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
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- あかり しげい
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1 A( ) x (x ) x 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
2 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 (mod N) (3) x y (mod N), y z (mod N) x z (mod N) (3) x y (mod N), y z (mod N) x y y z N x z = (x y)+(y z) x z N N, x, y, m N > 0 (1) x y (mod N) x mod N = y mod N (2) x y (mod N) mx my (mod N) (3) m 0 x y (mod N) mx my (mod mn) (4) x y (mod mn) x y (mod N) (1) x y (mod N) x = q 1 N + r 1, y = q 2 N + r 2 r 1 r 2 x y = (q 1 q 2 )N +(r 1 r 2 ). x y N r 1 r 2 N 0 r 1 r 2 < N r 1 r 2 = 0 x mod N = r 1 = r 2 = y mod N (2) x y N mx my = m(x y) N 2
3 (3) ( ) x y (mod N) x y = qn (q: ) m(x y) = mqn mx my mn ( ) mx my (mod mn) mx my = qmn (q: ) m(x y) = mqn m 0 x y = qn x y N (4) mn N mod N x x (mod N) y y (mod N) x + y x + y (mod N), xy x y (mod N) x x (mod N) y y (mod N) x x y y N (x + y) (x + y ) = (x x ) + (y y ) N x + y x + y (mod N). xy x y = xy x y + x y x y = (x x )y + x (y y ) N xy x y (mod N) m, d m d d m d m d n d m n m n gcd(m, n) m n 3
4 1.2.2 m, n > 0 (1) r = 0 gcd(m, n) = n m = qn + r (2) r 0 gcd(m, n) = gcd(n, r) (1) n > 0 n n r = 0 n m gcd(m, n) = n (2) d d m n d n r m n n r m n (1) r := m mod n (0 r < n ) (2) r = 0 n (3) r > 0 n r m := n; n := r (1) = = = = = = 0 4
5 57 gcd(2109, 1653) = gcd(1653, 456) = gcd(456, 285) = gcd(285, 171) = gcd(171, 114) = gcd(114, 57) = mod 1073 ( ) ( ) 3 = = = m, n 0 x, y mx + ny = gcd(m, n) 5
6 x, y 1 (1) = 2109 (2) = 1653 (3) ( 1) = 456 (1) (2) (4) 2109 ( 3) = 285 (2) (3) 3 (5) ( 5) = 171 (3) (4) (6) 2109 ( 7) = 114 (4) (5) (7) ( 14) = 57 (5) (6) gcd(m, n) = m, n 0 gcd(m, n) = 1 m n m, n > 0 (1) m n (2) mx + ny = 1 x, y (3) mx 1 (mod n) x (4) nx 1 (mod m) x (1) (2) (2) (1) d = gcd(m, n) (2) d m, n mx + ny 1 d > 0 d = 1 (2) (3) mx + ny = 1 ny 0 (mod n) mx 1 (mod n) (3) (2) mx 1 (mod n) mx 1 = qn q mx + n( q) = 1 (2) (4) 6
7 1.2.8 N > 0 m mx 1 (mod N) x mod N m ( ) x mod N (x N 0 (x mod N) < N) m 1 mod N m mod N m N mod N ( ) mod x 1 (mod 97) 97x 0 (mod 97) 2 x 2 x a (mod N) (1) 97x 0 (mod 97) (2) 23x 1 (mod 97) (3) 5x 4 (mod 97) (1) (2) 4 (4) 3x 17 (mod 97) (2) (3) 4 (5) 2x 21 (mod 97) (3) (4) (6) x 38 (mod 97) (4) (5) = 874 = (mod 97) ( )/97 = ( 9) = 1 Nx 0 (mod N) mx 1 (mod N) x x a (mod N) a mod N m 7
8 m n mx + ny = 1 x, y mod x + 22y = 1 x, y 1 96 mod cc 1300cc 100cc , 5, 7 x x a (mod 3), x b (mod 5), x c (mod 7) x = (70a + 21b + 15c) mod m n a, b mu+nv = 1 8
