2 2 ( M2) ( )
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- さみ うとだ
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1 2 2 ( M2) ( )
2 2 P. Gaudry and R. Harley, 2000 Schoof 63bit 2 8 P. Gaudry and É. Schost, bit 1 /
3 p: F p 2 C : Y 2 =F (X), F F p [X] : monic, deg F = 5, J C (F p ) F F p p Frobenius ϕ p : J C J C ϕ p χ J C (F p ) = χ(1) χ k N [χ(ϕ p ) mod 2 k ]D k = 0, for D k J C [2 k ] χ mod 2 k 2 k 3
4 Gaudry Harley D 1 J C [2] D i J C [2 i ] D i+1 s.t. [2]D i+1 = D i D k J C [2 k ] D i 2 D i =(X 2 + u 1 X + u 0, v 1 X + v 0 ), u 1, u 0, v 1, v 0 F q [2](X 2 + U 1 X + U 0, V 1 X + V 0 ) = D i U 1, U 0, V 1, V 0 U 1 U 0 V 1 V 0 Gröbner M 1 (U 1 ) = 0, U 0 = M 0 (U 1 ), V 1 = L 1 (U 1 ), V 0 = L 0 (U 1 ), M 1, M 0, L 1, L 0 F q [U 1 ], deg M 0, deg L 1, deg L 0 < deg M 1 = 16 M 1 (α) = 0 ( X 2 + αx + M 0 (α), L 1 (α) + L 0 (α) ) {D [2]D = D i } i F q 16 M 1 4
5 Gaudry Schost M 1 M 0, L 1, L 0 J C [2] M 1 D 0 := (X 2 + U 1 X + M 0 (U 1 ), L 1 (U 1 )X + L 0 (U 1 )), g J C [2] D 0 + g = (X 2 + U (g) 1 (U 1)X + U (g) 0 (U 1), V (g) 1 (U 1)X + V (g) 0 (U 1)) U (g) 1 F q [U 1 ]/(M 1 ) g M 1 J C [2] G 3 G 2 G 1 G 0 = (Z/2Z) 4 (Z/2Z) 3 (Z/2Z) 2 (Z/2Z) {0} s Gj (U 1 ) := g G j U (g) 1 (U 1) F q [U 1 ]/(M 1 ) G j M 1 s Gj (U 1 ) F q [J C [2] : G j ] 5
6 Gaudry Schost M 1 s G3 (U 1 ) F q 2 T 3 α 3 s G2 (U 1 ) F q (α 3 ) 2 T 2 α 2 s G1 (U 1 ) F q (α 3, α 2 ) 2 T 1 α 1 s G0 (U 1 ) F q (α 3, α 2, α 1 ) 2 T 0 α 0 α 0 M T 3, T 2, T 1, T 0 M 1 T 2, T 1, T 0 F q M 1 T 3, T 2, T 1, T 0 G 3, G 2, G 1 6
7 JANT 16 M M 1 2 G 3, G 2, G 1 T 3, T 2, T 1, T 0 Gaudry Schost M 1 F F p 2 2 T 3, T 2, T 1, T 0 M 1 7
8 M 1 1 M 1 F q [U 1 ] F q 2 i N, D i J C [2 i ]\J C [2 i 1 ] D pj i := ϕ p j(d i ), 0 j 3 F : F p J C [2 i ] = D i, D p i, Dp2 i, D p3 i F q : D i J C [2] J C [2 i ] J C (F q ) J C [2] J C (F q ) [2]D = D i D q2 = D {D [2]D = D i } = {( X 2 + αx + M 0 (α), L 1 (α) + L 0 (α) ) M 1 (α) = 0 } 8
9 M 1 2 M 1 F q [U 1 ] F q i N, D i J C [2 i ]\J C [2 i 1 ], D i+1 s.t. [2]D i+1 = D i, g J C [2] J C [2] J C (F q ) D i+1 D i+1 + g {D i+1 + g g J C [2]} = {D [2]D = D i } = {( X 2 + αx + M 0 (α), L 1 (α) + L 0 (α) ) M 1 (α) = 0 } 1, 2 M 1 = 16 {}}{ (X a 1 )(X a 2 ) (X a 15 )(X a 16 ) 8 {}}{ (X 2 + a 1 X + a 0 ) (X 2 + h 1 X + h 0 ) 9
10 2 2 M 1 F q [U 1 ] 2 T 3,T 2,T 1 : T 0 : M 1 10
11 T 0 G 3, G 2, G 1 T 0 F q T 1 α 1 F q s G1 (U 1 ) M 1 J C [2] = g 1, g 2, g 3, g 4 G 1 = g 1 G 2 = g 1, g 2 G 3 = g 1, g 2, g 3 U ( ˆD) 1 (U 1 ) = U q 1 mod M 1(U 1 ) ˆD J C [2] g 1 := ˆD U q 1 mod M 1(U 1 ) ˆD G 3, G 2, G
12
