Improved Weil and Tate pairings for elliptic and hyperelliptic curves
Full text
(2)
(3) .
(4) . .
(5)
(6) ½.
(7)
(8)
(9) . ¾
(10)
(11)
(12) ! ! "#
(13) . ! $
(14) %
(15) $ % ! & $ %
(16) $
(17) & $ % $ ' $
(18) ( &
(19) ) $ %
(20)
(21) $$ % %
(22) %
(23) $ $
(24) ( %
(25) ) ! $ %
(26) * #+ Æ
(27) ' ! $ %( %
(28) ) & $ % $
(29)
(30) Æ % % * , -
(31) * *
(32) $
(33) . )
(34) $$( %
(35) ) & $ % $ $
(36)
(37) /
(38) $ $ % % $ $
(39) %
(40) #
(41)
(42)
(43) 0+
(44) % ( . ! "#$ % &' & % ( ' ' % ) & Æ & & * . % "+,$ -. /' 0 "$ 1 ( &
(45)
(46) $$ .
(47) . /
(48) .
(49)
(50) %
(51)
(52) ( ! 1 ( 2*
(53)
(54) %%
(55)
(56) (.
(57) 2 % 3 % 4 ! "5$ ( ! * % -2 6 6 7,8 "$ . Æ 6 ( Æ 1 7,8 % 7 "9$ "+$ / "+$ : "7$ 6 ( . Æ "7$ ( % ; 1 % 4 ! & ( ! ' 6 ( ,8 7 6 * 0
(58) . Æ : . "7$ % <
(59) "=$ & 7 ( ( * . 7 ! ( . ( . > ( . = ' .
(60)
(61)
(62)
(63) .
(64) & & : & : - 2 - 2 Æ .
(65)
(66) - 2 & "++ +,5$ -2 ' & ! ( - 2 -2 -2 -2 )
(67) * - 2 ? . * - 2 ? . . . . . . . . -2 ? -- 22 . . . . %' ,
(68) ( - 2 -2 % . . - 2 ? . . - 2 @ -2. * * * . ? ? ? - 2 -2 . "++ ; = A5$ ' ( * 1 - ' "+$2 B . . . . . ? . .
(69)
(70) . -+2. C
(71) - 2
(72) - 2 % Æ "A$ .
(73) . D( 1 Æ 1 6 : 1 .
(74) ¾ %' B -2 . + E ! ( - - 22 % ? " $ ? -2 - 2 -72 - 2 - 2 . ? " + +$ B F + % @ ? - @ 2 @ ? - @ 2 7 % Æ
(75) - 2 ? @ - 2 @ - 2 % Æ
(76) - 2 ? @ - 2 @ - 2 > . ? - @ -2 @ -22 - @ - 2 - 22 ? - @ -2 -22 - @ - 2 @ - 22 / - @ 2 - @ 2 - 2 ( @ ? & ? ? * * ? -2 - 2 - 2 - 2 : * @ -2 @ -2 - 2 : ? ? ? - 2 : -2 ? + . . . . . . . . . . . . . ! " . #
(77) $ % .
(78)
(79)
(80) .
(81)
(82)
(83)
(84)
(85)
(86)
(87)
(88)
(89)
(90)
(91) . . . . . -2 - 2 ? - +2-2 - 2 - 2 . . .
(92)
(93) F? - @ 2 - 2 F? - @ 2 - 2 )
(94) F? - @ 2 - 2
(95) -2 ? ---- @@ 22 --2222 ? --@2 2 --@ 2 2
(96)
(97) - 2 ? - 2 -
(98) 2 ? - @ 2 - 2 ? ? - 2
(99) - 2 - 2 - @ 2 ?
(100) - @ 2 ? - @ 2 -2
(101) - 2 - 2 . - 2 ? - 2 - @ 2 - @ 2 . 1 - @ 2 - 2 -2 ? - 2 - @ 2 G - 2 - @ 2 - 2 - 2 ? - 2 - @ 2 - @ 2 ? - -@ 2 2 - @ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C -2 -2 - @ 2 - 2 - @ 2 - 2 ? - @ 2 - - 22 - @ 2
(102) F? ( - @ 2 - 2 - @ 2 -2 -2 ? 2 - @ 2 -2 - %' -2 H ? -2 . * * -2 * ? @ 0 ! ! -2 & * . * . -2 ? -2 . . . . . . . . . . . . . . . . .
