Криптология. Вчера, сегодня, завтра
.PDF!
" # $# % & #
$ $'
( ! )
53##+305))6*;4826)4#.)4#.);806*,48+8||60)),85;;]8*;:#*8+83(88)5*+;46(;
88*96*?;8)*#(;485);5*+2:*#(;4956*2(5*=4)8||8*;4069285);)6+8)4##;1(#
9;4881;8:81;48+85;4)485+528806*81(#9;48;(88;4(#?34;48)4#;161;:188#?;
( *
.
! " #$! %
.
! &
! ' ' (
) !
# % # %
$!
* !
& $! &
) ' $! !
+ # % ,
& ( #
% # $%
,
- !
! !
' !! #./01&.230%" $! (
*
$!
4 ! ,
4 ( !
( # %
$!
! ( (
#
% ( (
!
$! # %
, ( # ).
5 $ $!
(
(
$!
.
6 .272 (
8
$ 9
: ;333 < = = ' 5
$! (
#' % ! >.?
@ABCAD $! EFGHFI J $! =
Y X Z, |
(1) |
X & #AK3 LK. MK;1% Z & #
< = 0 = N & 7% Y -
$! ( ;O (23 3 = 0; 23 4 = 1, %
P : $! =
$ $
8 - Q (
, $! $ ( $!
* =
! R ' .2;O S J 6 ( N ! !
$!
: T ! 6 $! = " #.%
X,Y Z ! >3 .?
( ; (0 0=0, 0 1=1, 1 1=0% R
6 9
, ( $!
8 U U $! ,
$! 6 $!
,
,
V .272 ,
( '
-' .272 8 T
>;?
T ( $! 6 ' U
- $ T
.272
( ! ' .27/ R $ .2WO P ) !! V 4 X R ! >0? ) !! X
(
(
(
.
. 6 ( &
Z [Z1 , Z2 , ..., Zk ] (
)
|
|
|
X |
|
X |
Y |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Z |
|
|
|
|
|
|
|
|
|
Z |
|
|
|
|
|
|
|
|
Q |
|
|
|
|
|
|
|
|
|
5 . J
! ( $!
$!
( , .
k & , !
! Y3 .Z |
|
! ( X [ X1 , X 2 , ..., X m ] T ! |
|
Y [Y1, Y2 , ..., Yn ] |
! ' X Z 8 |
|
|
Y EZ ( X ) |
(2) |
Y ! ' $
X Z.
. $! (
|
|
|
X DZ (Y ) |
(3) |
|
( ! X ! ' Y, |
||
$ |
|
Z. |
Y ' |
|
|
|
||
|
|
|
X # % ' Z. |
|
|
6 . |
T .272 >;? |
|
Q ( X, Y Z & 6
X 9
Z
! . X Z
.
T !
$ J
" R (
[ #>;? \ 0O3% 5$
!
" U $!
$ -, T
( " (
( [ J
( (
.
- T
$! M X ( Z.
6
Y , ( $
$! " # T
X Y P(X=x | Y=y) = P(X=x) ( x [x1, x2 , ..., x M ] y. ) ( $ ' X
Y '
( -
T $
5 $! $!
( ! Y3 . ]&.Z U ] ^
_ `
^K_K` -(
P(Z |
z) |
L M |
LM ( z ' |
||||
$! ( " |
|
||||||
|
|
|
|
Yi |
X i Zi, |
i=1, 2, ... , M, |
(4) |
|
( L - ( |
||||||
xi |
|
X i |
|
yi Yi |
|
zi Zi zi |
( #7% P(Y y| X x) L M ( y ( x X X Y
$ % #
T % $
J 6 $ $! & $
5
( $! 6
! ( (
$! , !
T
J $! 6 $
X )
( , $! $ ( U # $! $ %
,
$!
( a
( $! 6
$
( U #
V &6$%
.