13 2 2 2 D i 2 D i+1 := (X 2 + U 1 X + M (i+1) 0, L (i+1) 1 X + L (i+1) 0 ) D i+1 2 D i+2 := (X 2 + U 1 X + M (i+2) 0, L (i+2) 1 X + L (i+2) 0 ) 3 i > 1 D i J C [2 i ]\J C [2 i 1 ] F q : D i [2]D i+1 = D i, D i+1 J C (F q 2)\J C (F q ) ˆD ˆD J C [2]\{0} : D i+1 + ˆD = D q i+1 D i+2 + ˆD = D q2 i+2 i > 1 J C [4] J C [2 i ] J C (F q ) D q i+1 D i+1 = D q2 i+2 D i+2 i T 3, T 2, T 1 T 0 T 3, T 2, T 1, T 0 13
14 2 14
15 CPU: Athlon64 2.4GHz Magma V p = , C: Gaudry Schost D 2 J C [4] 2 2 i D i (3 i 10) ( ) D 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 G j JANT
16 2 T j D 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 T T T T U q 1 mod M :G j : JANT : 16
17 2 T j D 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 T T T T :G j : JANT : 17
18 Algorithm 1 2 Input: C : Y 2 = F (X), F F p 5, m 3 Output: D m J C [2 m ]\J C [2 m 1 ] 1: D 1 J C [2] D 2 J C [4] 2: G 0 {0}, G 1 D 1, G 2 D 1, D p 1, G 3 D 1, D p 1, Dp2 1 3: flag false 4: for i = 2 to m 1 do 5: D i 2 M 1, M 0, L 1, L 0 6: if flag=false then 7: M 0, L 1, L 0 G i M 1 α 8: if α / F q then 9: flag ture 10: U ( ˆD) 1 (U 1 ) = U q 1 mod M 1 2 ˆD 11: G 1 ˆD, G 2 ˆD, ˆD p, G 3 ˆD, ˆD p, ˆD p2 12: else 13: M 0, L 1, L 0 G i M 1 α (T j ) 14: D i+1 (X 2 + αx + M 0 (α), L 1 (α)x + L 0 (α)) return D m 18
index calculus
index calculus 2008 3 8 1 generalized Weil descent p :, E/F p 3 : Y 2 = f(x), where f(x) = X 3 + AX + B, A F p, B F p 3 E(F p 3) 3 : Generalized Weil descent E(F p 4) 2 Index calculus Plain version Double-large-prime
:00-16:10
3 3 2007 8 10 13:00-16:10 2 Diffie-Hellman (1976) K K p:, b [1, p 1] Given: p: prime, b [1, p 1], s.t. {b i i [0, p 2]} = {1,..., p 1} a {b i i [0, p 2]} Find: x [0, p 2] s.t. a b x mod p Ind b a := x
Skew-Frobenius IISEC, JANT18 1
Skew-Frobenius IISEC, 2008 7 5 JANT18 1 Frobenius C/F p Jacobian J C (F p n) Frobenius ϕ p ϕ p Z[ϕ p ] End(J C ) ϕ p J C (F p ) J C (F p n) (J C (F p n)/j C (F p )) p g(n 1) g 2, p n J C (F p ) JANT18
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1 2 3 5 6 7 8 9 10 Act Plan Check Act Do Plan Check Do 11 12 13 14 INPUT OUTPUT 16 17 18 19 20 21 22 23 24 25 26 27 30 33 32 33 34 35 36 37 36 37 38 33 40 41 42 43 44 45 46 47 48 49 50 51 1. 2. 3.
SQUFOF NTT Shanks SQUFOF SQUFOF Pentium III Pentium 4 SQUFOF 2.03 (Pentium 4 2.0GHz Willamette) N UBASIC 50 / 200 [
SQUFOF SQUFOF NTT 2003 2 17 16 60 Shanks SQUFOF SQUFOF Pentium III Pentium 4 SQUFOF 2.03 (Pentium 4 2.0GHz Willamette) 60 1 1.1 N 62 16 24 UBASIC 50 / 200 [ 01] 4 large prime 943 2 1 (%) 57 146 146 15
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コンピュータ概論
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x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +
1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4
Copyright c 2008 Zhenjiang Hu, All Right Reserved.
