,1,3}, ∈ ( 2 ) Output: 1. ≔ ( , ) ( ) 2. 3 ≔ 3. if ( = 1) ℎ 4. ≔ 5. if ( = 3) ℎ 6. ≔3 7. for i form m-2 down to 0 do 8. ≔ ( ) 9. if =1 ℎ 10. ≔ ( , ) 11. if =3 ℎ 12. ≔ (3 , ) 13. ( = )

TABLE VI NAF VALUES DEC {0, 1, -1} NAF 0 00000000 1 00000001 2 00000010 3 00000010-1 4 00000100 5 00000101 6 0000010-10 7 00000100-1 8 00001000 9 00001001 10 00001010 . . . . . . 254 1000000-10 255 10000000-1

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[12632] = 2 2 2 2 2 (3 ) +

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The scalar multiplication denoted by [12632] using Algorithm X would be denoted as follows:

IV.

Pros and Cons of Reviewed Techniques and Algorithms

In this section the characteristics of each algorithm is discussed. This characteristic includes advantages and disadvantages of each individual algorithm.

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III.6. NAF-Block RecodingMethod

There are two main steps in NAF method to perform point multiplication operations.  First step is to compute NAF equivalent.  Second step is to perform point multiplication operation from obtained NAF in previous step. As [15] conclude the NAF method for point multiplication is more efficient than the binary method as it reduces the number of hamming weight from for a given input and accordingly increase the speed of multiplication, where l is the bit length of the scalar. Efficient NAF computation operation can improve the efficiency of point multiplication operation. They introduced block method that can improves the speed of NAF computation comparing with the standard method.

IV.1. Binary Method Advantages: - The use of restricted multipliers (e.g., with small Hamming weight) does speedup directly Algorithms [16]. Disadvantages: - High computational load for long key size [17]. IV.2. Modified Booth Recoding Advantages: - When the operands are unknown it guarantees a maximum of 1/2 or one half of the total computations compared to 2's complement [10].

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The advantage of modified booth recoding is that it has a maximum hamming weight of approximately l/2, where l is the length of input.

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Simple look up table.

Disadvantages: - Required computation in order to combine the blocks and compute the final result.

Disadvantages: - High complexity to compute the 2's complement [18].

V.

IV.3. Traditional NAF

Due to the latest advancements and rapid changes occurring in technology, enhancement in speed and security of encryption algorithms are potentially open issues for many digital devices and applicants. To achieve these objectives it has become inevitable to analyze and enhance the existing cryptographic techniques. In this review, advantages and disadvantages of the reviewed algorithms has been studied. According to section IV, each algorithm has its own strengths and weakness. Each algorithm is suitable to be used for different applications. It can also be seen that the adaptation of a recoding technique into another algorithm is practical. For instance, the {0, 1, 3}-NAF has many advantages in its hamming weight property, whereas complicated look up table is its main disadvantage. However, NAF-Block Recoding has a simple and less complex look up table. In this case, the authors’ recommendation is to further study and experiment on adopting the look up table of NAF-Block Recoding with {0, 1, 3}-NAF algorithm. It is determined that combination of these two algorithms will produce a faster algorithm. The new algorithm might inherit the advantages of simplicity in look up table and less required resources from NAF-Block Recoding and less hamming weight from {0, 1, 3}-NAF recoding. Therefore, more investigations and experiments on this hypothesis are suggested by authors’ as a future research direction.

Advantages: - Less hamming weight and less resources required compare with Binary method. - Has good performance in worst case [19].

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Disadvantages: - Possibly of Same Hamming weight in some case and higher complexity [20]. IV.4. Complementary Recoding in

IV.5. Joye and Yen (2000)

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Disadvantages: - The complexity of scalar multiplication, kP, depends on the Hamming weight of the scalar k [21].

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Advantages: - Only operation required for recoding Complementary method is bitwise subtraction.

Conclusion

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Advantages: - It does not have NAF property but still maintain a minimal Hamming weight and it is as efficient as the Reitweisner’s algorithm.

Disadvantages: - A principal disadvantage is that it uses right-to-left recoding to generate unsigned digit representations [The recoded string must be computed and stored before the left-to-right scalar multiplication] [22].

Acknowledgements This work was supported by Department of Computer Science, University Putra Malaysia, UPM Serdang, Selangor, Malaysia.

