elliptic curve cryptosystems using efficient exponentiation

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1 See discussios, stats, ad author profiles for this publicatio at: elliptic curve cryptosystems usig efficiet expoetiatio Article Jauary 007 CITATIONS 0 READS author: M. Saleh Birzeit Uiversity 5 PUBLICATIONS 39 CITATIONS SEE PROFILE Some of the authors of this publicatio are also workig o these related projects: weak trasitivity i devays chaos View project All cotet followig this page was uploaded by M. Saleh o 05 Jauary 07. The user has requested ehacemet of the dowloaded file. All i-text refereces uderlied i blue are added to the origial documet ad are liked to publicatios o ResearchGate, lettig you access ad read them immediately.

2 EFFICIENT ELLIPTIC CURVE CRYPTOSYSTEMS USING EFFICIENT EXPONENTIATION Kamal Darweesh ad Mohammad Saleh Mathematics Departmet & Scietific Computig Master Program Birzeit Uiversity Palestie Keywords: cryptography, elliptic curves, affie coordiates. 000 AMS Mathematics Subject Classificatio: 94A60, 68P5, G05 Abstract. Elliptic curve cryptosystems (ECC) are ew geeratios of public key cryptosystems that have a smaller key size for the same level of security. The expoetiatio o elliptic curve is the most importat operatio i ECC, so whe the ECC is put ito practice, the major problem is how to ehace the speed of the expoetiatio. It is thus of great iterest to develop algorithms for expoetiatio, which allow efficiet implemetatios of ECC. I this paper, we improve efficiet algorithm for expoetiatio o elliptic curves defied over F p i terms of affie coordiates. The algorithm computes ( P+Q) directly from radom poits P ad Q o a elliptic curve, without computig the itermediate poits. Moreover, we apply the algorithm to expoetiatio o elliptic curves with width-w Mutual Opposite Form (wmof) ad aalyze their computatioal complexity. This algorithm ca speed up the wmof expoetiatio of elliptic curves of size 60-bit about (.7 %) as a result of its implemetatio with respect to affie coordiates. Itroductio Elliptic curve cryptosystems, which were suggested idepedetly by Miller[7] ad Koblitz[5], are ew geeratio of public key cryptosystems that have smaller key sizes for the same level of security.

3 The elliptic curve cryptographic operatios, like ecryptio/decryptio schemes geeratio/verificatio sigature, require computig of expoetiatio o elliptic curve. The computatioal performace of elliptic curve cryptographic protocol such as Diffie-Hellma [3] Key Exchage protocol strogly depeds o the efficiecy of expoetiatio, because it is the costliest operatio. Therefore, it is very attractive to speed up expoetiatio by providig algorithms that allow efficiet implemetatios of elliptic curve cryptosystems [][4][6][8][9][]. There are typical methods for expoetiatio such as biary methods ad widowig methods[9]. These methods ca speed up expoetiatio by reducig additios, where additio of two poits ad doublig of two poits are performed repeatedly. Oe of the efficiet widowig methods is wmof[]. It is a base- represetatio which provide the miimal hammig weight of expoet. Its great advatage is that it ca be geerated from left-to-right which meas, that the recodig does t have to be doe i a separate stage, but ca be performed o-thefly durig the evaluatio. As a result, it is o loger ecessary to store the whole recoded expoet, but oly small parts at oce. Aother approach to speed up expoetiatio is by icreasig the speed of doubligs. Oe method to speed the doubligs is direct computatio of several doublig, which computes P directly from P E(F q ), without computig itermediate poits P, P,, -. Sakai ad Sakurai[] proposed formulae for computig P directly ( ) o E(F p ) i terms of affie coordiates. Sice modular iversio is more expesive tha multiplicatio, their formulae requires oly oe iversio for computig P istead of iversios i usual add-double method. I this paper, we improve efficiet algorithm for expoetiatio o elliptic curve defied over F p i terms of affie coordiates. We costruct efficiet formulae to compute ( P+Q ) directly from P, Q E(F p ), without computig itermediate poits P, P,, P, ( P+Q),, ( P+Q), where. Our formulae have computatioal complexity (4+0)M + (4+6)S +

