e d m m = D d (E e (m)) D d E e
m f c = f(m) m m m 1 f(m 1 ) = c m m 1 m c = f(m) c m c m b b 0, 1 b r f(b, r) f f(b, r) := y b r 2 n, n = pq ggt (p, q) = 1 p q y n f K f(x + y) = f(x) + f(y) f(x y) = f(x) f(y) x 2 = x x y ( n). x y n y y n
G H G H f : G Ñ H @g 1, g 2 P G : f(g 1 g 2 ) = f(g 1 ) f(g 2 ) G H H E c 1, c 2 m 1 + m 2 E(m 1 + m 2 ) = c 1 c 2 = E(m 1 ) E(m 2 ) c 1, c 2 m 1 + m 2 E(m 1 m 2 ) = c 1 c 2 = E(m 1 ) E(m 2 ) c r m r E(m r) = c r = E(m) r D n 2
c 1, c 2 m 1 + m 2 m 1 + m 2 = D(c 1 c 2 ) c 1, c 2 m 1 + m 2 m 1 m 2 = D(c 1 c 2 ) c r m r m r = D(c r )
1 2 1 n 1 2 n E e (m)?ˆ=e e (m 1 ) m ( ) Ee (m) D d = D d (E e (m m)) = m m = 0 E e (m) m m 1 m m 1 ( ) $ Ee (m) & D d E e (m 1 ) % ă 0 ñ m ă m 1 = 0 ñ m = m 1 ą 0 ñ m ą m 1
ID
E( ) Key 1 E( ) E( ) Key 1 E( ) Key
d i = D d (E e (P i ) E e (P i 1 )) E e (c i ) = E e (T i d i ) E e (C i ) = E e (C i 1 ) E e (c i ) e := d := i := P i := i d i := P i 1 P i c i := P i 1 P i E e := D d := T i := i C i := C i = ř i x=0 c x
C C Ð E e (0) Ź P l Ð E e (0) Ź P a Ð E e (0) Ź Ź P a Ð E e ( ) Ź d Ð D d (P a P l ) Ź T Ð Ź E e (c) Ð E e (d T ) Ź E e (C) Ð E e (C) E e (c) Ź C Ð D d (E e (C)) Ź C Ź C E e (C i ) C i = D d (E e (C i )) C i ζ x x = (E e (C i )) ζ x x d
D d (x) = D d ( E e (C i ) ζ)! = D d (E e (C i ζ)) = C i ζ ζ ζ = D d(x) C i ζ C i i i+k C i+k Ð C i C
T S i E e (P i ) E e (T i ) E e (c i ) d i P i E e (P i 1 ) d i = P i D d (E e (P i 1 )) c i E e (c i ) = E e (T i d i )
C E e (c i ) C c i E e (C) fl nź (E e (c i )) i=0 ) nź 0 =! D d (E e (C) (E e (c i )) i=0 0 E e (C) śn i=0 (E e(c i )) ζ nź E e (C) (E e (c i )) ˆ E e (ζ) i=0 ζ C c i ζ C c i ) nź ζ =! D d (E e (C) (E e (c i )) ˆ E e (ζ) i=0 E e (C) ś n i=0 (E e(c i )) T S x P x