RSA

RSA p q N=p*q(N)=(p-1)*(q-1) K

RSA M-

RSA :M=Ek=MKk=M l (N)+1=M*(M(N)) l= =M*1=M mod N

M(N)=1 mod N (M,N)=1, (M,N)>1a*b=1modpa*b=l*p+1

RSA : p q .N=p*q; (p-1)*(q-1) (K, (N)) - - k K - - , mod .

RSA k K - :Kk=1mod (N), .. (N), k .(N)=(p-1)*(q-1), .. p q k .p q N , .

RSA k K, M, E, N, f() g() : M=Ekmod N k: ME=MKmod N M. k=logEM mod N .

RSA M E : mod N^E0=E Ei=(Ei-1)Kmod NEk=E Ek E, Ek-1 :Ek=(Ek-1)Kmod N E=MKmodN , .

RSAE= MKmodN =MmodN, .. . :p=2p-1q=2q-1, p q .

RSA p q ( p q)

: p b , 0

: K (0

:M=amodp

:a=baKmodp ;a=MyKbaK=MbKbaK=M modp.

: K , , *. . , RSA.

*( , )

: y2 + axy + by = x3 + cx2 + dx + e (a,b,c,d,e ) :y2 + y = x3 - x2 : (0, 0), (1, -1), (1, 0) D (0, -1)

.. : 0, . , . , 0.

0 - ., 0 = -0 + 0 = . :S = (x, y) T = (x, -y). ,..:1 + 2 + 0 = 0 1 = -2.

P Q (. ) , . P Q, , S. P + Q + S = , P + Q = -SP + Q = T

.. - P Q, : S = P S = Q. Q: Q S . Q + Q = 2 Q = -S. . k (k>0) k .

.. G . : : k ( ): K=k*G

M ..A k .: E=(k*G, M+k*K)=(x,y):M=y-k*x[M+k*K]-[k*(k*G)]= = M+k*(k*G)- k*(k*G)=M

( ) ( , DES).

+ +

A

K1. :

K k2. :E0=f(K;KB)K= g(E0;kB)(E;E0)3. := g(E;K)E =f (M; K)KB