file0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 50 100 150 200 250 300 350 400 450 500

26

Transcript of file0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 50 100 150 200 250 300 350 400 450 500

�� �� �� � � �� � � � �� �� �

�� �� ��� � � � �� � � �� � � � �

�� �� ��� � � �� !" # �� �$

% &'( )*+ ,- '. * ,/0 + 1' + &'2 / * )/ 3 '

%( '4 ' ( 50 - ). / 3( 5. / * )/ + 0

678 9:; <8 = >?@ A :B 9: 7 CD D EF G GH H HIJK L CM M I MNO CD D EF G GH H HIJK P LQ MN O I MNO 678 9: ; <8 = >?@ R; 8 < =?S

T

�� �� � � ��

� �� �� #� " � � ��

� � � � � � � # �� � � ! �� #�� � � �� � �� � � � � � �

� � � " �� # � �� � � � �

� � � � � � #� � � � � �� � � � � ��

� � ! # #� �� � � ! � � � �� � �� � � ! � � � � $ � ! # #� ��

� ! "# " � �� %$ � � � � � # � � � � # ��&

�� � � ��� � � �� � � ��

�� � $ � #� � $ �� �� #� " � � � � � �� �� � � � � � � $ � � � �

� & � T�� � � � ��� � � � & � & � � � � � �� ��� �

�� � $ � #� � $ ! � � � � # � � � � $ �� � #� �� ! � � �� � � � � � � � ��� �� �

� ! #� � � � #�� � � �" ! #� � $ # � � � #� �� � � #� �

� � #! #� � $ � � � ! � � � � �� !"# $# !" % " !# &

� ! � � �� �� ' ! � � � #� � $ � ! � � $ ! $ � #� � � � # � � �� � ! �� �"

�� � � ��� � � �� � � �� �� �� �

−8 −6 −4 −2 0 2 4 6 8

−6

−4

−2

0

2

4

6

−6 −4 −2 0 2 4 6 8

−6

−4

−2

0

2

4

6

� &� � � � � & � � � � &� � � � � � �

� � � �� � �� � �� �� � � � � �� � � �� � � ��

−6 −4 −2 0 2 4 6

−4

−2

0

2

4

P ⊕ Q

Q

P

R

L′

L

−6 −4 −2 0 2 4 6

−4

−2

0

2

4

2P

P

L′

L

R

� � � � $ � #� � �� $ #� � �" � �� $ � ! �� $ #

� &� � � � � � �

� � � �� � �� � �� �� � � � � �� � � �� � � ��

� � #� T� � &� � � �� � #� � �� �� � � � � � �

� �� � T� � & ! $ � � T � & � T�� & � �� � � #� �$ � T� � &� � �

� �� � �

� T �� � &�

��

� � � � & � � T � �� & � � T �

� � � T� & � � & � T � �� & � � T �

� T� � &�

��

� � �� � & T �� & � T � � �� T � T � � � � T � T�� T � � �

� � � � � T � � � T �� � �� � � T � � � � T � T� T � � �

! � � �� $ # � �� � T� � &� � �� � �$ � �

� � � � & � T � �� & � � T � � &

� � � � � � � T �� � � �� �

�� � � ��� � � �� � � �� �� �� � �� �� � � � �� � �

�� � �� �� � � �� � � �� �� � � �� � � �� � � � �� � ������

� �� �

� �� �

� �� �� �� �

� �� �� �

� ���� �

��

��

��

��

��

��

��

��

��

��

��

��

��

! � � �� �� #� " � � � � &� � � � � � � � ��

� � � � ��� � � �� � � � � �� �� �� � � � � � � �� �� � � �� � ��

�� � $ � #� � $ � � # ��� � � � � �� �� #� " � � � � � �� $ � # � �� � � �

� ! ! � � �� $ # � � �� � � � � ! $ � �� " # � � � �� �

� � � � � �� � �$ � ! $ � � � �� " # �

� � $ ! � � � � � � # � � � � $ � # � �� �

� ! ! � � �� �� �$ # � � � � � � ! $ � �� " # �� � ���

� � � � �� � �$ � ! $ � � � �� " # �

� ! � ! $ � � ! # � �� � � #� � � � � ! � � � � $ � � � # � � � � � � � # #� $ �

� � �� � �� � �� � � � � � � � �� �� �� � � � � �� �

� �� � !" � #� � � � � ! � � �� � � �� � � �� � � � � �� � � � ��

� � � � �� � $ � �� �� ! $ �" � � � � �� � �� �� �T � �� �& � � � � �� � # � �$ ! $

� �� " � � � � � # � �" � �� �" � �� �� � �� � �

� � �� � � � � �" � � � ! � � � ! � � � �� � � � � �� � �� �

� � �� � ! � � � � � � � !� � � ! " ! $ � �� � ! � � � � $ � #! $ # � ! � � #� � �

