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Page 1: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

��� � � ��� �� � �� � �� �� �� �

��� ��� � �

��� � �� � !"� � #� $ �% & % ')( * + ��, � �.- / 0 �� � �

1324 56 7.89 :3; <8= ; 8 >?8 5 @9 74 8= 7

A8 ; B= <2 =DC E @ < F @ GHI I IC J?K9 @8 L

M @= 6 @9N OPC HI I Q

; R L < S < B @4

TVUWXY Z Z[ U \] [ ^`_ \ [baY ced [ \ fa _ \g a Z[ h`iX \ [aY j a k \ f[ U gX \W ] [X lX ] WX l la d W ZX U laY mX U \ fW i ano ] [ m f \Y a \ [ i W UXY Z \ f[ Un W ] g [ U U [baYX ] W g X [Y \X [Y W ZqpXY ZX U laY mX U \ fW k_ l l UW \ a k U l[ ZW U ] W g X [Y U i a g n lW \W�rs f[g aY t`u WY p v] a ] wX d [ \x p yaY [ s fX i fX ]XY Z z] ] v _Y {W lg XY gX ZW gX |a ] i aY \] [ ^`_ \ [baY U \a \ fW UW U l[ ZW Ur

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � �Y \] a Z_ i \ [baY c � j

Page 2: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

��� � � ��� �� � �� �� �

�8 ; 7 6 9 89 � R L < S < B @4 � ��� �� ���

�K K <K 7 @= 7� R L @ � S @9 � @= � ��� �� � ��

1 L @K K� A6 8 K � @N OI � GI O H� GIC A @6 !"

A6 7.2 9 < @ L� #?2 = � @N O$� GI O"� GIC A @6 !%

&9 89 8' 6 <K < 7 8 K�

OI Q O G Q #?2 �89 = � L)( 8 !9 @ E *+ ,�- � �. / � � �0 ��

I P Q Q O H &9 2 ! @ ! < L < 7N # * / ,. � � ,1 �

H G$ G Q G 1324 56 7 @ ! < L < 7N A B8 2 9N ,. � �. 2� + � ,� . ,

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � �Y \] a Z_ i \ [baY c � j

Page 3: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

� � � � ��� �� � �� �� � � �� � ��� �

�9 @ �8� " I � 8 � @4 C GI � 8 �89 ; <K 8 K

�8 � @4 ( 9 @ �8 K ! 8 L2 Q$ < L L= 2 7 ! 8 ; 24 ! <= 8 � < 7 B 7 B8 ( 9 @ �8 K 2 F 7 B8 8 �89 �; <K 8 K �

R � @4 � O � H � HI I Q�� 4 2 8 � S� O � G � HI I Q��

� � � 5 @( 8� �� ��� � � ��� � ��� ��� �� � � � ! " �� # �� !$ �&% �� '() * +) �

S?8 7 8 8= 2 7 B89 7 B <= ( KC 7 B8 K 8 K L < �8 K ; @= ! 8 F8 7 ; B8 � F9 24 7 B <K 5 @( 8 �

# @ < L <= ( � <K 7� ,8 ( <K 7 89 7 B9 2 6 ( B 7 B8 ; 2 6 9 K 8 � � � 5 @( 8 � � L L4 8 K K @( 8 K < L L ! 8

5 2 K 7.8 � 7 B9 2 6 ( B 7 B <K 4 @ < L <= ( L <K 7 � -. . / 021 354 6 0 / 71 / 0 8 4 94: ; / 0 4 94 3

0 < 0 =4 . ; / 0 �

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � > �Y \] a Z_ i \ [baY c � j

