(x) 1 x-1 . ( , : 14 15)

p:(p) = p-1 x y :(xy)=(x) (y): (35)=(5) (7)=4*6=24

a n :

a (n) =1 mod n

() a p p :

a p-1 =1 mod p

. ; n; ; .

: ( . ) ( . )

: :ax=a*a*a* * a mod n

, x

, :a16=a8*a8;a8=a4*a4;a4=a2*a2;a2=a*a .. 16 4.

~ 2*log x .

x=x0+2x1++2k-1xk-1

logay=x mod n;.. :y=ax mod n

:log37mod 17;

:

:

.pi , si >0, . .

( gcd)(x,y)=? x y . ,

() y>x, y x a1 b1:y =a1*x +b1x =a2*b1 +b2b1 =a3*b2 +b3...bk-2=ak*bk-1+bk

bk=0, bk-1

bk-1 = 1 .

. a, a-1:a* a-1 = 1 mod n

:a=2 n =5a-1 = 32*0=0 mod 52*1=2 mod 52*2=4 mod 52*3=1 mod 52*4=3 mod 5

4/2=4*2-1=4*3=2 mod 5

3/2=3*2-1=3*3=4 mod 5

, (a,n)=1, .. n , . , .2*?=1mod6,

2*1=2mod62*2=4mod62*3=0mod62*4=2mod62*5=4mod6

,

., ., n ~ n/log(n) ( n), .. . ( n-q n ~q/ln(n) ) n=10127 q=104 ~ 34. , .

, , () , .. .

() *. , .

. bin-1=1 mod n (n (n; bi)=1).., n , bi .

k b1 b2 bk ; bi

, . ()k. k- k=20 p~10-6 ( ). .

:ai n bi -

, RSA ( , )- ( )- ( ) ( )ECC ( )

RSA E=MKmod N M=Ekmod N