(103) C ? ( ? . ? 0
(104) + - 2 - - 22 ? - +2 * * C ' -2 & -2 - 2 -2 ? - +2 ->2 - 2 - 2 & F
(105)
(106)
(107) !
(108) F
(109) - 2 ? - 2 - 2
(110) -2
(111)
(112)
(113) -2 ? - +2 - 2 ? - 2 - 2 * + ? ? * ? -2 & * ! - 2 ? - 2- 2 * C 0 - 2 & ? -2 ? , ( + C
(114) - 2 ? --22 ? -- 22 --2 2 ? - 2 --22 ? - +2 --22
(115) -2 ? - +2 . . . . . . . . . . . . . . . . . . . . . . ' ( . : ( 1 & & & ( & : & ' & "9$ ! ' 1 E 1 ? - @ 2 - 2 ? - @ 2 - 2 4 . 7+ / . 77 - 2 ? - @ 2 - 2 - 2@-2 " A++$ ? / ( -0+2 " A+7$ -+2 - 2 - 2 ? - 2 - 2
(116)
(117) -=2 - 2. ) * . . . . . . . . . . . .
(118) < - 2 ! & ! . , - 2 & -2 ? --@2 2 --@ 2 2 ? --@2 2 --@ 2 2 % ? " $ ? - @ 2 - 2 -2 - @ 2 -=2 ' F ?
(119) - @ 2
(120) - 2
(121) - 2
(122) - @ 2
(123) - 2
(124) - @ 2 -A2
(125) - @ 2
(126) - 2 . . . . . '
(127) - @ 2
(128) - 2 @ ? - @ 2 + & 7 & .
(129) - 2 ? - 2 - 2 -- 2 - 22 & Æ
(130) - @ 2
(131) - 2 @ I &
(132) - 2 - 2 -- @ 2 2
(133) - @ 2 ? - @ 2 -- @ 2 2 4 -A2 6 9 & 7 & +, ' +9 7 . . . . . ( -72 ?
(134) -2
(135) - 2
(136) - 2
(137) - 2 -52
(138) - 2
(139) -2
(140) - 2
(141) - 2 ? " $ 7 7 +
(142) -2 ? -2 - 2 --2 - 22
(143) - 2 - 2 - 2 -- 2 - 22 . . . . . . . . . . . . . .
(144) + -2 ? - 2
(145) - 2
(146) -2 & +
(147) - 2 - 2 - 2 6 A 7 Æ ' -52 A & 6 ( +7 & 7 & +9 & 7 & 1 ? 7 & = 7,8 . .
(148)
(149)
(150)
(151) # .
(152)
(153) & +
(154) - 2" $ / - 2" $ - 2 ( -2 - 2" $ - 2 - 2 & "9$ -2 F? - - 22 I & -2 -2 F? - 2 ( ) -2 ? -2 . . . . . . . . %' B -2 / % ? " $ ? -2 -92 - 2 . ? " + +$ B F + % @ ? - @ 2 7 %
(155) - 2 ? @ - 2 @ - 2 . ? - @ -2 @ -22 ? - @ -2 -22 . .
(156) / - @ 2 - @ 2 ( @ ? ? ? * * -92 ? - +2 : * @ -2@ -2 - 2 : ? ? ? - 2 : . . . . . . " . - 2" $ - 2 ? -2 . -2 ? - 2
(157) + . # .
(158)
(159)
(160) .
(161)
(162)
(163)
(164)
(165)
(166)
(167)
(168). . ' E - +2 . -2 ? - @ 2 -2 ? - 2 - @ 2 . . . . 0 ? ( . . 7A * 1 E . 7A -A2 7 / + 7 7 A > 1 + +, % ( -52 7 / + 7 + 7 > ( + 5 .
(169) 6 ( 5 & + & +, & + & 1 ? & = 7,8 ; ( "7$ ( Æ : "7$ - ? 72 & ! Æ ' ! /
(170)
(171)
(172)
(173)
(174) .
