elliptic curve cryptosystems using efficient exponentiation
|
|
- Tyrone Hensley
- 6 years ago
- Views:
Transcription
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
Some Explicit Formulae of NAF and its Left-to-Right. Analogue Based on Booth Encoding
Vol.7, No.6 (01, pp.69-74 http://dx.doi.org/10.1457/ijsia.01.7.6.7 Some Explicit Formulae of NAF ad its Left-to-Right Aalogue Based o Booth Ecodig Dog-Guk Ha, Okyeo Yi, ad Tsuyoshi Takagi Kookmi Uiversity,
More informationOblivious Transfer using Elliptic Curves
Oblivious Trasfer usig Elliptic Curves bhishek Parakh Louisiaa State Uiversity, ato Rouge, L May 4, 006 bstract: This paper proposes a algorithm for oblivious trasfer usig elliptic curves lso, we preset
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationClassification of problem & problem solving strategies. classification of time complexities (linear, logarithmic etc)
Classificatio of problem & problem solvig strategies classificatio of time complexities (liear, arithmic etc) Problem subdivisio Divide ad Coquer strategy. Asymptotic otatios, lower boud ad upper boud:
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationParallel Vector Algorithms David A. Padua
Parallel Vector Algorithms 1 of 32 Itroductio Next, we study several algorithms where parallelism ca be easily expressed i terms of array operatios. We will use Fortra 90 to represet these algorithms.
More informationAnalysis of Experimental Measurements
Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,
More informationInformation-based Feature Selection
Iformatio-based Feature Selectio Farza Faria, Abbas Kazeroui, Afshi Babveyh Email: {faria,abbask,afshib}@staford.edu 1 Itroductio Feature selectio is a topic of great iterest i applicatios dealig with
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationCOMPARISON OF FPGA IMPLEMENTATION OF THE MOD M REDUCTION
Lati America Applied Research 37:93-97 (2007) COMPARISON OF FPGA IMPLEMENTATION OF THE MOD M REDUCTION J-P. DESCHAMPS ad G. SUTTER Escola Tècica Superior d Egiyeria, Uiversitat Rovira i Virgili, Tarragoa,
More informationThe Minimum Distance Energy for Polygonal Unknots
The Miimum Distace Eergy for Polygoal Ukots By:Johaa Tam Advisor: Rollad Trapp Abstract This paper ivestigates the eergy U MD of polygoal ukots It provides equatios for fidig the eergy for ay plaar regular
More informationA Block Cipher Using Linear Congruences
Joural of Computer Sciece 3 (7): 556-560, 2007 ISSN 1549-3636 2007 Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece &
More informationRecursive Algorithm for Generating Partitions of an Integer. 1 Preliminary
Recursive Algorithm for Geeratig Partitios of a Iteger Sug-Hyuk Cha Computer Sciece Departmet, Pace Uiversity 1 Pace Plaza, New York, NY 10038 USA scha@pace.edu Abstract. This article first reviews the
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationQuantum Computing Lecture 7. Quantum Factoring
Quatum Computig Lecture 7 Quatum Factorig Maris Ozols Quatum factorig A polyomial time quatum algorithm for factorig umbers was published by Peter Shor i 1994. Polyomial time meas that the umber of gates
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More information-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION
NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationFrequency Domain Filtering
Frequecy Domai Filterig Raga Rodrigo October 19, 2010 Outlie Cotets 1 Itroductio 1 2 Fourier Represetatio of Fiite-Duratio Sequeces: The Discrete Fourier Trasform 1 3 The 2-D Discrete Fourier Trasform
More informationDesign and Analysis of Algorithms
Desig ad Aalysis of Algorithms Probabilistic aalysis ad Radomized algorithms Referece: CLRS Chapter 5 Topics: Hirig problem Idicatio radom variables Radomized algorithms Huo Hogwei 1 The hirig problem
More informationIntensive Algorithms Lecture 11. DFT and DP. Lecturer: Daniel A. Spielman February 20, f(n) O(g(n) log c g(n)).