IV.6. {0, 1, 3}-NAF Recoding

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Advantages: - Better cost and lesser hamming weight for scalar multiplication. - At average case, less complexity compare with Traditional NAF [14].

References [1] [2]

Disadvantages: - High hamming weight at worst case [14]. - High algorithm complexity to compute the X in look up table. - Longer running time required

[3]

IV.7. NAF-Block Recoding Method

[5]

[4]

N. Koblitz, “Elliptic curve cryptosystems,” Math. Comput., vol. 48, no. 177, pp. 203–209, 1987. M. Bafandehkar, S. Md Yasin, R. Mahmod, and Z. M. Hanapi, “Comparison of ECC and RSA Algorithm in Resource Constrained Devices,” IT Convergence and Security (ICITCS), 2013 International Conference on. pp. 1–3, 2013. R. K. Kodali, K. H. Patel, and N. Sarma, “Implementation of Energy Efficient Scalar Point Multiplication Techniques for ECC.,” Int. J. Recent Trends Eng. Technol., vol. 9, no. 1, 2013. A. Rezai and P. Keshavarzi, “A New Left-to-Right Scalar Multiplication Algorithm Using a New Recoding Technique,” Algorithms, vol. 8, no. 3, 2014. X. Huang, P. G. Shah, and D. Sharma, “Minimizing hamming weight based on 1’s complement of binary numbers over GF (2 m),” in Advanced Communication Technology (ICACT), 2010 The 12th International Conference on, 2010, vol. 2, pp. 1226–1230. S. U. Nimbhorkar and D. L. G. Malik, “A Survey On Elliptic

Advantages: - Less iteration compare with NAF (faster). - less resources required compare with NAF [23].

[6]

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Mohsen Bafandehkar, Bafandehkar, Sharifah Sharifah Md Yasin Yasin,, Ramlan Mahmod

[12]

[13] [14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22] [23]

[24]

[25]

[26]

[27]

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[11]

Dr. Sharifah Md. Yasin received her bachelor degree BSc. (Hons) in Mathematics and Statistics from University of Bradford, England in 1991. Her master degree is MSc. in Information Technology from the UniversitiKebangsaan Malaysia in 2002. She graduated her PhD degree in comput computer er security from Universiti Putra Malaysia, 2011. She was a lecturer at University Putra Malaysia from 2004 to 2011. Currently, she is a senior lecturer at University Putra Malaysia. Her PhD research work related to elliptic curve cryptography (ECC). She ddeveloped eveloped new mathematical formula and algorithms in ECC. Her research interest is in cryptography and computer security. She is a member in Malaysian Security for Cryptology Research (MSCR) since 2006. She is also a member in Information Security Research Group for Faculty of Computer Science and Information Technology, University Putra Malaysia since 2011.

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Mohsen Bafandehkar obtained diploma in Mathematics and Physics in 2000. Then he received his Pre Pre-University University certificate in Mathematic in 22001. 001. When he discovered his interest in computer science, he attend a course and graduated with ass associated ociated degree in computer hardware engineering in 2004. After few years of working in industry as hardware engineer and software developer, he decided to get back to academia. He started a course in computing and graduated with First class honors from St Staffordshire affordshire University, UK. He completed his program in 2011. Later on he continued his Master degree in University Putra Malaysia (UPM) in Faculty of Computer Science and Technology (FSKTM) (FSKTM). His research interest are includes but not limited to security, cryptography, ryptography, AI and machine llearning. earning.

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Authors’ information

Profes or Dr. Ramlan Mahmod obtained his Professor Bachelor degree in Computer Science from Western Michigan University, USA in 1982 and his Master degree in Comp Computer uter Science from Central Michigan University, USA in 1984. He completed his PhD in Artificial Intelligence from Bradford University, UK in 1994. He is currently a Professor at Universiti Putra Malaysia. His research interest includes Cryptography, Informa Information tion Security, Computer Graphics and Intelligent Computing. He has published more than 200 articles in academic journals and conference proceedings, more than ten software copyrights, more than ten patents, and several chapters of the books.

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[28] Aloy Anuja Mary, G., Chellappan, C., Elliptic curve cryptography (ECC) based four state quantum secret sharing (QSS) protocol, (2013) International Review on Computers and Software (IRECOS), 8 (8), pp. 1970 1970-1979. 1979.