) $ (
# , % & ' ! ' T >7? R & , (
! G#X|Y) #
X Y% ( !
b#X=x|Y=y%
|
|
( x y X Y R
H(X|Y)=H(X)+H(Y|X% $
#) , (
>7? ! ' %
P #;% #0% (
H(Y| X, Z ) 0, |
(5) |
H(X| Y, Z ) 0, |
(6) |
G#X|Y, Z% 3 Y Z
X a $ (
H(X| Y) H(X), |
(7) |
, X Y
)
H(X| Y) H(X, Z| Y) H(Z| Y) H(X| Y, Z ) H(Z| Y) H(Z ). |
(8) |
Q #O% ! $ U
( $ 4
$ #W% #/%
H(Z ) H(X). |
(9) |
R #/% & , & ' # (
)
4 K ! U Lz
H(Z ) log(LK ) K log L |
(10) |
|
z |
z |
|
$
N
H(X) M log Lx |
(11) |
# Lx & U ! %
Lx Lz #
$! 6 % ' T #2%
#.3% #..%
K M |
(12) |
8 ( 9 (
' $! 6
(
8 "
$! $!
$!
$! # % ( "
6
-
$!
( $! (
# % Q
( ' ' (
X$ DCA # Dcdefg Chijck Almejin% 5 '
( A L
-( "
& A
& A
& L
& L
a )
A ( ` L
` -
! ' (
Y & $! M a
L ) $! ' # M)
Y:
- L M A
$!
A - EA .
!.
DCA $!
$! ' 3 N&. N 6
DCA ( ! ' !
!
Q M N e (
M e mod N o ' ! '
! - N, M e , ! ' ( (
6 DCA ! ( $
# ;33& % ( (
( )
A
$ P Q (
N 6
N ' e
& P Q.
p N ' $!
N R $ A
P Q '
$!
* P Q A ( ! ' 8 ( ' . N,
N - N=PQ
(N) (P 1)(Q 1)
6 $ 8
$ Ra)#x,N%
x ( N ) 1(mod N )
!
x k ( N ) 1 x(mod N )
Q A ( >W? d
e d 1(mod (N ))
(
e d k (N) 1
4 |
M e mod N |
d |
M |
|
|
|
|
|
" #$%.
6 ( , '
! '
S GF(q) q , >q & # Y3 . q -
.Z q%? ! (
4
Y X mod q, 1 < X < q - 1,
& ! , GF(q% #
, . ; q & . GF(q%% X ! Y
GF(q):
X log Y GF(q), 1 < X < q - 1.
* X Y 4
$ 2 log2 q ' ( R
|
|
|
|
|
|
|
|
2 |
2 |
|
|
|
|
|
|
|
|
2 |
2 |
|
2 |
2 |
|||
37 |
|
32 4 1 |
|
|
2 |
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
J X Y (
>/9 29 .3?
4 q ! q
U '
L(q) e ln q ln ln q
!
J ! ' L(q) ' (
( (
) A
X A ' . ; q & . 8 X A A
( L
|
YA |
|
X A mod q . |
|
|
|
|
||
N |
|
L |
|
|
|
X B |
|
||
A YB - , A L |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
KAB |
X A XB |
mod q |
|
|
|
|
||
, ) |
K AB , |
||||||||
A YB & L & X A : |
|||||||||
KAB |
YBXA mod q |
|
|
XB XA mod q |
XB XA |
XA XB |
mod q |
||
L K AB |
" |
|
|
|
|
||||
|
K |
AB |
Y X B mod q . |
|
|
|
|
||
|
|
|
A |
|
|
|
|
|
|
R A L X A |
X B -, |
||||||||
( K AB YA |
|
YB 8 , |
! $
! ) $
!
4 q .333 YA X A
;333 ( .333& J
! GF(q% 2100 #
1030 % '
N ! , '
4 ( ( S 2n ,
' >.. .;?
$
( '
! '