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2 1 UTF Youtube ( ) / 30

3 4 5 ad bc = (a, b, a x + b y) (c, d, c x + d y) (1, x), (2, y) ( ) / 30

4 ( ) / 30

5 mod ( ) / 30

6 2,000 ( ) ( ) / 30

7 ( ) / 30

8 ( ) https ( ) ( ) / 30

9 p p 0 1, 2,..., p 1 p p = , 4, 8, 5, 10, 9, 7, 3, 6, p = 7 2 2, 4, , 2, 6, 4, 5, 1 6 ( ) / 30

10 p p x x, x 2, x 3,..., x p`1 = 1 1, 2, 3,..., p 1 ( 1 x ) x k p 1 x k(p`1) = (x p`1 ) k = 1 p a p a p`1 1 (mod p) Fermat p = 6, a = (mod 6) p = 6 ( ) ( ) / 30

11 n 1, 2, 3,..., n n φ(n) n = 20 1, 3, 7, 9, 11, 13, 17, 19 φ(20) = 8 n 1, 2,..., n 1 n φ(n) = n 1 p a p a p`1 1 (mod p) n a n a (n) 1 (mod n) n = 20 a = (mod 20) ( ) / 30

12 Euclid n, m n < m n, m 1 n = 0 m 2 m n m 0 m, n n, m 0 (1) m, n d mx + ny = d x, y m, n d = 1 mx + ny = 1 x, y ( ) / 30

13 n = 39, m = = 4 15, = 2 9, 15 9 = 1 6, 9 6 = 1 3, 6 3 = m, n ( ) / 30

14 = 15, = 9, = 6, = 3, = 3. ( ) / 30

15 RSA Rivest, Shamir, Adleman ( ) p, q 2 n = pq, L = (p 1)(q 1) 3 L L e 4 de 1 (mod L) d (3) (4) Euclid (e, n) d ( ) / 30

16 n = pq, L = (p 1)(q 1), de 1 (mod L) n M M e (mod n) n = pq 1, 2,..., n n p q q p 1 p + q 1 φ(n) = pq p q + 1 = (p 1)(q 1) = L a n = pq a L 1 (mod n) ( ) / 30

17 n = pq, L = (p 1)(q 1), de 1 (mod L) mod n d M M n = pq M L 1 (mod n) de = Lt + 1 (t ) M de (M L ) t M M (mod n) M e d d ( ) / 30

18 n = pq, L = (p 1)(q 1), de 1 (mod L) M n = pq M p, q p M q M q`1 1 (mod q) de = Lt + 1 M de (M q`1 ) (p`1)t M M (mod q) M de M p, q M de M (mod n) d d ( ) / 30

19 d e d e e = 1000 e = e = M M 2, M 4, M 8, M 16,... M 1000 = M 512 M 256 M 128 M 64 M 32 M 8 e = = ( ) / 30

20 p, q p = 11, q = 13 n = = 143, L = = 120 e = 7 7d 1 (mod 120) d = 103 M = 9 M (mod 143) (mod 143) M ( ) / 30

21 n M e n = pq L = (p 1)(q 1) de 1 (mod L) d n = pq n ( ) p, q (n ) n = pq M e M ( ) ( ) / 30

22 ( ) p, q p 2002 Agrawal, Kayal, Saxena 3 ( ) / 30

23 Fermat p p p 0 ( ) / 30

24 ( ) ( ) / 30

25 ( ) x f ( ) f(x) ( ) 1 f(x) 2 y f(x) = y x 3 x 1, x 2 f(x 1 ) = f(x 2 ) ( ) / 30

26 RSA RSA M e d M ed M (mod n) M d M d e M de M d M m = f(m) m e M f(m) m de ( ) / 30

27 Diffie-Hellman p p {1, 2,..., p 1} = {x, x 2,..., x p`1 } x x p 2 k y = x k p 2 r x r y y r = x kr x r k x rk ( ) / 30

28 p x, x k x r k x kr mod p x y y = x k k x, y 1, 2,..., p 1 ( p x ) ( ) / 30

29 p = 11 x = 2 {x, x 2, x 3,..., x 10 } {1, 2,..., 10} k = (mod 11) r = (mod 11) (mod 11) (mod 11) ( 5 ) ( ) / 30

30 ( ) / ( ) / ( ) 2 ( ) / 30

2008 (2008/09/30) 1 ISBN 7 1.1 ISBN................................ 7 1.2.......................... 8 1.3................................ 9 1.4 ISBN.............................. 12 2 13 2.1.....................

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15 mod 12 = 3, 3 mod 12 = 3, 9 mod 12 = N N 0 x, y x y N x y (mod N) x y N mod N mod N N, x, y N > 0 (1) x x (mod N) (2) x y (mod N) y x A( ) 1 1.1 12 3 15 3 9 3 12 x (x ) x 12 0 12 1.1.1 x x = 12q + r, 0 r < 12 q r 1 N > 0 x = Nq + r, 0 r < N q r 1 q x/n r r x mod N 1 15 mod 12 = 3, 3 mod 12 = 3, 9 mod 12 = 3 1.1.2 N N 0 x, y x y N x y

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