SEA Journal of Software. Vol.13, No /2002/13(06) ,2, , ); , )

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1 /00/(06) Joura of Software Vo No6 SEA ( 45000) ( 0009) E-ma zyf06@saom htt//wwwhomesa SEA(Shoof Ees At) (sogey yes) Mora SEA F SEA SEA SEA Frobeus SEA TP9 A 985 N Kobtz VS Mer [~5] () ( ) ( ) RShoof [6] Shoof At [7] Ees [8] SEA(Shoof Ees At) MoraLerer [9~] SEA SEA F SEA SEA (sogey yes) t mod SEA (9900) 97 (G ) (000400) (96 ) (976 ) (94 )

2 56 Joura of Software 00(6) SEA F a b F a b 0 = 4a + 7b 0 E( F ) = {( x F E y = x + ax y = x + ax } { O} E F j(e) E j_ ( x _th SEA Φ (<<) E[] E _torso E Frobeus φ ( x ( x φ tφ + = 0 t Frobeus Hasse t y ) Tate T (E) E( F ) = + t t < () φ ( P) + [ ]( P) = [ t ] ( P) P E[] () t mod > (CRT) E( F ) t 4 φ _th f ( ( deg = ( ) / ) E[ ] \ { O} x () F [ x y]/( y = x + ax + f b f ( Shoof O (og 8 ) E[] Frobeus F Ees At ( ( x j( E)) [] ) Ees Ees Isogey ( φ ) h deg = ( ) / λ φ (P) = [λ](p) mod ( y = x + ax h ( λ φ t λ + λ t t ( ζ + + ζ ) mod ζ F F h mod t mod At Frobeus r (r ( x j(e)) [] ) t mod T SEA Ees At t mod t mod CRT BSGS(baby ste ad gat ste) SEA Mora [9] Ees E I I E E E E E I I I I I I E E E I I I E E E _ th f g t mod deg g = ( ) b mod T ( ) j t a mod b {0 } t a + + j 0 b j [4] } Vrtua T t mod Φ * T + j Φ = {a a Mora SEA (ab)ees max t mod max

3 SEA 57 E E E ) ε( x y _th f Frobeus α for = to max E E ) ε ( x y I E E (I I * ) E _th f h E h I I I f h 4 λ ( 0 λ < ) α = α λ ( x 4 retur t α + α mod y + E ) = [ α ]( x mod ( y = x + ax (*) 4 F [ x y]/( y = x + ax x ( y ) O (og ) O(deg F g( α (*) ( x y ) = [ α ]( x mod ( y = x + ax g( h ( g( (**) α t mod At α (**) ( ) / ( ) / ( mod ( ( [56] ) g h g I I I g g I g I I I ( 0) deg g ( < deg (**) (*) /4 40 (ab)ees dmax ( t mod ) E ε ( x ) h ( y α E ε ( x ) I E E (I I * ) + + y E f + h h + + f deg h ( > d max goto ste 6 5 g( f = 0 goto ste 5 5 g( I goto ste 5 5 λ ( 0 λ < ) α = α λ ( x y ) = [ α ]( x mod ( y +goto ste 6 = { α + α T 7 Retur ( T ) mod } + = x + ax g( t mod max Ees = a = b = 6 = (**)

4 58 Joura of Software 00(6) 5 Petum450Mhz t mod 4 ( 7 ) (5 ) = max=5 deg ( ) * = 4 5 = 5 g = 8 t mod SEA At x modφ ( x j( E)) deg (x j(e))=+ Ees Ees t mod t mod ( x y ) mod ( y = x + ax ( ) deg = ( ) At At t mod T T t mod T v SEA Ees At Vrtua Frobeus t t T BSGS T t T T () Ees () At t mod BSGS () Vrtua [4] Iteget Chose System( ICS) ICS SEA 4 Ees = BSGS Vrtua / = T () T > AtBoud z t mod ) ) > 4 s GF( 60 +7) 00 Peum 64M Φ T s t mod T ( s ) 4 (ImaxVrBoudAtBoudz) () Imax Ees () ( 0 v s) (a) (b) () v < 4 s v s > 4 v T < VrBoud s Vrtua T s v v T < AtBoud s + j ( T 450MHzw98 A (ImaxVrBoudAtBoudz)=( ( T ) [4] Tabe Comarso of the effees Fed Cofgurato Sheme Average tme (s) Mmum tme (s) Maxmum tme (s) Petum 450MHz A Petum 00MHz ICS

