楕円曲線暗号の計算機実験

2009SE295

  

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2

ˆŠ‰R‹

F

p

ŒŠ=Ž k=‘

5

’2“ (+”Gz

p

O

a, b ∈ F

p

I0•–y w—74˜™

E(p; a, b) =

(x, y) ∈ F

p

× F

p

| y

2

= x

3

+ ax + b

∪ {O}

(1)

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F

p

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O

5Wœ6WžŸ O¢¡ £ L @c¤¥„P0e'”N32i¦Y§7

a, b ∈ F

p

5

4a

3

+ 27b

3

6= 0

(2)

<8¨'B …)© ()O … @0j ª « qRrtGu

E(5; 1, 1)

“ (4Ÿ"¬­AG®

F

5

I'¯ €Nw'7

f (x) = x

3

+ x + 1

(

x = 0, 1, 2, 3, 4

3"(4°"7

y =

0, 1, 2, 3, 4

(

y

2

(0°p<c~G … @JO±7

x

0 1 2 3 4

f (x)

1 3 1 1 4

y

0 1 2 3 4

y

2

0 1 4 4 1

OWP"@:jG² H <+³´ … @)Oµ7

x = 1

’¶ (

x

I· €'w 5:7

y

2

= f (x)

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y

¸8¹º … @NMNO ¸8»6¼ @0j½"¾ £ 7

x = 0

(pO4|c7

f (x) = 1

)Y§7

y = 1

O

y = 4

3

y

2

= f (x)

¸8¿ YÁÀ · j

E(5; 1, 1)

5:7

9

Ÿ ¼¦Â P2@

E(5; 1, 1) =

(0, 1), (0, 4), (2, 1), (2, 4), (3, 1),

(3, 4), (4, 2), (4, 3) ∪ {O}

(W˜™632i'@0j

y

ÃÄ <

4 = −1, 3 = −2

OÆŦ|:Ç y w—7

E(5; 1, 1)

<

xy

È8É6“ I4ÊWË'… @6OÌ74Í Ê (2 ^ I P@?j

-

_x

1 2 3

0

1

2

−1

−2

6

y

u

u

u

u

u

u

u

u

3

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E = E(p; a, b)

O¢Õ4Ö … @0j

(i)

×Ø (0Ÿ

P ∈ E

O̜„0žŸ

O

Oc(0Ùp<

P + O = O + P = P

(3)

I „Y Ô C'@cj … P8Ú"ÛW7

O

<c72M+(cÒÓ IÜ+… @

0

«ÞÝ ß'à ¬RO … @0j

(ii)

œ„0žŸ

O

’—¶ ( ×Ø (0Ÿ

P = (x, y) ∈ E

I0•–y wG7

P

0

_{= (x, −y)}

3 ÔAá @WŸs5

E

(:Ÿ_OcPN@+j

P

O

P

0

(0Ùp<

P + P

0

_{= P}

0

_{+ P = O}

₍₄₎

O Ô C@vj … P0ÚNÛv7

P

0

<

−P

O … @?jƤ I 7

P = (x, 0)

(" ^ P+Ÿ ¸ i L £ « i'@JOc5  P€8¬â7

P + P = O

32i'@0j ã2ä « Ò"Ó=( Ô2å

(iii) ) P

1

= (x

1

, y

1

) ∈ E, P

2

=

(x

2

, y

2

) ∈ E

I'· €6w27

(a) x

1

6= x

2

7 á B"57

(b)

x

1

= x

2

, y

1

= y

2

6= 0

¸¿ YÆÀ ·¦© (=O … @4j6æ'( O8|?7

λ =















y

2

− y

1

x

2

− x

1



(a)

(Wç™



3x

2

1

+ a

2y

1



(b)

(Wç™



(5)

x

3

= λ

2

− x

1

− x

2

, y

3

= λ(x

1

− x

3

) − y

1

(6)

3 Ô)á @cŸ

P

3

= (x

3

, y

3

)

5

E

(We'” O?P2@0j ª « qArt8u

E(5; 1, 1)

“ (8Ÿ)(+ÒNÓ2¬

F

5

(

2

è O›é zs5:7

x

0 1 2 3 4

x

2

0 1 4 4 1

x

1 2 3 4

1/x 1 3 2 4

(N ^ I P'@cj

P = (0, 1) ∈ E(5; 1, 1)

I8· €Gw87

2P =

P + P

<c~G … @0j

x

1

= 0, y

1

= 1

O y w—7Gêë

(5)

(

(b)

(Wç™kOÌêë

(6)