4 3 I, where M, S ad I deote multiplicatio, squarig ad iversio respectively i F p, ad = +. Moreover, we show i which way this ew algorithm for direct computig ( P+Q ) ca be combied with wmof expoetiatio method []. We also implemet wmof expoetiatio with ad without these formulae ad discuss the efficiecy. The result of this implemetatio shows that.7% speed icrease i wmof expoetiatio with these formulae o elliptic curve of size 60-bit. Let F p deotes a prime fiite field with p elemets. We cosider a elliptic curve E give by Weierstrass o-homogeeous equatio: E: y = x 3 + ax + b where a, b F p, p >3, ad 4a 3 + 7b 0(i.e. E is smooth). Let P = (x, y ), P ( x, y ), P = P = (x, y ) E(F p ). Let the elliptic curve poit additio ad doublig be deoted by ECADD ad ECDBL, respectively. Let M, S ad I deote multiplicatio, squarig ad iversio, respectively i F p, where S = 0.8M, as it is customary owadays. This paper is orgaized as follows: I Sect., we give some defiitios ad otatios. I Sect., we summarize pervious work. I Sect. 3, we will describe our algorithm for direct computig of ( P+Q ) i terms of affie coordiates. I Sect. 4, we use this algorithm i expoetiatio with wmof method, ad show i what way these ew derived formulae ca improve the speed of the expoetiatio. I Sect. 5 timig of our implemetatio will be give. Fially coclusios will be give i Sect. 6. Previous work I this sectio, we summarize the kow algorithms for poit additio, poit doubligs, ad direct doubligs.. Poit additio I terms of affie coordiates, poit additio ca be computed as follows:

5 4 Let P = (x, y ), ad Q = (x, y), where deotes the poit at ifiity, the P ( x, y ) ca be computed as follows x = - x - x y = (x - x ) - y (y - y ) = (.) (x - x ) The formulae above have computatioal complexity S + M + I. []. Poit doublig I terms of affie coordiates, poit additio ca be computed as follows: Assume Let P = (x, y ) O where O deotes the poit at ifiity, the P= P = (x, y ) ca be computed as follows x = - x y = (x x ) - y 3x a (.) y The formulae above have computatioal complexity S + M + I [].3 Direct Doublig Oe method to icrease the speed of doubligs is direct computatio of several doubligs, which ca compute P directly from P E(F q ), without computig the itermediate poits P, P,, - []. Guajardo ad Paar[4] suggested icrease doublig speed by formulatig m algorithms for direct computatio of 4P, 8P, ad 6P o elliptic curves over F i terms of affie coordiates. Sakai ad Sakurai[] proposed formulae for computig P directly ( ) o E(F p ) i terms of affie coordiates. These formulae require oly oe iversio for computig P istead of iversios i regular add-double method. I affie coordiate, direct computatio requires oly oe iversio for computig P istead of iversios i regular add-double method. Therefore direct computatio of several doubligs may be effective i elliptic curve

6 5 expoetiatio i terms of affie coordiate, sice modular iversio is more expesive tha modular multiplicatio [] 3 Direct Computatio of ( P + Q) i affie coordiate I this sectio, we derive formulae for computig ( P+Q ) directly from a give poit P, Q E(F p ) without computig the itermediate poits P, P,, P, ( P+Q),, ( P+Q), where, i terms of affie coordiate. These formulae ca work with wmof expoetiatio method[]. We begi by costructig formulae for small,, the we will costruct algorithm for geeral,. As a example, let =, =, let P = (x, y ), Q = (x, y), P ( x, y ) E(F p ) the for a elliptic curve with Weierstrass form i terms of affie coordiates P = P = (4P +Q) = (x, y ) ca computed as the followig: ) Computig 4P as i [] 4P = P 4 = (x 4, y 4 ) ca be computed as follows. Let C 0 = y A 0 = x 0 B =3x +a 0 A =B 0 0-8A C C =-8C - B (A -4A C ) 4 0 B 3A 6aC A =B - 8A C 4 C =-8C - B (A -4A C ) The 4P = P 4 = (x 4, y 4 ) ca be computed as follows.