9 u, v { x a (mod m) x b (mod n) x nva + mub (mod mn). 1 mn (0 mn 1) 1 x a (mod m) x b (mod n) nx na (mod mn) mx mb (mod mn) nvx nva (mod mn) mux mub (mod mn) (nv + mu)x nva + mub (mod mn) nv + mu = 1 x nva + mub (mod mn) x nva + mub (mod mn) m mn x nva + mub (mod m) mu 0 (mod m) mu+nv = 1 nv 1 (mod m) mub 0 (mod m) nva + mub a (mod m) nva + mub b (mod n) x a (mod 3), x b (mod 5) = 1 x 6b 5a (mod 15) x c (mod 7), x d (mod 15) 9
10 7 ( 2) + 15 = 1 x 15c 14d (mod 105) d = 6b 5a x 15c 14(6b 5a) (mod 105) x 70a 84b + 15c (mod 105) (mod 105) x 70a + 21b + 15c (mod 105) x a (mod 3), x b (mod 5), x c (mod 7) 10
11 N > 0 Z N = {a Z : 0 a < N, a N } Z N φ(n) φ(n) Z 12 = {1, 5, 7, 11}, φ(12) = 4. Z 10 = {1, 3, 7, 9}, φ(10) = 4. Z 7 = {1, 2, 3, 4, 5, 6}, φ(7) = N > 0 (1) a N a mod N Z N. (2) a Z N a 1 mod N Z N. (3) a, b Z N ab mod N Z N. (1) a N ab 1 (mod N) b a = a mod N 0 a < N a a (mod N) a b ab 1 (mod N) a N a mod N = a Z N (2) a Z N a N b = a 1 mod N ab 1 (mod N) 0 b < N b Z N (3) a, b Z N aa 1 (mod N) bb 1 (mod N) a, b c = ab mod N 0 c < N c ab (mod N) c(a b ) ab(a b ) = (aa )(bb ) 1 (mod N) c Z N N > 0 N 11
12 a a φ(n) 1 (mod N) a Z N Z N = {a 1,..., a m } m = φ(n) i = 1,..., m a i = aa i mod N a i Z N {a 1, a 2,..., a m} = {a 1, a 2,..., a m } i j a i a j a i = a j aa i mod N = aa j mod N aa i aa j (mod N) b = a 1 mod N baa i baa j (mod N) ba 1 (mod N) 1 a i 1 a j (mod N) a i a j (mod N) 0 a i, a j < N a i = a j a i aa i (mod N) a 1a 2 a m = a 1 a 2 a m a m a 1 a 2 a m = (aa 1 )(aa 2 ) (aa m ) a 1 a 2 a m (mod N) a m a 1 a 2 a m a 1 a 2 a m (mod N) b i = a 1 i mod N b m b 2 b 1 a m (a 1 a 2 a m )(b m b 2 b 1 ) (a 1 a 2 a m )(b m b 2 b 1 ) (mod N) a i b i 1 (mod N) (a 1 a 2 a m )(b m b 2 b 1 ) 1 (mod N) a m 1 (mod N) 12
13 a N a = a mod N a φ(n) 1 (mod N) a a (mod N) a φ(n) a φ(n) 1 (mod N) Z 10 = {1, 3, 7, 9} (mod 10) (1) (mod 10) (2) (mod 10) (3) (mod 10) (4) (3 1)(3 3)(3 7)(3 9) (mod 10) 3 4 ( ) (mod 10) (3 7 9)(9 3 7) (3 7 9)(9 3 7) (mod 10) (3 7 9)(9 3 7) 1 (mod 10) (mod 10) p Zp = {1, 2,..., p 1} p ( ) a p a p 1 1 (mod p) a a p a (mod p) 13
14 p p Z p 1 (mod p) a Z p Z p = {a i mod p : i = 1, 2,..., p 1} a Z Z p φ(p 1) a Z p l > 0 l p 1 al l p 1 lj 1 (mod p 1) j > 0 lj = 1 + m(p 1) (m 0) (a l ) j = a lj = a 1+m(p 1) = a(a p 1 ) m a 1 m = a (mod p). a l Z p Z p 1.5 RSA ( ) 0 1 (2 ) ( )
15 RSA RSA p, q 2 L p 1 q 1 ( ) e L e = 41, 257, N = pq : e N : x < N y = x e mod N : d = e 1 mod L x = y d mod N RSA y = x e mod N ed 1 (mod L) ed = 1 + ml y d x ed (mod N). N = pq y d x ed (mod p), y d x ed (mod q) 15
16 L p 1 x ed = x 1+mL = x(x p 1 ) m x p x ed = x(x p 1 ) m x 1 m x (mod p) x p 0 x ed x (mod p) y d x (mod p) y d x (mod q) ( y d x p q ) y d x (mod pq) 0 < x < N = pq y d mod N = x. e, N d N = pq p q ( 1000 ) N p N d d 1000 y d mod N