(175)
(176) Æ ? -2 ? 7
(177) @+ & & 7 &
(178) ? - 2 3 : ? -2 F? - 2 4 4 ! "
(179) " E
(180) : ! "
(181) ? -2 " F? - 2 " E
(182) % ! ! "
(183) " E
(184) "
(185) -# $ 2 & # -2 * " $ -2 -# -22 ? $ -2 & " . . . . . # . %' % F - 2" $ - 2 - 2 ? % & "5 95+$ '
(186) ! - 2" $ - 2
(187)
(188) -
(189) 2 ? ! -! 2 F?
(190) - 2 ½ % ! &
(191) - 2 & . . . . . .
(192) I " 1 / % 4 ! "5 95795$ 3 ' ; ">$ : . >> ; 1 ' ) . # + # #
(193)
(194) " ! ".
(195) ! ? @ @ @
(196) ? @ @ @
(197) -! 2 F?
(198) - @ @ @ 2 -! 2 ? -! 2 -! 2
(199)
(200)
(201)
(202)
(203)
(204) . . . . . . . . 4 ? - 2 - 2 . ! ? @ @ @
(205) ! ? @ @ @
(206) % -7! 7 2 6 ? @ @ @ 7 3 @ ? - @ 7 2 @ @ - @ 7 2
(207)
(208) -
(209) 2 ? @ @ 7
(210)
(211)
(212) ¼ . . . . . . . . . . . . .
(213) ? @ @ .
(214) ¼ -
(215)
(216) ¼ 2 7 ! 3
(217)
(218) ¼ -7! 7 2 :
(219) -2 ?
(220) ¼ - 2 . . . C. 2 ?
(221) - 2
(222) - -7! 7 2 ?
(223) ¼ - 2
(224) - 2 ?
(225) - 2. . -7! 7 2 ? -! 2 -! 2 ? -
(226) - @ @ 22 . . . . . . . .
(227) % .
(228) ! %
(229) -
(230) 2 ? " " - +2
(231) . ! " ?
(232)
(233) -
(234) 2 ? "
(235)
(236) & B " " ; 1 & " & ( -& 2 ? " @ " "
(237)
(238) - 2 % ? +
(239) + . " "
(240) - 2
(241) - 2
(242) &
(243) - 2 ?
(244) - 2
(245) - 2 & - 2 . . . . . . .
(246) ! % % ? "" $ ! " 7 ?
(247) - 2
(248) -2
(249) - 2
(250) -2
(251) ? "" + +$ B -# $ 2 -# $ 2 " " < ; 1 -# $2 " @ "
(252) -2 ? -# -2 # -2 $ -2 @ $ -22 # $ : -2 ? + #-2 ? # -2# -2 $-2 -$2 ' -#2 ? $-2 & " " "> ##$ -#$2 -J# J$2 #J ? - $2# J$ $ - #J2 -J$2 ' -J#2 -J#2
(253) : 7 "> ##$ " :
(254) ? 7 -#-22 7 : - 2 ? $--#-22-@22- 2 -#2 & - 2 F? - 2 -- 22 . . .
(255).
(256) -& 2 ? " @ " " 7 & - 2 ? #-- 22 $-- 22 @ - 2 -- 22 ? $-- 22 @ - 2 & - 2 #-- 22 $-- 22 @ - 2 -- 22 $-- 22 @ - 2
(257) F? -$ @ 2- 2 -$ @ 2-2 -+,2 F? -$ @ 2- 2 -$ @ 2-2 # ( "5$ " :
(258) ? 7 -#-22 7 & - 2 ? -- 22 : F? F? " .
(259) 7 : ( & -#2 & & F? -- 22 , - : ! : " 6 ( ! )-2 -)-2 -22 ! . .
(260). ! ( . G ; 1 3 7 F ? -2 = < > 7 / 7A 7 Æ <E & K & * E ' & E +
(261) ! F? @ 7
(262) @ 7 @ "5$ ? "" $ ?
(263) - 2
(264) - 2
(265) - 2
(266) - 2 ' ? & - 2 & -2 & -2 & -2 . . . . .