�� � �� � # �� � � ��� � �

� � �� � � � � �" � � � � ! � �" # � T �� ! $ � � � �� �� � � � � � � ! � � � �� � � �

� � �� # � �$ �� ! � ! # #! �� � �� � " � "# ! $� � $ �

� � �� � �� � �� � � � � � � � � � � �� � � � � � �� �

� � �$ �� � � $ � � !� � # � $ � ! $ � # � $ � � � � � � " � � " � � � � � � $ ��� � � # �

� � � � � �� $ � # � �� � � ��� ��� �� ! � � � � #� $ # � � �� � � #� � � �� � � � �

� � � � �" �� �� $ �" � ! � " � � � � � ! � � � �

� � �� ! � � $ �� ! � #$ � ! # � � $ � � ! �� � � � �� � � $ �� ! � �" � " � � �� � ! � �

� ��� � �� � � # � �$ � � � � ! $ � � � �� � � � � $ �� $ � ! � #� � �

� � �� � � �� �$ # �� � � � � $ � � � � �� � ��� �� � #� � �� � � #� �$ � �� �

� $ � #! $ �� �� � � � � ! $ � � � �� � �� � $ �" � $ �� � $ �$ #� ! � #� � �

� � � � � � � � � �� � $ � � #� � $ �

� � $ � �" � � " � � �� � � �� � � ! $ � � &� � � # � �� � �

T�

� �� � � � �� �� � �� � � � � � � � � � � � �� � �� �

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

200

250

300

350

400

450

500

Keylength conventional systems RSA and DSA

Key

leng

th e

llipt

ic c

urve

sys

tem

� � � � �$ � # � � � $ � � #� � �� � �" � � ! � �$ # � � # � �� ! � � � #� �$ � # �

� � ��

" � � � � � �

T T

� � � � � ��� � � � � � � � � � � �� �� �� ��

� � � � � �� � �� �$ # � � � � � #� � � �

� � � � � � � � �%$ � �� �� ! $ � � � � # � ! � " �� �� � � ! #� � $

� � � � � � � �$ �� �� $ �" $ � ! $ � # � $ � � � � � # � ! � " �� �� � � ! #� � $

� �� � $ ! #" � � � �� � � #� � � �

� � " � � � �� � $ "" � �� �� � $ ! #" � ��

� � � �� � � � �� #! � �� � $ ! #" � � � � � ��� #��

� � $ � � #� � $ � �� � � #� � ��

� � � � � � �$ # � �� ! # � � � $ � � #� � $ � � �� �

T &

� � � � � �� �� � � �� �� � � � � � � � � �� �� � � � � � � �

� � � �� � � �� �� #� " � � � ! $ ! � � � �� � ��

� # � �� # � � � �� � � � ! #! ��� �� $ !" #� �$ #� ! #� � $ � � ! #! � $ # � �� � # � ! $ �

$ � $ $ � � " � � ! #� � $

� � #! $ � ! � � � � �� � �� $ � " � � $ � � � � � � � � � � �

� � � � � � �� � � � � �� �

� � �! � � � � � �� $ �

� � � � � �� � � $ � � � �

� � � � � � � �� � � � $� � � � �� $� � � � �� �

� � � �

T �

� � � � � � � �� � �� �� �� � ��

�� �� !� $ ! � !� � # �� �

� �� �� #� " � � � � � �� $ � # � �� � � �

� � �� $ # � � ��� � � � � �� � � � � ! $ � �� ! # �� � � � �� � �

� � �� � ! # � ! $ � " � �� � � �

� � �� � # � ! $ � �� � $ # � � �� �� $ #� �� $ # �� � ! � � � � � � �

� �� " # � � � �� �

� �" � �� � � �� � � � � � � ! # � � � �� � �

T �

� � � � � � � �� �� � � � � �� �� �� � ��

! � �� � $ ! � � � � ! � �� � � # � � � �� � �� � $ �

�� � �� � # ! � ! $ � �� � $ # � � �� � � � # � � � � �

� � �� " # � �� � � �� T � � T � ! $ � � � � T� � � � �� �� � � � # � � # � �

� � �� " # � ��� T� � �

� � �� " # �� � �� � � �� �

� � �� " # � � � �� T �� � � �� � � � �� �� � � � # � � # � �

� � ! � � �� � $ ! #" � � � �� # � �� �� � ! � �� � � � � � � �

T �

� � � � � � � �� �� � � � �� �� � �� �� ��

� �� � ��� � ��� � �� � � � � � � � � � � �� �� � � � � ���

� ! � � � � �� � � �� � � �� � � � � � � �� � � �� � � � � � � � � � �#" $ � �

% & ' ( � � �� ) �� � � �� �

* & ' ( � � �+ , �� - ' � "

. & ' ( � � �/ - , � + ' � " � � � / 0 , �+ ' � "

� & ' ( � � �/ -21 � 3 / 0 1 4 ) �65 - � 7 - �� � �8 , 5 - ' � "

9 : ; ; � ( � � �� � �� � � � � � � � � � � 8 ) <- =

� �� �� �� � � � � � � � � �� � �

�� �� ���� � ��� ��� ��� � ���

!"# ��$ %�� & $ �' &� �� � & �( �� � ' " "

) � *,+ �( �� $ ��� ( - .0/ /1 1 .0/23 3 4 - -

� 56 78 7� %( � ( % - .09 - : 3 . :; : < 4 :3

� 56 78 = �� �> *� ( % 3 .2 : - / .; 2 ; 2 4; 2

1 : < ;@? A� � ' � B � � �C

!"# ��$ %�� & $ �' &� �� � & �( �� � ' " "

! 78 ) � *+ �( �� $ ��� ( D D 1 E . < <1

! 78 7� %( � ( % - - - .09 / E 1 ; ; .13 E < < : 4 E 9

! 78 = �� �> *� ( % FHGI 3 J - . :; 2 1 .3 91 < 41

! 78 = �� �> *� ( % FHGI 1LK M N - J 3 . E :; 2 .0923 -3 4 < /

# � � � � ( > � OP � � Q( � � $ � 4 O &+ & � ( 5� ( C �� $ � ( � B R � � � � �C C 6 � �� ��C

- S

� � � �� �� �

� � �� � � � �� � � ; �� � � � ;� � � �� � � � � �� � � � � ( � �� �

� )�

� -� � � � �� ��� � �

� � � � � � � �� � � � � � � ( �� � �� � � � �

� � � � ' � � � � � � � � � ;� � � � � � ��� � ��� � �� �� � � � � � � �� � � !� �" !