5 SEA 59 SEA SEA E ( F ) E ( F ) E Isogey Ees SEA t mod E( F ) E( F ) = + t mod E F Ees _th F x x + ax = E( F) E x x + ax = ( x x + ax ) E( F ) E ( F ) E( F) t mod ( x x F x mod ( x y ) = [ λ]( x mod ( y = x + ax y = x + ax a b ab 0 = 4a + 7b 0 At t mod F Ees F x x + ax = goto ste t mod + t mod goto ste E ( F ) E ( F ) goto ste y = x + ax 4 y = x + ax a b ab 0 = 4a + 7b 0 F t mod (a) At t mod (b) Ees <4 t mod () Ees >4 F

6 60 Joura of Software 00(6) x x + ax = goto ste t mod + t mod goto ste E ( F) E ( F) E ( F ) goto ste 4 goto ste 4 y = x + ax 4 (ab) = F = a= b= order= (x=( ) Referees [] ANSI X The Et Curve Dgta Sgature Agorthm (ECDSA) Pub Key Crytograhy for the Faa Serve Idustry Amera Baers Assoato 998 [] ANSI X Key Agreemet ad Key Trasort Usg Et Curve Crytograhy Pub Key Crytograhy for the Faa Serve Idustry Amera Baers Assoato 999 [] IEEE P6 Stadards for Pub-Key Crytograhy Isttute of Eetra ad Eetros Egeers 999 [4] SEC Et Curve Crytograhy Stadards for Effet Crytograhy Grou 999 [5] FIPS 86- Dgta Sgature Stadard Federa Iformato Proessg Stadards 000 [6] Shoof R Coutg ots o et urves over fte feds Joura of Theore des Nombres de Bordeaux 99579~54 [7] At AO The umber of ots o a et urve moduo a rme Seres of e-ma to the NMBRTHRY mag st 99 [8] Ees ND Et ad moduar urves over fte feds ad reated omutatoa ssues I Bue DA Tetebaum JT eds Coutatoa Persetve o Number Theory AMS/Iteratoa Press 998 ~76 [9] Couveges J-M Mora F Shoof s agorthm ad sogey yes I Adema LM Huag MD eds ANTS-I LNCS 877 Srger-Verag 994 4~5 [0] Leer R Mora F Coutg the umber of ots o et urves over fte feds strategy ad erformaes I Guou LC Qusquater JJ eds Proeedgs of the EUROCRYPT 95 LNCS 9 Srger-Verag ~94 [] Dewaghe L Remars o the Shoof-Ees-At Agorthm Mathemats of Comutato ~5 [] Lerer R Fdg good radom et urves for rytosystems defed over I Fumy W ed Proeedgs of the EURO-CRYPT 97 LNCS ~9 [] Lehma F Maurer M M er V et a Coutg the umber of ot o et urves over fte feds of haraterst greater the three I Adema LM Huag MD eds ANTS-I LNCS 877 Srger-Verag ~70 [4] Tetsuya Izu Kogure J Noro M et a Effet Imemetato of Shoof s Agorthm I Ohta K Pe DY eds Proeedgs of the ASIACRYPT 98 LNCS ~79 [5] Shou V A ew oyoma fatorzato agorthm ad ts memetato Joura of Symbo Comutato ~97 [6] Cohe H A Course Comutatoa Agebra Number Theory Graduate Texts Mathemats 8 New Yor Srger-Verag 996 F

7 SEA 6 Effet Imemetato of SEA Agorthm ZHU Yue-fe GU Chu-xag PEI Dg-y (Deartmet of Networ Egeerg Zhegzhou Iformato Egeerg Uversty Zhegzhou Cha) (State Key Laboratory of Iformato Seurty Graduate Shoo The Chese Aademy of Sees Bejg 0009 Cha) E-ma zyf06@saom htt//wwwhomesa Abstrat The ore of hoosg a seure et urve for et urve rytosystems s the auato of the order of a radomy seeted et urve It s ow that SEA (Shoof Ees At) agorthm s reety the most effet method to auate the orders of et urves over F Isogey yes method made by Mora s a mortat oa otmzed tehque to mrove SEA agorthm I ths aer sogey yes method s ehaed ad a sheme of more otma ombato of the varous tehques SEA agorthm s rovded Furthermore some dsussos are made o how to seed u the seeto of et urves wth rme order ad a effet memetato of SEA agorthm over F s desrbed Key words et urve Frobeus edomorhsm SEA (Shoof Ees At) agorthm sogey ye Reeved August 000 aeted Deember 5 00 Suorted by the Natoa Natura See Foudato of Cha uder Grat No9900 the Natoa Grad Fudameta Researh 97 Program of Cha uder Grat NoG the Foudato of He a Prove of Cha for Dstgushed Youg Shoars uder Grat No (NASAC00) ~6 ( ) () () 6 (A4) () PDF PS htt//wwwjosorg E-ma wqx@suedu htt//wwwseuedu/aadem/nasachtm

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