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Curve Cryptography (ECC),” International Journal of Advanced Studies in Computers, Science and Engineering Engineering,, vol. 57, no. 11. pp. 1443 1443–1453, 1453, 20 2012. B. Ansari and M. A. Hasan, “High “High-Performance Performance Architecture of Elliptic Curve Scalar Multiplication,” Computers, IEEE Transactions on on,, vol. 57, no. 11. pp. 1443 1443–1453, 1453, 2008. M. Li and A. Miri, “1 Analysis of the Hamming Weight of the Extended wmbN wmbNAF,” AF,” 2011. A. Faz-Hernández, Faz Hernández, P. Longa, and A. Sánchez, “Efficient and secure algorithms for GLV GLV-based based scalar multiplication and their implementation on GLV GLV– –GLS GLS curves (extended version),” J. Cryptogr. Eng. Eng.,, pp. 1–22, 1 22, 2014. D. Villeger and V. G. Ok Oklobdzija, lobdzija, “Analysis of booth encoding efficiency in parallel multipliers using compressors for reduction of partial products,” in Signals, Systems and Computers, 1993. 1993 Conference Record of The Twenty Twenty-Seventh Seventh Asilomar Conference on on,, 1993, pp. 781 781–784. 784. G. W. Reitwiesner, “Binary arithmetic,” Adv. Comput. Comput.,, vol. 1, pp. 231 231–308, 308, 1960. R. Katti, “Speeding up elliptic cryptosystems using a new signed binary representation for integers,” in Digital System Design, 2002. Proceedings. Euromicro Symposium on on,, 2002, pp. 380 380–384. 384. S. Klavžar, U. Milutinović, and C. Petr, “Stern polynomials,” Adv. Appl. Math. Math.,, vol. 39, no. 1, pp. 86 86–95, 95, 2007. S. Md Yasin, “New Signed Signed-Digit Digit {0, 1, 3} 3}-NAF NAF Scalar Multiplication Algorithm for Elliptic Curve Over Binary Field,”.PhD Field,”.PhD Thesis,University Thesis,University Putra Malaysia, 2011. H. K. Pathak and M. Shanghi, “Speeding up computation of scalar multiplication in elliptic curve cryptosystem,” Int. J. Comput. Sci. Eng. Eng.,, vol. 2, no. 4, pp. 1024–1028, 1024 1028, 2010. J. López an and d R. Dahab, “Fast multiplication on elliptic curves over GF (2m) without precomputation,” in Cryptographic Hardware and Embedded Systems Systems,, 1999, pp. 316 316–327. 327. B. Wang, H. Zhang, and Y. Wang, “An efficient elliptic curves scalar multiplication for wirel wireless ess network,” in Network and Parallel Computing Workshops, 2007. NPC Workshops. IFIP International Conference on on,, 2007, pp. 131 131––134. 134. D. Villeger and V. G. Oklobdzija, “Evaluation of Booth encoding techniques for parallel multiplier implementation,” El Electron. ectron. Lett., Lett. vol. 29, no. 23, pp. 2016–2017, 2016 2017, 1993. H. K. Pathak and M. Sanghi, “Speeding up Computation of Scalar Multiplication in Elliptic Curve Cryptosystem,” vol. 02, no. 04, pp. 1024 1024–1028, 1028, 2010. K. Wu, D. Li, H. Li, T. Chen, and F. Yu, “Pa “Partitioned rtitioned Computation to Accelerate Scalar Multiplication for Elliptic Curve Cryptosystems,” in Parallel and Distributed Systems (ICPADS), 2009 15th International Conference on on,, 2009, pp. 551 555. 551–555. B. Wang, H. Zhang, Z. Wang, and Y. Wang, “Speeding Up Scalar Multiplication Using a New Signed Binary Representation for Integers,” in Multimedia Content Analysis and Mining SE - 35, 35 vol. 4577, N. Sebe, Y. Liu, Y. Zhuang, and T. Huang, Eds. Springer Berlin Heidelberg, 2007, pp. 277 277–285. 285. M. Joye, “Exponentiation “Exponentiation method resistant against side side-channel channel and safe safe-error error attacks.” Google Patents, 03 03-Jun Jun-2014. 2014. H. Brar and R. Kaur, “Design and implementation of block method for computing NAF,” Int. J. Comput. Appl., Appl., vol. 20, no. 1, pp. 37 37–41, 41, 2011. Alkhatib, M., Al Salem, A., Efficient hardware implementations for tripling oriented el elliptic liptic curve crypto crypto-system, system, (2014) International Review on Computers and Software (IRECOS), 9 (4), pp. 609 609-617. 617. Sharifah, M.Y., Rozi Nor Haizan, N., Jamilah, D., Zaitun, M., {0, 1, 3} NAF representation and algorithms for lightwei lightweight ght elliptic 3}--NAF curve cryptosystem in Lopez Dahab Model, (2014) International Review on Computers and Software (IRECOS), 9 (9), pp. 1541 15411547. Tripathy, P.K., Biswal, D., Multiple server indirect security authentication protoco protocoll for mobile networks using elliptic curve cryptography (ECC), (2013) International Review on Computers and Software (IRECOS), 8 (7), pp. 1571 1571-1577. 1577. Sumathi, D., Kathik, S., Proof of retrievability using elliptic curve digital signature in cloud computing, (2014) International Review on Computers and Software (IRECOS), 9 (9), pp. 1526 1526-1532. 1532.

