Download - Лекция_N11_(Модульная арифметика RSA).ppt
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(x) 1 x-1 . ( , : 14 15)
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p:(p) = p-1 x y :(xy)=(x) (y): (35)=(5) (7)=4*6=24
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a n :
a (n) =1 mod n
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() a p p :
a p-1 =1 mod p
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. ; n; ; .
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: ( . ) ( . )
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: :ax=a*a*a* * a mod n
, x
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, :a16=a8*a8;a8=a4*a4;a4=a2*a2;a2=a*a .. 16 4.
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~ 2*log x .
x=x0+2x1++2k-1xk-1
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logay=x mod n;.. :y=ax mod n
:log37mod 17;
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:
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:
.pi , si >0, . .
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( gcd)(x,y)=? x y . ,
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() y>x, y x a1 b1:y =a1*x +b1x =a2*b1 +b2b1 =a3*b2 +b3...bk-2=ak*bk-1+bk
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bk=0, bk-1
bk-1 = 1 .
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. 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
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4/2=4*2-1=4*3=2 mod 5
3/2=3*2-1=3*3=4 mod 5
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, (a,n)=1, .. n , . , .2*?=1mod6,
2*1=2mod62*2=4mod62*3=0mod62*4=2mod62*5=4mod6
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,
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., ., n ~ n/log(n) ( n), .. . ( n-q n ~q/ln(n) ) n=10127 q=104 ~ 34. , .
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, , () , .. .
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() *. , .
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. bin-1=1 mod n (n (n; bi)=1).., n , bi .
- k b1 b2 bk ; bi
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, . ()k. k- k=20 p~10-6 ( ). .
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:ai n bi -
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, RSA ( , )- ( )- ( ) ( )ECC ( )
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RSA E=MKmod N M=Ekmod N