17 1.5.4 d 2 α x d mod N α mod N 2 α 0 α0 α 2 1 α1 α x α mod N = a x α0 mod N = a 2 mod N, x α1 mod N = a 2 x mod N d 2 x d mod N 3 37 mod = = 3, 3 10 = 3 2 = 9, = (3 10 ) 2 = 81, = (3 100 ) 2 3 = (mod 100) = ( ) (mod 100) = ( ) (mod 100) 1.6 ElGamal RSA ElGamal RSA ElGamal 1 ( ) 17
18 IC ElGamal Z n b b x x b b x = b x x b b Diffie-Hellman p Z p g ga mod p g b mod p g ab mod p ElGamal q Z q ( )q 1 q := ( ); (q ) 1 < g < q 1 g 1 (g ) g Z p 1 < a < q 1 (a ) h = g a mod q; (h ) x (0 < x < q): (g k mod q, xh k mod q) : (y, z) x = zy a mod q y = g k mod q y a g ka (mod q) y = (y a ) 1 mod q y g ka 1 (mod q) z = xh k xg ak (mod q), zy x (mod q) 18
19 2 ( ) 2 RSA ( ) x 3 + 4x 2 + 2x + 5 3x 2 + 7x + 6 x
20 x 3 + 7x 2 + 9x x x x x x + 30 x 1 Z[x] mod Z 10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Z 10 x Z 10 [x] Z 10 [x] 2 2 (0 1) 1 2 Z 2 = F 2 = {0, 1} F 2 x F 2 [x] F 2 [x] 1011 x 3 + x + 1 x 2 10 x 2 = 10 2 = 100 x 3 = 10 3 = F 2 [x]
21 2.1 F 2 [x] F 2 [x] ( ) = x 3 + x n F 2 [x] n 1 f F 2 [x] f deg f f F 2 [x] x a F 2 = Z 2 ( mod 2 ) f(a) f = 1011 = x 3 + x + 1 f(0) = ( ) mod 2 = 1 f(1) = ( ) mod 2 = 3 mod 2 = f F 2 [x] f 0 g F 2 [x] g = qf + r, 0 deg r < deg f q, r F 2 [x] 1 q g/f r r g mod f F 2 = Z (mod 2) )
22 1011 = x 3 + x = x 2 + x + 1 F 2 [x] 0 1 F 2 [x] f = f(x) F 2 [x] x F 2 a F 2 f(a) F 2 = {0, 1} f(0) f(1) f(a) = 0 a f f, g F 2 [x] f = gh h F 2 [x] g f ( ) g f f mod g = a F 2 x a ( 2 1a) f F 2 [x] f(a) = 0 (a f ) f(x) = (x a)q + r r 0 0 F 2 f(a) = r r 0 f(a) = F F 0 F 2 [x] f, g F 2 [x] f g F ( 0) f g (mod F ) f g F mod F mod F F, f, g, h F 2 [x] F 0 (1) f f (mod F ) (2) f g (mod F ) g f (mod F ) 22
23 (3) f g (mod F ), g h (mod F ) f h (mod F ) F, f, g, h F 2 [x] (1) f g (mod F ) f mod F = g mod F (2) f g (mod F ) hf hg (mod F ) (3) h 0 f g (mod F ) hf hg (mod hf ) (4) f g (mod hf ) f g (mod F ) f f (mod F ) g g (mod F ) f + g f + g (mod F ), fg f g (mod F ) F 2 1 f, g, h f = gh g(h ) f ( ) h f h g h f g f g gcd(f, g) f g f, g F 2 1 g 0 f = qg + r (1) r = 0 gcd(f, g) = g (2) r 0 gcd(f, g) = gcd(g, r) F 2 f g 0 (1) r := f mod g (deg g > 0 0 deg r < deg g ) (2) r = 0 g 23
24 (3) r 0 g r f := g; g := r (1) (1) deg r = 0 r = (= x 3 + x 2 + x + 1) 1010 (= x 3 + x) 101 (= x 2 + 1) = = f, g 0, f, g F 2 [x] fu + gv = gcd(f, g) u, v F 2 [x] gcd(m, n) = f, g 0, f, g F 2 [x] gcd(f, g) = 1 f g f, g 0, f, g F 2 [x] (1) f g (2) fu + gv = 1 u, v F 2 [x] (3) fu 1 (mod g) u F 2 [x] (4) gv 1 (mod f) v F 2 [x] f F 2 [x] deg f 1 deg g < deg f 24
25 g F 2 [x] g 0 g f f f f F 2 [x] 1 deg f p Z p = {1,..., p 1} mod p Z p Z p Z p 1 (mod p ) 2.3 F 2 [x] F 2 [x] F 2 [x] 1 ( ) 1 1 f = 101 f(1) = = m n (m + n) ( ) , 11 (1 ) ( ) ,