Page 4: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

� /, �, � +� �.� � �, � � �� � � �� -� � /. + �. 2- �. � �� �� , � / �

� � ��� � ,� �� /. � �� � 2 � �+

�� � � + � � �� �� 0� , , 2 0 � � � �. /� 2

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � �Y \] a Z_ i \ [baY c � j

Page 5: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

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

�8 ; 7 6 9 89 � R L < S < B @4 � ��� �� ���

�� ; 8� A @6 ! $ O H

�� ; 8 E?2 6 9� A6 8 K � @N P� GI OI � HI �

& B2 = 8� Q GI

� � �� B 7 7 5� � � � ; K � 7.8 ; B= <2 = � @; � < L �% ! < B @4 �

& L8 @K 8 ; 2 = 7 @; 7 5 89 K 2 = @ L LN 2 9 !N 5 B2 = 8 � B8= 8 89 5 2 K K < ! L8 � � 2 < � 84 @ < L�

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � �Y \] a Z_ i \ [baY c � j

Page 6: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

�� � � � �

J= 79 2 �6 ; 7 <2 = 7.2 19N 5 7.2 L2 (N

: 6 !K 7 < 7 6 7 <2 = 1 < 5 B89 K

: B @= = 2 = � K A B8 2 9 N 2 F :38 ;9 8 ;N :N K 7.84 K

S L2 ; �; < 5 B89 K

> < �89 8= 7 < @ L 19 N 5 7 @= @ LN K <K

E @K B <= ( @= � �= 8 � A <4 8 : < ( = @ 7 6 9 8 K

#?89 � L8 � K & 6� � L8 K

J= 79 2 �6 ; 7 <2 = 7.2 � 6 4 ! 89 A B8 2 9N

& 6 ! L <; �8N 19N 5 7.2 ( 9 @ 5 BN

> <� 8 � E?8 L L4 @= �8N R �; B @= ( 8

, : �, @ ! <= � K � @9 < @= 7

,8 L @ 7.8 � @ L)( 2 9 < 7 B4 K

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � �Y \] a Z_ i \ [baY c � j

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�� � � � � � �� � ��� �

> � � � ! @K 8 � K < ( = @ 7 6 9 8 K ; B84 8 K

�89 2 � �= 2 L8 � ( 8 &9 2 7.2 ; 2 LK

� < @ 7 � : B @4 <9 < �8= 7 < �; @ 7 <2 = K ; B84 8

:38 ;9 8 7 : B @9 <= (

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � �Y \] a Z_ i \ [baY c � j �

Page 8: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

�� � � � � � � � ��� �� �

� � 9� � 0 < : 9 � � =�� A B8 @; 7 2 9 @9 7 2 F 9 < 7 <= ( <= K 8 ;9 8 7 ; B @9 @; 7.89 K �

� � 9� � 0 � 6 �. � / ; /� A B8 @= @ LN K <K @= � �8 ; < 5 B89 <= ( 2 F K 8 ;9 8 7 9 < 7 <= ( K �

� � 9� � 0 < . < : �� �8 !K 7.89 � K � 7 B8 K ; <8= 7 < �; K 7 6 �N 2 F ;9N 5 7.2 ( 9 @ 5 BN @= � ;9N 5 7 �@= @ LN K <K �

