RS, 1977. .
( ). () Q, N = * Q (N) = (-1)(Q-1).
:
(N), (, (N)) =1
D :
D < N, * D ≡ 1 (mod (N)).
(, N) . . D .
RS 6.3.3.1.
6.3.3.1 - RSA
, . ( , , ) - h() m:
m = h().
S , - m D:
S = mD (mod N).
(, S) - , S, S D.
(, S) - . - m', S :
m' = SE (mod N).
, - h():
m = h().
, ..
SE (mod N) = h(),
(,S) . , D S , D , N .
, , S , S D, . "" .
RS:
1. N D RS , . , . .
2. 3. RS . , RS D , .
|
|
, 1, 2 3, -
m1 = h(1), m2 = h(2), m3 = h(3),
m3 = m1 * m2 (mod N).
, 1 2
S1 = m1D (mod N) S2 = m2D (mod N).
S3 3, D:
S3 = S1 * S2 (mod N).
,
S1 * S2 (mod N) = m1D * m2D (mod N) = (m1m2)D (mod N) = m3D (mod N) = S3.
1984 . -. 1991. - .
- (GS)
GS Ų Gm Signatur lgorithm ( -). GS , , , - . , - RS, .
-. ( - ), G, G < . (~10308 ~21024) G (~10154 ~2512), .
X, 1 < ≤ (-1),
Y =GX mod .
Y , . Y .
. , - h() m:
m = h(), 1<m<(-1),
, 1 < < ( -1), , (-1) . :
= GK mod
, , b
m = * + * b (mod (-1)).
(,b) S:
S = (, b),
.
(, , b) , (, ) . (, , b) ,
S = (, b)
.
m = h(),
.. .
|
|
= Ya ab (mod )
, ,
= Gm (mod ).
,
Ya ab (mod ) = Gm (mod ).
, , , S= (, b) X, Y. , , X, , .
, - , . - , , .
: =11, G=2 = 8. :
Y = GX mod = Y = 28 mod 11=3.
, - m = 5.
, - m = 5, = 9. , ( -1) . , (9,10)=1. b :
= GK mod = 29 mod 11 = 6,
b , :
m = * + * b (mod (-1)).
m = 5, = 6, = 8, = 9, = 11
5 = (6* 8+9* b)(mod 10)
9* b= -43(mod 10).
: b = 3. : = 6, b = 3. . Y = 3, - :
m = 5,
:
1) Yaab (mod ) = 36 * 63 (mod 11) =10 (mod 11);
2) Gm (mod ) = 25 (mod 11) =10 (mod 11).
, .
, - , (, b). .
- RS:
- , , 25% , .
- , (-1) (.. ).
- - ( RS).
- RS. , 1,5 , , , .