2008 10 27 Copyright c 2008 Zhenjiang Hu, All Right Reserved. (Bool) True False data Bool = False True Remark: not :: Bool Bool not False = True not True = False (Pattern matching) (Rewriting rules) not
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Contents
Contents 2 6 8 10 11 12 13 14 16 18 20 26 28 29 30 31 34 36 38 40 43 44 48 49 50 54 55 56 57 200,000 150,000 100,000 185,848 192,986 190,450 85,33745.9% 27,74014.9% 1,4890.8% 23,15212.5% 23,56912.7% 24,55813.2%
セアラの暗号
1 Cayley-Purser 1 Sarah Flannery 16 1 [1] [1] [1]314 www.cayley-purser.ie http://cryptome.org/flannery-cp.htm [2] Cryptography: An Investigation of a New Algorithm vs. the RSA(1999 RSA 1999 9 11 2 (17
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Ÿ ( ) Ÿ 540,000 980,000 300,000 700,000 1,200,000 1,100,000 1,300,000 980,000 400,000 220,000 280,000 400,000 300,000 220,000 1,300,000 460,000 260,000 400,000 400,000 340,000 600,000 1,500,000 740,000
企業考動報告書2010 JR西日本 CSRレポート
H17.9 H17.11 H18.3 H19.2 H19.6H18.10H17.5 H17.6 H18.3 H17.5 H17.6 H17.9 H18.1 H17.10 H18.3 H18.4 H18.7 H19.4 H19.6 H18.10 H18.10. 4 H 17. 25 H19.10H20.10 H21.9 H21.10H22.3H20.10H21.9H20.4 H20.5 H21.4 H21.10
AJAN IO制御コマンド コマンドリファレンス
- 1 - Interface Corporation 1 3 2 4 2.1...4 2.2...8 2.3...9 2.4...19 2.5...20 3 21 3.1...21 3.2...23 3.3...24 3.4...28 3.5...29 30 31 Interface Corporation - 2 - 1 AJANI/O Linux Web site GPG-2000 http://www.interface.co.jp/catalog/soft/prdc_soft_all.asp?name=gpg-2000
2 ID POS 1... 1 2... 2 2.1 ID POS... 2 2.2... 3 3... 5 3.1... 5 3.2... 6 3.2.1... 6 3.2.2... 7 3.3... 7 3.3.1... 7 3.3.2... 8 3.3.3... 8 3.4... 9 4... 11 4.1... 11 4.2... 15 4.3... 27 5... 35... 36...
1 911 9001030 9:00 A B C D E F G H I J K L M 1A0900 1B0900 1C0900 1D0900 1E0900 1F0900 1G0900 1H0900 1I0900 1J0900 1K0900 1L0900 1M0900 9:15 1A0915 1B0915 1C0915 1D0915 1E0915 1F0915 1G0915 1H0915 1I0915
y = x 4 y = x 8 3 y = x 4 y = x 3. 4 f(x) = x y = f(x) 4 x =,, 3, 4, 5 5 f(x) f() = f() = 3 f(3) = 3 4 f(4) = 4 *3 S S = f() + f() + f(3) + f(4) () *4
Simpson H4 BioS. Simpson 3 3 0 x. β α (β α)3 (x α)(x β)dx = () * * x * * ɛ δ y = x 4 y = x 8 3 y = x 4 y = x 3. 4 f(x) = x y = f(x) 4 x =,, 3, 4, 5 5 f(x) f() = f() = 3 f(3) = 3 4 f(4) = 4 *3 S S = f()
9BBH3A8_P0000
02 Yamaha CSR Report 2007 03 Yamaha CSR Report 2007 04 Yamaha CSR Report 2007 Yamaha CSR Report 2007 05 06 Yamaha CSR Report 2007 Yamaha CSR Report 2007 07 08 Yamaha CSR Report 2007 09 Yamaha CSR Report
,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = , ( ) : = F 1 + F 2 + F 3 + ( ) : = i Fj j=1 2
6 2 6.1 2 2, 2 5.2 R 2, 2 (R 2, B, µ)., R 2,,., 1, 2, 3,., 1, 2, 3,,. () : = 1 + 2 + 3 + (6.1.1).,,, 1 ,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = 1 + 2 + 3 +,
「東京都子供・子育て支援総合計画」中間見直し版（案）第2章 子供と家庭をめぐる状況
1 (1) (2) (3) (4) (5) (6) (7) (8) 2 (1) (2) (3) (4) (5) (6) (7) 10,000 100% 8,000 78.7% 78.1% 79.3% 78.3% 74.1% 75.4% 73.2% 80% 6,000 46.6% 47.5% 50.9% 51.5% 49.9% 51.8% 52.2% 60% 4,000 2,000 3,713
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16 4 1 17 1 50 -1- -2- -3- -4- -5- -6- -7- 1 2-8- -9- -10- -11- Web -12- (1) (2)(1) (3) (4) (1)()(2) (3)(4) -13- -14- -15- -16- -17- -18- -19- -20- -21- -22- -23- (2)(1) (3) -24- -25- -26- -27- -28- -29-