J= 2 6 9 ; 2 = 7 8 � 7 � 9� � 0 < . < : � <K 7 B8 K ; <8= 7 < �; K 7 6 �N 2 F 59 2 7.8 ; 7 <2 = 2 F <= F2 9 4 @ 7 <2 = �

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � �Y \] a Z_ i \ [baY c � j

Page 9: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

� � � � ��� � � � � � � � � � � � �

19N 5 7 2 ( 9 @ 5 BN K 6 5 5 2 9 7 K 7 B8 F2 L L2 <= ( K 89 <; 8 K�

O � 132 = � �8= 7 < @ L < 7N ,. � �. 1 �

H � J= 7.8 ( 9 < 7N ,. / � 2 �

G � �6 7 B8= 7 <; @ 7 <2 = ,. /� �

Q�� J �8= 7 < 7N ,. � � �

% � A <4 8 L <= 8 K K � / � 0. � , �

$ � &9 2 2 F 2 F 2 = 89 K B < 5 ,. � � � , + �. � �

R @; B B @K @9 <2 6 K � < �89 8= 79 8' 6 <9 84 8= 7 K <= � < �89 8= 7 ; <9 ; 6 4 K 7 @= ; 8 KC @= � 8 @; B <K

K 6 5 5 2 9 7.8 � !N @ < �8 @9 <8 7N 2 F K ; B84 8 K �

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � �Y \] a Z_ i \ [baY c � j

Page 10: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

� � � � � �� � �� � �

O � 1324 4 6 = <; @ 7 <2 = K �8= ;9 N 5 7 <2 = 2 9 @6 7 B8= 7 <; @ 7 <2 = �

H � � < L8 @= � � @ 7 @ ! @K 8 K 8 ; 6 9 < 7N

G � R L8 ; 79 2 = <; F6 = �K 79 @= K F89

Q�� R L8 ; 79 2 = <; 1324 4 89 ; 8

% � > < ( < 7 @ L ; @K B

$ � 132 = 79 @; 7 K < ( = <= (

" � R L8 ; 79 2 = <; 4 @ < L

� �6 7 B8= 7 <; @ 7 <2 = � & @K K 2 9 �KC & J � K

P � :38 ; 6 9 8 < �8= 7 < �; @ 7 <2 =C �; ; 8 K K ; 2 = 79 2 L

OI � :38 ; 6 9 8 59 2 7 2 ; 2 LK

O O � &9 2 2 F 2 F �= 2 L8 � ( 8

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � � �Y \] a Z_ i \ [baY c � j

Page 11: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

� � � � � �� � �� � � � �� � ��� �

O H � 132 = K 79 6 ; 7 <2 = !N ; 2 L L @ ! 2 9 @ 7 <= ( 5 @9 7 <8 K K 8 ;9 8 7 K B @9 <= ( �

O G � 132 5N 9 < ( B 7 59 2 7.8 ; 7 <2 =

O Q�� 8 7 ; �

i } t l[ ~ [ fX g�� �XY _X ]o �� p �� � � � � �Y \] a Z_ i \ [baY c � j

Page 12: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

� �� � � � � � �� � � �

�4� 0 8< < ��

: 7 <= K 2 =DC �� � �� � �� � ��� � ��� � � � �� �� � � � ��C 1 , 1 59 8 K KC OP P % �

� 6 3: 7 <= K 2 =DC �� � �� � �� � ��� � ��� � � � �� � � � � � ��C K 8 ; 2 = � 8 � < 7 <2 =DC 1 B @ 54 @= E @ L L � 1 , 1CHI I H � �

� 0 =4 9 �< < � / � /4 3 ; 6 0 =4 5< 1 9 /4�

S < B @4 C : B @4 <9C ! � "� � � � � �� # �� � �� �� # $ � $ % � ��� !� �� & � � � � � � � ' �� �)(�� � �C : 59 <= ( 89 �89 L @(C �8 *2 9 �C OP P G �

#?89 � L8C '� � � � � � +-, � ��� � � � �� � � �� � � � � , . # � � /� ' $ �� 0 $21 3 #4 ,657 58 9 9 :

;9 57 7 1 <= > �

?A@ BDCE CFGH I C IJK JGH LM CE CH KE H CN E F BDCO CEQP CQR SRP TU V-W ?YXH I VZ TP []\ K ^X F _E` X M JG \E G K BDC M KG LJF E LM CE CH K C I JH K BDC abME K C IJK JGHcE C F M C K E BX M JH SP deP f J g]C W U C ^ ^O XH P C K F R hR @ BC L M CE CH K X K JGH G i f Z T XH I IJ jC M CH K JX ^F Mk LK XH X ^k E JE JH K BDC a ME K C IJK JGH JE F ^G E C M KGK BC LM CE CH K X K JGH JH G \ M FG \ ME CQR