7 6 ) Computig (4P +Q) x A 4 (3.) ( 4C0C ) C y 4 (3.) 3 ( 4C0C ) Assume 4P = (x 4, y 4 ) -Q, recall from Sect.., the poit additio the P ( x, y ) = (4P +Q) i term of affie coordiates, ca be computed as follows: Now let 3 C - (4C0C ) y = (4C 0 C )(A - (4C 0 C ) x) (3.3) T =C 0 3 ( 4C C ) y, S A ( 4C C ) x, we get: 0 T = (3.4) (4C 0 C )S Substitutig, ad x 4 ito the expressio for x, we fid x = T S (A (4C0C ) x (4C0C ) S (3.5) Let M A ( 4C C ) x, we get : 0 x = T MS 0 (4C C ) S (3.6) Let A0 T MS ad, substitutig, ad x ito the expressio for y, we get: y = 3 3 (4C0C ) ys T(A 0 (4C0C ) xs ) 3 3 (4C0C ) S Let C ( 4C C ) ys T( A ( 4C C ) xs ), we get: 3) Computig (4P +Q)= P C y = 0 (4C C ) S (3.7) (3.8)

8 7 Recall from Sect.., the poit doublig, the affie coordiates, ca be computed as follows: 4 4 3A = 0 a(4c0c ) S C 0 (4C0C )S 4 4 P = P = (x, y ) i term of (3.9) Now, let B0 3A0 a( 4C0C ) S ad, substitutig, ad x ito the expressio for x, we fid: x = B 0-8A 0C (C ) (4C C ) S (3.0) Let A =B0-8A0 C 0, ad substitutig, y, x ad x ito the expressio for y, we fid y = Let C =-8C - B (A -4A C C - B (A 4A C ) (C ) (4C C ) S ), we get fially: C y = (C 0 ) (4C0C ) S The formulae above have computatioal complexity 8S + M + I (3.) (3.) 3. The formulae Computig ( P + Q) i Affie Coordiate From the above formulae for direct computig (4P +Q), we ca easily obtai geeral formulae that allow direct computig ( P +Q) for. Algorithm 3. describes these formulae. Algorithm 3. Direct Computatio of ( P + Q) i affie coordiate, where, ad P, Q E(F p ). INPUT: P = (x, y ), Q = (x, y) E(F p ) OUTPUT: P = P = ( P +Q)= (x, y ) E(F p ). Compute A 0 ad C 0 ad B 0

9 8 C 0 = y A 0 = x 0 B =3x +a. For i from to Compute A i, C i, for i from to - Compute B i i i- i- i- A =B - 8A C i 4 i- i- i i- i- C =-8C -B (A -4A C ) i- i 4 i i j j=0 B =3A +6 a( C ) 3. Compute the N, V, W, Z the A 0, C 0 - N A ( C i ) x i=0 - V A ( C i ) x i=0-3 W=C ( C i ) y i=0 k - k Z ( C i )N i= A W VN C Z y W( A Z x ) 4. if ( > 0) Compute B B 3A az For i from to Compute A i, C i, for i from to - Compute B i i i- i i A =B - 8AC 4 i i- i- i - i i C =-8C - B (A 4AC )

10 9 i- i 4 4 i i j j=0 B 3A 6 az ( C ) Compute Z Z = - Z( C ) i=0 5. Compute x k, y k A x Z C y Z 3 i Theorem 3. describes the computatioal complexity of this formula. Theorem 3. I terms of affie coordiates, there exits a algorithm that computes ( P +Q) at most [4(+) +] M, [4(+) + ]S, ad I i F p for ay poit P,Q E(F p ) where M, S ad I deote multiplicatio, squarig ad iversio respectively, ad = +. The proof is give i Appedix A. 3. Complexity Compariso For applicatio i practice it is highly relevat to compare the complexity of our algorithm for direct computig of ( P +Q) with regular add-double method which requires ( + ) separated doubligs ad oe additio, ad with Sakai- Sakuri algorithm[] for computig P ad Q. The performace of the ew method depeds o the cost factor of oe iversio relatively to the cost of oe multiplicatio. For this purpose, we itroduce, as [4], the otatio of a "break eve poit." It is possible to express the time that it takes to perform oe iversio i terms of the equivalet umber of multiplicatio eeded per iversio.