<WìN€N@JO±7

λ =

3x

2

1

+ 1

2y

1

=

3 · 0

2

+ 1

2 · 1

=

1

2

= 3

x

3

= λ

2

− 2x

1

= 3

2

− 2 · 0 = 4

y

3

= λ(x

1

− x

3

) − y

1

= 3 · (0 − 4) − 1 = 2

OµP@?j y B ¸ ovw87

2P = (4, 2)

3i@cjí Â I 7

3P =

P + 2P

<c~G … @0j

x

1

= 0, y

1

= 1, x

2

= 4, y

2

= 2

O y w—74êë

(5)

(

(a)

(Wç™kOÌêë

(6)

<WìN€N@JO±7

λ =

y

2

− y

1

x

2

− x

1

=

2 − 1

4 − 0

=

1

4

= 4

x

3

= λ

2

− x

1

− x

2

= 4

2

− 0 − 4 = 1 − 4 = 2

y

3

= λ(x

1

− x

3

) − y

1

= 4 · (0 − 2) − 1 = 1

¸

3P = (2, 1)

’

4P = P + 3P = (3, 4), 5P = P + 4P = (3, 1),

6P = P + 5P = (2, 4), 7P = P + 6P = (4, 3),

8P = P + 7P = (1, 4)

(6 ^ I ~—=í L @Wj

4 = −1

"Yð7

8P = (1, 4)

5

−P

32i'@0j y B ¸ o+w—7

9P = P + 8P = O

O?P2@0j

-

_x

1 2 3

0

1

2

−1

−2

6

y

u











:

A

A

AU

6

H

_HH

j

X

X

X

X

X

X

y

u

u

u

u

u

u

u

4

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E(F

p

) = E(p; a, b)

<

F

p

“ (Wqsr8t0u O y 7âþÿ I { |:P
z

n

3—7"5WCGw

nP = O

O?P2@

P ∈ E(F

p

)

< 4jJM:(

P

< WOv¡4j25 » ( <07"!$#5
» ( %ò<07„æ L&8L 7

Σ

n

= {0, 1, ... , n − 1}

¼RÂ '47$()N5

A =

P

<W7"! #G5

B =

%

P

<c~G y w—7—S} I+*,N… @0j   ¸ 7-

B

<W7"! # ¸ 7.%

A

<c~G … @JO±7

B =

%

A =

/%

P

(7)

O?P6o+w—70J(+1­

K =

/%

P

¸32_Â L @0j

#

"

!



A =

P

<c~G

-A



B

E(F

p

), P

« ê54—¬ 687 H "#

#

"

!

! #

B =

%

P

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X ∈ E

< `:a'x y w GH I* YvB€=O … @:j! # 5:7 2(0ê54I

P , A

O8 » ( %Š<?bAo+w—7

B =

%

P, Y = X +

%

A

(8)

<c~G y 7  I+*,N… @0jJKc[=o4B25

X = Y + (−

B)

(9)

33L axN… @0j

#

"

!



A =

P

<c~G



B, Y

E(F

p

), P, A

« ê54—¬ 687 H "#

'

&

$

%

! #

B =

%

P

Y = X +

%

A

<c~G

5

MONQPSR q)r—t+u

(31; 2, 17)

“ (0Ÿp<?C¾N@0j8­„+®

F

31

I—¯ €4w87

f (x) = x

3

+ 2x + 17

(„O+|µ7µqJr8tcu

E(31; 2, 17)

(0e2”z)5

]E(31; 2, 17) = 41

I PN@+j

P = (10, 13)

3 i¦Y§7

P, 2P, ..., 40P

á 3?TpÛU y w$VWX I4… @JO

’ Í (" ^ I P2@0j

0

5

10

15

20

25

30

0

5

10

15

20

25

30

Ê

1

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24

O y 7!"#—5 » (]8 p<

29

O y B:j 25

A = 24P = (17, 29)

<W7"! # 5

B = 29P = (23, 4)

<c~G y w8SU I+*,–y i ^ j  ¸

24B

<?! # ¸

29A

<c~G y w—7

24 B = 29 A = (10, 18)

(10)

OcP6o+wG7s(12­

K = (10, 18)

¸2Z L @+jpM MW3 •=N… @I>I@ A B @ `0a <?C¾N@Wj! # ¸4È F

X =

(30, 13) ∈ E

< `0ax–y w5 I+* @0j0! #G5

P , A ,

%Z<?bAo+w—7

B =

%

P, Y = X +

%

A = (30, 13) + (10, 18) (11)

<c~G y 7  I+*,N… @0jJKc[=o4B25

X = Y + (−

B) = (24, 1) + (10, −18)

(12)

33L ax 3J|?7

X = (30, 13)

<^JC2@NMNO ¸ 3J|—@0j

6

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