F l Z ^J m J BX O W nXH \ X M k ?oP pq q r ? p sH K MG I\ F K JGH c ? h

Page 13: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

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

�� � � � � ��� � ����

� 5� 5� 57 1 �8 � ��� 9 7 9 :� � 1 �8 � 7 � � � 5 1 "!# $ .&% % ' % ( +*) ) +,.- $ /102 ) 3 %4 0 ! ) 52 1 67 6

;9 57 7 1 <= = 8:9

;< = =� � 7 1 /% # 3- 0) % 0 ! 02 /02 ) 3 % +%4 2 > 3 5- ?1@ , - # @ - % (A # (% 0 0 ! 3 , % # A # 3- 4 0 , 32 14B B B C9 57 7 1 < = = < 9

F l Z ^J m J BX O W nXH \ X M k ?oP pq q r ?D sH K MG I\ F K JGH c ? h E

Page 14: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

� � � � � �� � � � �� � � � � � � � �

< 9 ; � 5 8 � � 9 8 ; :�� � < �<� <� �� 9 =8 �< � � � � � � ��� 99 � ; � � 9 8 ; :< 9 = 5 � :� �7 9

�8 � �9 < �<� � < � :8 � <� �<7 < �� 5 <� �9

� � � �9 < �<� <� 8 � :< � � 5� ; �8 9 5 �7 � 9 :8 7 � : 5 � 58 7 �7 < � < � 98 � � �< �7 � � � : 5

9 8 � � 5 �5 �7 � � ;<�� 5 �7 <� 8 7 98 � � 5 � ;< 9 � �9 5 � 9

> 9 �� �7 7 � 9 <8 � �<� ��� !" # $&% � � 9 � � 5 � � � : 5 : 58 � � �8 7 �8 �5 1 8 � �7 5� � :< =

� :9 � � : � : 5 5� 5 =� '7 � <� 57 1 8 � � 59 � : 5 :8 <9 � 8 7 9 � � � 9

F l Z ^J m J BX O W nXH \ X M k ?oP pq q r ? r sH K MG I\ F K JGH c ? h

Page 15: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

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

� 9 �<9 7 � 99 � ; � � 9 8 ; :< 98 � � 5 = ; �7 � � 59 5 =<8 : � �� % � � ��

�8 � � 59 5 =<8 : 1 > � 1 > �� � � �� � �� % � �

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

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

� � � � 59 5 =<8 : 1 � < 1 � <� � �" %

�" � �� % � �

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

� 9 � � 59 5 =<8 : 1 � < 1 <� � �" �" � �� % � �

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

� 9 �<9 7 � 99 � ; � � 9 8 ; :< 98 � � 5 = ; �7 � �8 � < 5 � � $ " �� � �� � :8 � � � 9 � � 5 8 9< ; : 59 �� 9

� : 5 �<� � � �7 7 � 9 <8 1 � � � � � � � � � 9� �� � 9 5 �58 � � : 5 = 58 � <� 1 � < � � �8 � < 5 �

�9 8 � 7 �8 � 5 � � : 5 9< ; : 59 9

F l Z ^J m J BX O W nXH \ X M k ?oP pq q r ?� sH K MG I\ F K JGH c ? h

Page 16: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

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

� 9 68 57 8 9 9< ; : 59 ��� �� � � � $ � � 9

�B � 7 �8 � � 5 � 8 � 6< 9 59 � � 5 = 1 < � 5 = 8 � �8 =< � <8 9 57 �� = 57 �< 9<7 � 5

9 5 � �7 1 <� � �< � �7 1 7 < � �8 � 9 9 �� �< �7 ; 59 � 59 5� �8 59 8 � � 1 ; 59 � � �8 7

7 99 < ;7 < � 1 < � 57 �7 < 97 �9 � 9 � � � < � � 59 8 9 � = � 9 �<� 5 1 � �� �� �8 = �59 � � =