11 I geeral let = +, let us deote the direct computig of ( P +Q) by symbol DECDBL(). The our formulae ca outperform the regular double ad add algorithm if the followig relatio to hold: Cost( separate ECDBL + ECADD) > Cost( DECDBL() ) Calculatio Method Complexity Break-Eve ( P +Q) S M I Poit where + = 3 DECDBL(3) M < I 3 doubligs + additio = 4 DECDBL(4) M < I 4 doubligs + additio = 5 DECDBL(5) M < I 5 doubligs + additio 6 + = DECDBL() (3.6 +) M I doubligs + additio Table 3. Complexity compariso: direct computig of ( P +Q) vs. Idividual ( + ) doubligs ad oe additio. Calculatio Method Complexity Break-Eve ( P +Q) S M I Poit where = 4, =0 DECDBL(4) 6 4. M < I Sakai-Sakuri algorithm = 3, = DECDBL(4) M < I Sakai-Sakuri algorithm =, = DECDBL(4) 6-3 M < I Sakai-Sakuri algorithm = DECDBL(4) M I Sakai-Sakuri algorithm 4(+ )+3 4(+ +) 3

12 Table 3. Complexity compariso: direct computig of ( P +Q) vs. direct computig of P ad Q. Igorig squarigs ad additios ad expressig the Cost fuctio i terms of multiplicatios ad iversios, we have: ( M + S + I + M + S + I ) > ( 4( +)M + 4(+)S +M +S + I) We defie r = I/M (the ratio of speed betwee a multiplicatio ad iversio), ad assume that oe squarig has complexity S = 0.8 M[]. We also assume that the cost of field additio ad multiplicatio by small costats ca be igored. Oe ca rewrite the above expressios as: Solvig for r i terms of M oe obtais: r M > (M + 8M +.6 M + 4M) (3.6 +) r > As we see from Table 3., if a field iversio has complexity I > 7.6 M, direct computatio of 3 doubligs ad oe additio may be more efficiet tha 3 separate doublig ad oe additio. Moreover, our algorithm for direct computig of ( P +Q) ca outperform Sakai-Sakuri algorithm for computig P ad Q if: Cost(direct computig of P simply addig the two) > Cost( DECDBL( + ) ) ad direct computig of Q ad the I case, we igore squarigs ad additios ad expressig the Cost fuctio i terms of multiplicatios ad iversios, we have: [(4+) M + (4+)S + (4 +) M + (4 +)S+ 3I + M + S ] > [ 4( +)M + 4(+)S +M +S + I] After simplificatio we ca rewrite the above expressios as: I > 6M +3S - 4 S - 4 M Solvig for r i terms of M oe obtais: r >

13 As we see from Table 3., if a field iversio has complexity I > 4. M, direct computatio of 4 doubligs ad oe additio by usig our algorithm is more efficiet tha 4 doubligs by usig Sakai-Sakuri algorithm ad the performig oe additio. Also, it clear from the table ad the above discussio that DECDBL() is differet from the Sakai-Sakuri algorithm for computig P ad Q. ( P +Q). 3. Expoetiatio with Direct Computatio of ( P + Q) By usig our previous formulae for direct computatio of ( P+Q ), where, ad P,Q E(F p ), we ca improve algorithm B. [] for elliptic curve expoetiatio with wmof by chage the step 3. of algorithm B. [] with a ew step that compute ( P+Q) directly as i the followig algorithm. Algorithm 3. Expoetiatio with wmof Usig Direct Computatio of ( P + Q) INPUT a o-zero t-bit biary strig k, P E(F p ), ad the multiple of the poit P, 0...tw ad 0...tw, the precomputed table look-up. OUTPUT expoetiatio kp.. i t. Q 3. While i do the followig 3.. if (k i XOR k i- ) = 0, the do the followig 3... Q ECDBL(Q) 3... i i else do the followig 3... idex ((k >> (i - w)) & ( w+ - )) - w if ( i < w) Q i -(w- idex) + ( w- idex Q + idex P) 3..3 else if ( i w) Q idex ( w- idex Q + idex P) i i - w 4. If i = 0 do the followig

14 3 4.. Q ECDBL(Q) 4.. If k 0 = the Q ECADD(Q,-P) 5. retur Q I algorithm 3., for each widow width w of wmof, Step 3. performs direct computatio of i-(w- idex) + ( w- idex Q + idex P) if (i < w) otherwise Step 3. performs direct computatios of idex ( w- idex Q+ idex P) if (i w), where idex = 0,, w-, idex P ={±, ±3,..., ±( w- - )}. 3.3 Complexity Aalysis of the wmof Method I this subsectio, we perform a aalysis of wmof method whe it used i cojuctio with the ( P+Q ) formulae. I additio, we compare the complexity of wmof method, with ad without formulae. Moreover we derive a expressio that predicts the theoretical improvemet of the wmof method by usig the formulae, i terms of the ratio betwee iversio ad multiplicatio times. Theorem 3. describes the complexity of algorithm B. [] for computig expoetiatio with wmof. Theorem 3. I terms of affie coordiate, let P E(F p ), t-digits expoet i wmof, the the complexity of algorithm B. [] (w+4 )t (w+3 )t (w+ )t average M + S + I w+ w+ w+ multiplicatio, squarig ad iversio respectively. The proof is give i Appedix A. for computig kp requires o,where M, S ad I deote Now Theorem 3.3 describes the complexity of algorithm 3. for computig expoetiatio with wmof by usig ( P+Q ). Theorem 3.3 I terms of affie coordiate, let P E(F p ), ad t-digits expoet i wmof, the the complexity of algorithm 3. for computig kp requires o