� # �' � �$ � � �� �� � � � � � � �� � ' � � � �� � � ; � �� � � � � � % %

- &

� � � �� �� � � � � �� � � � ��� �

B2

B1

b1

b2- �

� � � �� �� � � � � �� � �� � � �

� � � � � � � � ! � ; � �� � � � �' � � !� �� � �� �� � � � �� � � ; � � � � �

� ��� � �� / � � � �� � �/ � � ' � ' � �

� � !� � � � � � � � �� � � �� �� � � ' � � � � � � � ; � � � �

� � � � � � !� � ( � � � ' � � � ' � � � � � � � � ; � � � %�� �� - �� 0

� : � �� � � � � �� � ; � � !� � � ' � %�� � � �

� & � � � � � ! � ; � �� � � � �' � & !� �� � �� �� � � � �� � � ; � � � � �

� ; � � 8 � �� � � � / � � � �� � �/ $ 8 � � ' � ' � �

� & !� � � � � � � �

� �� �� � � � � & !� � ( � � � ' � � � ' � � � � � � � � ; � � � % � � 0

� �� � � � � � � � � �� � ; � & !� � � ' � %�� � �� � � � �

0 �

�� �� � � � � � �� � � � � � � � �� �� ��

� � #� � �� � � � � ��� � �� � � � ) � � � �� � � � $ � �� �

� ) � � � � � - ��� � � � �� - � )� -

� � � � � �

� � � � � ( � ;� � � � � � � � � �� � " !� " �� � �" ! ( � � � ; �

�� � � � ' � � � � � � � � � � - � � 0 � � � � � - ; �� ; �� � � � � � � � �� �

� 0 ; �� � ;�� � � �� � � � � �� $ �� � � � �� � � � �� ��

� �� �� �� � � ' � � � � � "! �� ; � �# � � �� �� � � � $ � % � � % �

0 -

�� �� � � � �� �� � � � � � � �� � �� � �

� � �# �� ' # � �� ( �

� �� (�� �� �� � ��� � �� � � � ! � � � � ; � � ��� � ��

� �� �# � ��� � � � � � � ��� � � � �� � � � � � � � �! � � � � � � � � � �

� ! � ! � ��� � �� � � � �� ��" # � �%$ � � " & �! � � � � � � � � � � �� � #' � "( � � 9 �

� �� �*) � �� �! �� � � �

�� �� +! �� $ � �� � � � � ! � � , � � �

�� � � � ;$ � �� -. � / 0 , � � �

� 1! � � � �� �� � � 2 � +� � - � � � � � � � ! � � , � � � � � ��

� 0 -. , � � �

3 3

�� �� �� � � � �� � � � � � � � � �� � �

� � � � ; � � � � � � # # ! 2� � � �� � �� �� � �� �� � # �� �#

� � � � �# � �! � � � � < � �

� � � � � � �� �. � 0 < 0 - � � � � �

� � �� ; � � � � ; � � � � � �� � � � � ��� � �� �� � �� �� � # �� �#

� � � � � � �� �. � 0 � . � 0 < 0 , � � 0 � � � �

� � � � � � �� � 0 � /. � 0 � 0 � /. � � � � �

� #� � ! �$ � � � # # ! 2� # �! ;� �� � � � � �� ! # ; � � � � � �� � � � �� �

� � �� � �� � 3 � � � �� � �� �! �� � �# � � � � � # # ! 2� � � ;$ � � � � 2 ��

# � : � � : ! � � ! � �� � � �� � � � � �� � �� �! �� � �#

3 �

�� �� � � � � �� � �

�� ; � �� � � � � � � � �

"�� � �' � �( $ � # � " #

" & � � � � � " & $ � � " ' � &

" / � & " # � � " & $ #' "( � �

" 3 &� #' $ " # � $ " � � � " # #

� #� � ! �� # $ � �# � � � �� � � �� � ! +� # �$ � � � � � � � � � �

� �� � � �� � � � � $ # � � � " !� !� � � � � �

� #� � �� # � � � � �� + � �� �� � �� � � �� � � � � �� � � � 2! � ! � � � �! �� � �� �# �

� � �� �# � �� $ � � #' $ � � ! �� � � � � � �� #

3

� � �� � � � � �� � ��

� � � � �� �# � � �� ! � � � � ! � � � � � �# � �� �� � �� � ! � �� ��

� � � � � � � � ��� �

� � � � ��� � � � � ��� ���� ��� � � � ��� ��� ��� � � � ���

� � � � � � � � �

� � � � � �

� � � � � �

��� ��� � � � ��� ��� ��� � � � ���

� � � � � �� ������������������

� � � �� �� � � ��� � , �"! # $&%' � � � 0 -. , � � � (

� � � �) � + � � � � ) % ' � # � � ) ) � % � � � � � � � �% %' * � � )' + � �� + � �%

3�

�� � �� � � �� � �

� �� �' ��' � � ) � � % � ) � �' � � � �' � *� � �� � � ) ��' � � � � � �� � � �� �

� � � % *�� �� � � � �' �� �' � � �� � �)' �� � � � % �� � � � � � ) �� � � �� �

� � � �� � � ) + � � %' � � % $ � �% � � ) � � � � �� � � �� $ � �) � � � ) � � ) � ��

� ! � �' " �� � � �� ' " ��' � % � � � � #% $ �� � � % $% � �) � " �) �% $ $ $ $

� � �� �' " � % � � � � �% � % � % ' ") � �� � � ) � �� � �

� & � ) � % " " � % % � � � � % � � �% ") � �� � �� �%' % � � ��

� � �� �' " � * �% � � ") � � � �% �% � � � % � �' � � � ) * ) � + � �) �� � ) � "�' " ��

� � ! �' � �% � � ) �� � � � $ * � % � " )' � � % �' � � "� � �) ( � � ) � �� )

* +