5 � 9< ; � 7 7 5 ��� � �8 5 7 < � �< <� �57 �< 8 9 5 5 � ; 59 7 5 � �< �5 � < � 1 � �8 9 � 5 =

5 � 5 = 5� � � 9 � = � < � � 59 8 = 1 < � 57 � � ;9 � � 5 � ; 59 <� � 5 9 5 � < � �8 7 9� =�

= � � 5 � 9 �

�� : 59 5 8 9 5 8 �7 � � 5 � � 59 7 � � :<7 � � �� < �7 68 57 8 9 '7 � � � 6< 9 59 �� 8 7 � 5 � �

8 7 � � :<7 <� �< =8 � 57 � � ;9 < �8 � 5 8 8 <9 7 � 8 � � <� � : 5 �8 � � 59 � < � : 5 :8 �

8 � � � :<� 9� � � � 5� �<8 � � � 7 8 �� : 5 � 9 � � 5 < � <� 9< ; : 59 � � :8 � <7 � � �

7 � 9 :8 � <� � : 5 � 9 � 59 � � � : 5 � 5 � � 59 7 � � � : 5 8 � ; :8 � 5 �� � :8 �� � �8

� � 9 � 9� �� � � 5 =8 � 5 � � � 9 4 �8 � � � � 5 � <7 : 57 � � � 5 9< ; : 59 � : 57 5�8 � � 5 �8 � � : 5 <9 = 58 � <� � : 5 = �7 �7 � �7 �< � � � 5 � : 5 �� �9 � : � 5 � � 59

� � � : 5 8 � ; :8 � 5 �� � 8 = 5 � � �� �� 9 �� 8 � �7 � � < � : � : 5 � � : 59 7 9 �

� ; � 5 � � � < �7 � � � 5 �< �8 68 57 8 9 � = �� � < �� � 9 � 9

F l Z ^J m J BX O W nXH \ X M k ?oP pq q r ? sH K MG I\ F K JGH c ? h

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

� 9 < = ' � : 9 5� � �9 � 8 � � � 5 <� � <� � �> � ' � : 9 5� � �9 � � � : 5 � < � 5 �7 5 � � � 5 � 5 9 8 ; :

�8 � �7 5 =8 ; :� 9 57 � =8 � 5 5� 99 � ; �< � � � 5 9 57 7 8 9 � � �9 8 � 7 ; � 7 < �< � � 8 � �7 � �7 �<�

� � �< � � 9< ; : 59 7 9

8:9 �� 9 � �� 8 9 4� � < � 5 �7 5 � � 99 � ; � � 9 8 ; :�� 9 69 � ; �8 � 8 � � 7 <7 �8 �7 � �8 9 � � � 99 � ; ��

8 � 8 � � 7 <7 � � < � 5 � � 8 5 9 � 5 � � : 5 � 8 9 9 � : 5 �< = = �9 = �� � � �� � 9 � = 9

� 9 < = �� '��� B � < = � �� � � � : �9 9 � � � 9 = �� :<� �� 9

= 9 �� 9 � � � �9 � �� B � �� � < � �9 �� � � � �9 � ; � � 9 � ; :� �� � �9 � ; � �� �� � � < � 9

< � 9 � < � � < = 8� '�� �� � �� � � �� � � � � � � �9 � = �� �� �� � �9 =< �� 9 � �9 � � < =< � � � ; � � � < �

9 �� � �9 � �� � � � � �� � ; = �� � 9 �� � � � � � � ; � �� < � ;9 < = �9 < � � �� 9 � �<� � �� 9

< < 9 < = 8� '��� � � < � �9 �� � �� � � � �� � � 9

F l Z ^J m J BX O W nXH \ X M k ?oP pq q r ? � sH K MG I\ F K JGH c ? h

Page 18: cs236506/03/slides/crypto... · 0 132 4 5 6 7.8 9:3; < 8=

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

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