15 4 4( w+3 )t 4( w+ )t t average M + S + I, where M, S ad I deote w+ w+ w+ multiplicatio, squarig ad iversio respectively. The proof is give i Appedix A. Relative Improvemet Let us deote the times it would take to perform expoetiatio by usig algorithms B.[], ad 3. by symbols T Regular method, T Formula method respectively. Accordig to theorems B.[], ad 3., we ca derive expressios for the time it would take to perform a whole expoetiatio with wmof as: ( w+4 )t ( w+3 )t ( w+ )t T Regular method = M + S + I (3.) w+ w+ w+ 4( w+3 )t 4( w+ )t t T Formula method = M + S + I (3.) w+ w+ w+ From equatios 3., ad 3., oe ca readily derive the relative improvemet by defiig r = I/M (the ratio of speed betwee a multiplicatio ad iversio) as: Relative Improvemet = T - T Regular method T Regular method Formula method (3.3) By usig (3.) ad (3.) Relative Improvemet = wi - [( w+ 8) M + ( w+ 5) S] ( w+ ) I + [( w+ 4) M + ( w+ 3) S] (3.4) I our implemetatio S M ad r =.6, let w = 4, the Relative Improvemet is = 4() r - 9 6() r + 3 (3.5) Relative Improvemet is 4(. 6) - 9 = 00 =.7 % (3.6) 6(. 6) Implemetatio ad Results I this sectio, we implemet our methods ad others, which have bee give i previous sectios to show the actual performace of expoetiatio. Implemetatio of a ECC system has several choices. These iclude selectio of elliptic curve domai parameters, platforms [].

16 5 4. Elliptic Curves domai parameters ad Platforms Geeratig the domai parameters for elliptic curve is vary time cosumig. It cosists of a suitably chose elliptic curve E defied over a prime fiite field F p, ad a base poit G E(F p ). Therefore we select NIST-recommeded elliptic curves domai parameters i [0]. We implemet 4 elliptic curves over prime fields F p, the prime modulo p are of a special type (geeralized Mersee umbers) with logp =60, 9, 4, 56. We call these curves as P60, P9, P4, or 56 respectively. The ECC is implemeted o a Petium 4 persoal computer (PC) with.0 GHz processor ad 5 MB of RAM. Programs were writte i Java laguage for multi-precisio iteger operatios, ad are ra uder Widows XP. 4. Timigs aalysis of wmof Expoetiatio Method We performed timig measuremets o the idividual k doubligs ad oe additio operatios ad the correspodig formulae for direct computatio of oe additio adjoit with k doubligs. I additio, we developed timig estimates based o the approximately ratio of speed betwee a multiplicatio ad iversio I/ M i prime filed F p as preseted i Table 4.. Curves Average Timig Average Timig Average Timig r = I / M ( sec) for M ( sec) for S ( sec) for I P P P P Table 4. The ratio of speed betwee a multiplicatio ad iversio i prime filed F p 4.. Optimal Widow Size To show the actual improvemet of wmof method with our ew formula, we must fid out the most efficiecy proper widow size, where the legth of iput biary form is 60-bits, 9-bits, 4-bits, or 56-bits. Figures ( ) illustrate the relatio amog the widow size w, the speed of the evaluatio ad pre-

17 Time of computatio i mesc Time of computatio i mesc Time of computatio i mesc Time of computatio i mesc 6 computed processes. We ca otice from these Figures that whe the widow size icreases, time of the evaluatio will decrease, while time of the precomputatio will icrease, ad the optimal w is 4 whe the iput is 60-bits, ad the optimal w is 5 whe the iputs is 9, 4 or 56-bits. So all the tests i this paper will be processed for w = 4 i 60-bits iput ad w = 5 for 9, 4, or 56-bits. 60 Precompute evalutaio sum 60 Precompute evalutaio sum Widow Size (w) Widow Size (w) Figure 4. Pre-compute ad evaluatio with 9-bits iput Figure 4. Pre-compute ad evaluatio with 60-bits iput 60 Precompute evalutaio sum 60 Precompute evalutaio sum Widow Size (w) Widow Size (w) Figure 4.4 Pre-compute ad evaluatio with 56-bits iput Figure 4.3 Pre-compute ad evaluatio with 4-bits iput