Itesive Algorithms Lecture 11 DFT ad DP Lecturer: Daiel A. Spielma February 20, 2018 11.1 Itroductio The purpose of this lecture is to lear how use the Discrete Fourier Trasform to save space i Dyamic
More information1 Hash tables. 1.1 Implementation
Lecture 8 Hash Tables, Uiversal Hash Fuctios, Balls ad Bis Scribes: Luke Johsto, Moses Charikar, G. Valiat Date: Oct 18, 2017 Adapted From Virgiia Williams lecture otes 1 Hash tables A hash table is a
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationExact Horadam Numbers with a Chebyshevish Accent by Clifford A. Reiter
Exact Horadam Numbers with a Chebyshevish Accet by Clifford A. Reiter (reiterc@lafayette.edu) A recet paper by Joseph De Kerf illustrated the use of Biet type formulas for computig Horadam umbers [1].
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationResearch Article A New Second-Order Iteration Method for Solving Nonlinear Equations
Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationSeed and Sieve of Odd Composite Numbers with Applications in Factorization of Integers
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-75X. Volume 1, Issue 5 Ver. VIII (Sep. - Oct.01), PP 01-07 www.iosrjourals.org Seed ad Sieve of Odd Composite Numbers with Applicatios i
More informationCS / MCS 401 Homework 3 grader solutions
CS / MCS 401 Homework 3 grader solutios assigmet due July 6, 016 writte by Jāis Lazovskis maximum poits: 33 Some questios from CLRS. Questios marked with a asterisk were ot graded. 1 Use the defiitio of
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationEfficient Reverse Converter Design for Five Moduli
Joural of Computatios & Modellig, vol., o., 0, 93-08 ISSN: 79-765 (prit), 79-8850 (olie) Iteratioal Scietific ress, 0 Efficiet Reverse Coverter Desig for Five Moduli Set,,,, MohammadReza Taheri, Elham
More information7. Modern Techniques. Data Encryption Standard (DES)
7. Moder Techiques. Data Ecryptio Stadard (DES) The objective of this chapter is to illustrate the priciples of moder covetioal ecryptio. For this purpose, we focus o the most widely used covetioal ecryptio
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationAlgorithm of Superposition of Boolean Functions Given with Truth Vectors
IJCSI Iteratioal Joural of Computer Sciece Issues, Vol 9, Issue 4, No, July ISSN (Olie: 694-84 wwwijcsiorg 9 Algorithm of Superpositio of Boolea Fuctios Give with Truth Vectors Aatoly Plotikov, Aleader
More informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationGUIDELINES ON REPRESENTATIVE SAMPLING
DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue
More informationLainiotis filter implementation. via Chandrasekhar type algorithm
Joural of Computatios & Modellig, vol.1, o.1, 2011, 115-130 ISSN: 1792-7625 prit, 1792-8850 olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais
More informationRandom Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.
Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationQuadratic Functions. Before we start looking at polynomials, we should know some common terminology.
Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationChapter 7: The z-transform. Chih-Wei Liu
Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationTHIS paper analyzes the behavior of those complex
IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 Itrisic Order Lexicographic Order Vector Order ad Hammig Weight Luis Gozález Abstract To compare biary -tuple probabilities with o eed to compute
More informationSAMPLING LIPSCHITZ CONTINUOUS DENSITIES. 1. Introduction
SAMPLING LIPSCHITZ CONTINUOUS DENSITIES OLIVIER BINETTE Abstract. A simple ad efficiet algorithm for geeratig radom variates from the class of Lipschitz cotiuous desities is described. A MatLab implemetatio
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationOPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES
OPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES Peter M. Maurer Why Hashig is θ(). As i biary search, hashig assumes that keys are stored i a array which is idexed by a iteger. However, hashig attempts to bypass
More information1 Duality revisited. AM 221: Advanced Optimization Spring 2016
AM 22: Advaced Optimizatio Sprig 206 Prof. Yaro Siger Sectio 7 Wedesday, Mar. 9th Duality revisited I this sectio, we will give a slightly differet perspective o duality. optimizatio program: f(x) x R
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationA New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem
This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: +852-2766-7410
More informationModified Logistic Maps for Cryptographic Application
Applied Mathematics, 25, 6, 773-782 Published Olie May 25 i SciRes. http://www.scirp.org/joural/am http://dx.doi.org/.4236/am.25.6573 Modified Logistic Maps for Cryptographic Applicatio Shahram Etemadi