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Int. Journal on Communications Anten Antenna na and Propagation, Vol. 5, N. 4

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International Journal on Communications Antenna and Propagation (IRECAP) Aims and scope The International Journal on Communications Antenna and Propagation (IRECAP) is a peer-reviewed journal that publishes original theoretical and applied papers on all aspects of Communications, Antenna, Propagation and networking technologies. The topics to be covered include but are not limited to:

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Communications and Information theory, multimedia signal processing, communication QoS and performance modelling, crosslayer design and optimization, communication software and services, protocol and algorithms for communications, communication network security, cognitive radio communications and networking, hardware architecture for communications and networking, emerging communication technology and standards, communications layers, internet protocols, internet telephony and VoIP, fading channel, mobile systems, services and applications, indoor communications and WLAN; superhighways, interworking and broadband VPN, spread spectrum communication. Wireless communications and networking, coding for wireless systems, multiuser and multiple access schemes, mobile and portable communications systems, real-time transmission over wireless channels, optical wireless communications, resource allocation over wireless networks, security, authentication, and cryptography for wireless networks, signal processing techniques and tools, wireless traffic and routing, ultra wide-band systems, wireless sensor networks, wireless system architectures and applications, wireless adhoc and sensor networking, cooperative communications and networking, bio-inspired wireless communications systems, broadband wireless access, broadband networking and protocols, internet services, systems and applications, P2P communications and networking, satellite and space communications, vehicular networks, emerging wireless communication technology and standards. Antenna analysis and design, antenna measurement control and testing, smart reconfigurable and adaptive antennas and multiple antenna systems for spatial, polarization, pattern and other diversity applications, novel materials for enhanced performance, Propagation, interaction of electromagnetic waves with discrete and continuous media, radio astronomy and propagation and radiation aspects of terrestrial and spacebased communications, theoretical and computational methods of predicting propagation and sensing, propagation and sensing measurements in all media, interaction of electromagnetic waves with biological tissue, characterization of propagation media and applications of propagation, multipath interference, channel modeling and propagation.

Instructions for submitting a paper

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The journal publishes invited tutorials or critical reviews; original scientific research papers (regular papers), letters to the Editor and research notes which should also be original presenting proposals for a new research, reporting on research in progress or discussing the latest scientific results in advanced fields; short communications and discussions, book reviews, reports from meetings and special issues describing research on all aspects of Communications, Antenna, Propagation and networking technologies. All papers will be subjected to a fast editorial process. Any paper will be published within two months from the submitted date, if it has been accepted. Papers must be correctly formatted, in order to be published. Formatting instructions can be found in the last pages of the Review. An Author guidelines template file can be found at the following web address: www.praiseworthyprize.org/jsm/?journal=irecap

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Manuscripts should be sent via e-mail as attachment in .doc and .pdf formats to: [email protected]

The regular paper page length limit is defined at 15 formatted Review pages, including illustrations, references and author(s) biographies. Pages 16 and above are charged 10 euros per page and payment is a prerequisite for publication.

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Autorizzazione del Tribunale di Napoli n. 17 del 22/03/2011

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