18 7 4.. The performace of improved wmof method Usig Table 4., we ca readily predict that the timigs for performig a expoetiatio with ad without the formulae preseted i Algorithm 3.. I additio, usig the complexity show i equatios (3., 3.) ad the timigs show i Table 4. we ca make estimates as to how log a expoetiatio with wmof will take usig both doubligs with formulae ad idividual doubligs. Measured % Improvemet Predicted Curves Method Timig Timig Predicted Measured P 60 P 9 P 4 P 56 Table 4. wmof with formulae (w = 4 ) wmof (w = 4) wmof with formulae (w = 5 ) wmof (w = 5) wmof with formulae (w = 5) wmof (w = 5) wmof with formulae (w = 5 ) wmof (w = 5) Average time compariso required to perform a expoetiatio without precomputatios stage of a radom poit i mesc (Petium IV.0 GHz).

19 8 5 Coclusio I this paper, we costructed efficiet algorithm for expoetiatio o elliptic curve defied over F p i terms of affie coordiates. The algorithm computes ( P +Q ) directly from radom poits P ad Q o a elliptic curve, without computig the itermediate poits. We showed that our algorithm for computig ( P +Q ) is more efficiet tha Sakai-Sakuri algorithm for computig P ad Q. A compariso was made based o otatio of a "break eve poit," which is the cost factor of oe iversio relatively to the cost of oe multiplicatio. Moreover, we applied the algorithm to expoetiatio o elliptic curve with wmof ad aalyze its computatioal complexity. This algorithm ca speed the wmof expoetiatio of elliptic curve of size 60- bit about (.7 %) as a result of its implemetatio with respect to affie coordiates. Ackowledgmet. The authors would like to thak the referees for their valuable commets ad suggestios that improved the writig of this paper Bibliography [] M. Brow, D. Hakerso, J. Lopez, ad A. Meezes, Software Implemetatio of the NIST Elliptic Curves Over Prime Fields, Topics i Cryptology - CT-RSA 00, LNCS 00, 50-65, (00).

20 9 [] H. Cohe, A. Miyaji, T. Oo, Efficiet Elliptic Curve Expoetiatio Usig Mixed Coordiates, Advaces i Cryptology ASIACRYPT 98, LNCS 54, Spriger, 5-65, 998 [3] W. Diffie, ad M. Hellma, New directios i cryptography, IEEE Trasactios o Iformatio Theory, vol. IT-, o. 6, , (976) [4] J. Guajardo, C. Paar, "Efficiet Algorithms for Elliptic Curves Cryptosystem", Advaces i Cryptography-CRYPTO'97, LNCS, 94, Spriger-Verlage, , (997). [5] N. Koblitz, "Elliptic curve cryptosystems," Mathematics of Computatio, vol 48, , (987). [6] K. Koyama, ad Y. Tsuruoka, Speedig Up Elliptic Curve Cryptosystems usig a Siged Biary Widows Method, Advaces i Cryptology-CRYPTO 9, LNCS740, , (99). [7] V. Miller, Use of Elliptic Curves i Cryptography, Advaces i Cryptology - CRYPTO 85, LNCS 8, Spriger, 47-46, (986). [8] A. Miyaji, T. Oo, ad H. Cohe, Efficiet Elliptic Curve Expoetiatio, Iformatio ad Commuicatio Security - ICICS 997, LNCS 334, Spriger, 8-9, (997). [9] B. Moller, Improved Techiques for Fast Expoetiatio Iformatio Security ad Cryptology - ICISC 00, LNCS 587, Spriger, 98-3, (003). [0] Natioal Istitute of Stadard ad Techology, Digital Sigature Stadard, FIPS Publicatio 86-, February, (000).

