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
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