(267) & " " " > 7 . 7A 7 & - 2 ? $--#-22-@22- 2 - 2 & E #-- 22 7 -( ( 2 $-- 22 - 2 % = . A, 7 A> 7 ( ! @ 7 @ 7 > / " " " # . . . . . ? & - 2 & - 2 & -2 & -2 / -+,2 #-- 22 % ? + 7 $-- 22 $-- 22 % . +7 . 9 7 >A 7 > 7=8 . 8
(268) . ! . !. . : ( = 3 7 & +
(269) & Æ ? -2 -2 ? @ + @ 7 @ > @ ++ @ + 3 * ? +,>,
(270) ! = 3 ! ? " @ 7 @ += + @ 79$
(271) 3 7A, ? " @ > @ 7 7# @ 7,$ ( ! -! 2 ? > @ 7 @ +=2
(272) ? - @ 7A2 - @ +# @@A7 @ #@+A@7+@@+#2.
(273) ! ' 7! ? " @ 7= @ # +, @ A$ ! ? " @ 7= @ # 7+ @ 7=$ 7 ? " @ @ 7A @ $ -7! 2 ? +A ? -! 2
(274) ? - @ 7A2- @ -+#@@A7 @ #@+A@ 7+@@+#22- @ 7= @ #2 -! 7 2 ? +A ? -! 2
(275) / -! 2 7 -! 2 - +2 @ +# @ 7 @ +A @ +#2 - @ 7= @ #2
(276) ? - @ 7A2- -,. @ 7= @ 77 @ +, @ 2. "
(277)
(278)
(279)
(280) 3( (-(( , 4 5( 6 , - ( Æ %'
(281) $ %'*
(282) $
(283)
(284)
(285) ( ( 5%
(286)
(287) $%
(288) 0708( -9 $ % ': % ##( #( (-(( , , - ( %' ; 2$
(289) ( < (
(290) ( =
(291)
(292)
(293) -9
(294) $ % ': % #( 0( , . ;1 ( '*
(295) $ ! $ %(
(296) 0#>0?<8783 > ? #0( ( 2( 9( 9 $ % * $ $ (
(297) >3@@?<"733 3"@( ( . (
(298) 4%' 1 - ( & % $ 4$ ' $ 9
(299) ¾ A B (
(300) $%
(301) 33373#0 -9 $ % ': % #0( 8( 6
(302)
(303) C% 6
(304) - -( % ( ;
(305) $ $ ! $ % ( .
(306)
(307) !
(308) " #
(309) $ ! % !
(310) ! &! ''(! ! ) $%
(311) 0070. -9
(312) $ % ': % #0( @( 2 ; 4
(313) '2 % C1( 1 % '
(314) *
(315) %
(316)
(317)
(318) %$
(319) (
(320) 8#>#8?<87 @ 3""( ( 2* 6 4
(321) ( $ % & ' %( 9
(322) ; 1 ( 6
(323) *+ ! , ! *-! ! ! ('! $%
(324) 0#700@( -9
(325) $ % ': % ##( "( D( & ! & $ %
(326)
(327) * % *1
(328) $* 1 $'
(329)
(330)
(331) >
(332) ?( 9
(333) ; 1 ( 6
(334) * + ! , ! *-! ! ! ('! $%
(335) #70#( -9 $ % ': % ##(.
(336) . 3( : ( ( $%
(337)
(338)
(339) ( E$*
(340)
(341) $ 3"8( 33(
(342) $ ( $ .
(343) ( 2&
(344) $ % ': % 3"8(.
(345)
Related documents
“Surface forces between cellulose surfaces in cationic polyelectrolyte solutions: The effect of polymer molecular weight and charge density,” Nordic Pulp Paper Res.. “Flocculation
The focuses of this study is to analyses the Iterative Local Gaussian Clustering (ILGC) algorithm and implement the algorithm to the spatio-temporal data, which is crime data..
For this project, stream origin locations were determined using softcopy aerial photography, a Digital Raster Graphic (DRG), a Wake County Digital Line Graphic (DLG), terrain
i. To translate the work process of landscape management & maintenance practised by Pejabat Harta Bina, Universiti Teknologi Malaysia into an information system that will
To set the radio to VOX Mode, repeatedly press the button until the VOX icon flashes on the LCD display.. The VOX icon will appear on the LCD display if the radio is set
Thus, this project is important to the digital crime investigators to identify the information regarding thumb drives that had been connected to a Windows based computer machine
A comprehensive framework simulation test and analysing data from implemented simulation test are to benchmark the privacy and security issue of service delivery model