21 [] K. Okeya, K. Schmidt-Samoa, C. Spah, T. Takagi, Siged Biary Represetatios Revisited, Advaces i Cryptology CRYPTO 004, LNCS 35, Spriger, 3-39, (004). [] Y. Sakai, K. Sakurai, Efficiet Scalar Multiplicatios o Elliptic Curves with Direct Computatios of Several Doubligs. IEICE Trac.Fudametals, E84-A No., 0-9, (00). Appedix A: Computatioal Complexity I this Appedix, we give proof of theorems 3., 3., 3.3. I the followig proofs, we igore the cost of a field additio ad a subtractio, as well as the cost a multiplicatio by small costats. A. Theorem 3. Proof The complexity of step ad step (the same as i [, Algorithm] ) ivolve (M + 3S) + (M+S)( -) + S - I step 3, we first compute Ci which takes - multiplicatio. Secodly, i=0 we perform oe squarig to compute multiplicatio to compute - ( C i ) i=0 - ( C i ) i=0. Next, we perform oe x. The we obtai N, ad V. Next, we - perform two multiplicatios, oe multiplicatio to compute ( C i ) y ad i=0 other to compute ( C i )( C i ) y ( C i ) y i=0 i=0 i=0. The we obtai W. Third, we perform two squarig to computew,n, ad oe multiplicatio to computevn. The we obtai A 0. Fourth, we perform oe multiplicatio to compute - ( C )N i=0 i. The we obtai Z. Next, we perform two squarig to

22 compute Z, 4 Z,ad oe multiplicatio to compute 3 Z. Next, we perform two multiplicatios to compute Z x, 3 z y, Fially, we perform oe multiplicatio to compute W( A0 Z x ). The we obtai C 0. The complexity of step 3 ivolves ( -)M + 9M +5S. I step 4 we perform oe squarig to compute multiplicatio to compute 4 az, where A 0. Next we perform oe 4 Z is computed i step 3. The we obtai B 0. The complexity of step 4. ivolve M + S ad the complexity of step 4. ivolves (M + 3S) + (M+S)( -) as step. I step 4.3 we compute - C i which takes - multiplicatios. Secodly, i=0 - we perform oe multiplicatio to compute Z( C i ). The we obtai ew i=0 value for Z. the complexity of 4.3 ivolves M. Hece, the complexity of step 4 ivolves 4 M + 4 S. I step 5, we perform oe iversio to compute - Z ad the result is set to T. Next, we perform oe squarig to compute T. Next, we perform oe multiplicatio to compute. The we obtai x A T. Fially we perform two multiplicatios to compute. The we obtai y. The complexity of C T T step 5 ivolves 3M + S + I. So the complexity of above computatios ivolve [4(+) +] M, [4(+) + ]S, where = +. A. Theorem 3. Proof We oticed that algorithm B. [] performs a ECADD operatio each time the curret digit is o-zero, recall from theorem 4 [] that the average o-zero desity of wmof is asymptotically w+ also, oe ECDBL operatio is performed i each iteratio (where i 0) to double the itermediate result. The

23 o average, algorithm B. [] for computig expoetiatio with wmof requires t ECDBL + t ECADD w+ Recall that the computatioal costs for doublig ad additios operatios i affie coordiate. The we ca rewrite previous expressio as: t (M + S + I )t + (M + S + I ) w+ We ca rewrite previous expressio i terms of M, S, ad I as: ( w+4 )t ( w+3 )t ( w+ )t M + S + I w+ w+ w+ A.3 Theorem 3.3 Proof Recall from theorem 4 [] that for t-digits expoet k i its wmof, if t the average o-zero desity of wmof is asymptotically of k is ifiity. Log sequece costituted from two types of blocks:. b = (0), legth of this block is ;. b = (0 i * 0 w-i- ), legth of this block is w ad 0 i w - ; The the umber of block b equals w+ w+ ad wmof because every block b has a o-zero bit, ad the umber of block b equals amout of 0s i wmof the amout of 0s i b which equals w t - ( w - )( ) t = w+ w+ t w+ Now, step 3. of algorithm 3. performs t w+ blocks b ad step 3. performs t w+ block b the algorithm 3. for computig kp requires o average

24 3 t t ECDBL + DECDBL( w) w+ w+ Recall the computatioal costs for doubligs ad additios operatios i affie coordiate. The we ca rewrite previous expressio as: (M+S+I + 4( w +)M +4( w +)S+M +S+I ) w+ We ca rewrite previous expressio i terms of M, S, ad I as: 4( w+3 )t 4( w+ )t t M + S + I w+ w+ w+ View publicatio stats

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