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More information10.2 Infinite Series Contemporary Calculus 1
10. Ifiite Series Cotemporary Calculus 1 10. INFINITE SERIES Our goal i this sectio is to add together the umbers i a sequece. Sice it would take a very log time to add together the ifiite umber of umbers,
More informationConfidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More informationt distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference
EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The
More informationA Note on Matrix Rigidity
A Note o Matrix Rigidity Joel Friedma Departmet of Computer Sciece Priceto Uiversity Priceto, NJ 08544 Jue 25, 1990 Revised October 25, 1991 Abstract I this paper we give a explicit costructio of matrices
More informationHigher-order iterative methods by using Householder's method for solving certain nonlinear equations
Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem
More informationSums, products and sequences
Sums, products ad sequeces How to write log sums, e.g., 1+2+ (-1)+ cocisely? i=1 Sum otatio ( sum from 1 to ): i 3 = 1 + 2 + + If =3, i=1 i = 1+2+3=6. The ame ii does ot matter. Could use aother letter
More informationRandom assignment with integer costs
Radom assigmet with iteger costs Robert Parviaie Departmet of Mathematics, Uppsala Uiversity P.O. Box 480, SE-7506 Uppsala, Swede robert.parviaie@math.uu.se Jue 4, 200 Abstract The radom assigmet problem
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationVector Permutation Code Design Algorithm. Danilo SILVA and Weiler A. FINAMORE
Iteratioal Symposium o Iformatio Theory ad its Applicatios, ISITA2004 Parma, Italy, October 10 13, 2004 Vector Permutatio Code Desig Algorithm Dailo SILVA ad Weiler A. FINAMORE Cetro de Estudos em Telecomuicações
More informationADVANCED SOFTWARE ENGINEERING
ADVANCED SOFTWARE ENGINEERING COMP 3705 Exercise Usage-based Testig ad Reliability Versio 1.0-040406 Departmet of Computer Ssciece Sada Narayaappa, Aeliese Adrews Versio 1.1-050405 Departmet of Commuicatio
More informationGenerating Functions. 1 Operations on generating functions
Geeratig Fuctios The geeratig fuctio for a sequece a 0, a,..., a,... is defied to be the power series fx a x. 0 We say that a 0, a,... is the sequece geerated by fx ad a is the coefficiet of x. Example
More informationChapter 6. Advanced Counting Techniques
Chapter 6 Advaced Coutig Techiques 6.: Recurrece Relatios Defiitio: A recurrece relatio for the sequece {a } is a equatio expressig a i terms of oe or more of the previous terms of the sequece: a,a2,a3,,a
More informationANALYSIS OF EXPERIMENTAL ERRORS
ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationStochastic Simulation
Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso
More informationEE260: Digital Design, Spring n Binary Addition. n Complement forms. n Subtraction. n Multiplication. n Inputs: A 0, B 0. n Boolean equations:
EE260: Digital Desig, Sprig 2018 EE 260: Itroductio to Digital Desig Arithmetic Biary Additio Complemet forms Subtractio Multiplicatio Overview Yao Zheg Departmet of Electrical Egieerig Uiversity of Hawaiʻi
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationApplied Mathematics Letters
Applied Mathematics Letters 5 (01) 03 030 Cotets lists available at SciVerse ScieceDirect Applied Mathematics Letters joural homepage: www.elsevier.com/locate/aml O ew computatioal local orders of covergece
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationResearch Article A Unified Weight Formula for Calculating the Sample Variance from Weighted Successive Differences
Discrete Dyamics i Nature ad Society Article ID 210761 4 pages http://dxdoiorg/101155/2014/210761 Research Article A Uified Weight Formula for Calculatig the Sample Variace from Weighted Successive Differeces
More informationApproximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece
More informationRandom Matrices with Blocks of Intermediate Scale Strongly Correlated Band Matrices
Radom Matrices with Blocks of Itermediate Scale Strogly Correlated Bad Matrices Jiayi Tog Advisor: Dr. Todd Kemp May 30, 07 Departmet of Mathematics Uiversity of Califoria, Sa Diego Cotets Itroductio Notatio
More informationA New Method to Order Functions by Asymptotic Growth Rates Charlie Obimbo Dept. of Computing and Information Science University of Guelph
A New Method to Order Fuctios by Asymptotic Growth Rates Charlie Obimbo Dept. of Computig ad Iformatio Sciece Uiversity of Guelph ABSTRACT A ew method is described to determie the complexity classes of
More informationsubject to A 1 x + A 2 y b x j 0, j = 1,,n 1 y j = 0 or 1, j = 1,,n 2
Additioal Brach ad Boud Algorithms 0-1 Mixed-Iteger Liear Programmig The brach ad boud algorithm described i the previous sectios ca be used to solve virtually all optimizatio problems cotaiig iteger variables,
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More information