26 , 11001, = F 2 [x] F 2 [x] ( 8 4 ) n 1 F 2 [x] n n 1 n n F 2 [x] 2 {0, 1, 10, 11} F 2 [x]( ) f, g 2 f g fg mod 111 ( 111 ) F 4 = {0, 1, 10, 11} F 4 F 2 [x] = (mod 111) a 0 ax = 1 x 0 ( ) F
27 4 q q n F F 2 [x] n q = 2 n F q = F 2 [x] n 1 F 2 [x] f, g F q fg f g F F q F q q n 1 n 1 n n n F 2 [x] 0 f F q F F 2 [x] F f fu + F v = 1 u, v F 2 [x] g = u mod F g n 1 fg 1 (mod F ) F q fg = 1 3 F q q F q F q = F q {0} 1 27
28 2.4.2 q F q α F q F q = {α i : i = 1, 2,, q 1} α q 1 = 1 α F q x F q xq 1 = 1 F q [y](f q ) F 2 [x] mod 1011 F 8 (8 = 2 3 ) 10 α = 10 α = 10, α 2 = 100, α 3 = (mod 1011), α 4 = 110, α 5 = (mod 1011), α 6 = (mod 1011), α 7 = (mod 1011). f mod f 10 = x f F 2 [x] 10011, mod (11 ) 28
29 2.5 F 2 n Z/pZ (p ) q q F q q 1 F q 1 GF (q) p F p = Z/pZ = {0, 1,..., p 1} + p p n 2 F p [x] n f F p [x]/(f) F p n ( ) F p f F p [x]/(f) (0 ) F p [x]/(f) n F p = {0, 1,..., p 1} p n F p n = F p [x]/(f) (f n 1 ) F q 1 F q [y] a 0 y 3 + a 1 y 2 + a 2 y + a 3 = (a 0, a 1, a 2, a 3 ) F q F q m (a 0, a 1, a 2,..., a m 1 ) (a i F q ) ( ) F m q Fm q (m 1) 1 ( ) 29
30 2.5.2 f = (a 0, a 1,..., a m 2, a m 1 ) F m q F q y y f(y) = a 0 y m 1 + a 1 y m a m 2 y + a m 1 f = (1, 2, 3) R 3 f(y) = y 2 + 2y + 3 f(1) = 6 f = (1, 0, 0, 0) f(y) = y 3 F q [y] F 2 [x] K f K[y] deg f > 0 g K[y] g = qf + r, 0 deg r < deg f q, r K[y] 1 q g/f r r g mod f K = F 4 = F 2 [x]/(111) (111 x 2 + x + 1 ) F 4 = {0, 1, 10, 11} 10 2 = 11, 11 2 = 10, = 1 f = (11, 1, 10) = 11y 2 + y + 10, g = (1, 11, 10, 1, 0, 1) = y y y 3 + y
31 ) = F 2 [x] f, g, F K[y] f g mod F = 0 f g (mod F ) K K 1 f, g, h f = gh g h f (, factor, divisor) g f f 0 (mod g) h f h g h f g f g 1 gcd(f, g) f g K a K y a f K[y] f(a) = 0 f(a) = 0 a f ( ) f(y) = (y a)q + r r 0 0 K f(a) = r r 0 f(a) = 0 31
32 2.5.7 K f(y) K[y] n f(y) K (f(y) = 0 ) n a 1 K f(a 1 ) = 0 f(y) = (y a 1 )f 2 (y) a 2 K f(a 1 ) = 0 a 2 a 1 f(a 2 ) = (a 2 a 1 )f 2 (a 2 ) = 0 (a 2 a 1 ) 1 = f 2 (a 2 ) = 0 f 2 (y) = (y a 2 )f 3 (y) f(y) = (y a 1 )(y a 2 )f 3 (y) a 3 K f(a 3 ) = 0 a 3 a 1, a 2 f(a 3 ) = (a 3 a 1 )(a 3 a 2 )f 3 (a 3 ) = 0 a 3 a 1 0 a 3 a 2 0 K f 3 (a 3 ) = 0 f 3 (y) = (y a 3 )f 4 (y) 1 n n f, g K[y] g 0 f = qg + r, q, r K[y] (1) r = 0 gcd(f, g) = a 1 g (a g ) (2) r 0 gcd(f, g) = gcd(g, r) K f g 0 (1) r := f mod g (deg g > 0 0 deg r < deg g deg g = 0 g K g 1 K ) (2) r = 0 a 1 g a g (3) r 0 g r f := g; g := r (1) 32
33 (1) deg r = 0 r K 0 33
II Time-stamp: <05/09/30 17:14:06 waki> ii
II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................
1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C
0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,
, 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 =
x n 1 1.,,.,. 2..... 4 = 2 2 12 = 2 2 3 6 = 2 3 14 = 2 7 8 = 2 2 2 15 = 3 5 9 = 3 3 16 = 2 2 2 2 10 = 2 5 18 = 2 3 3 2, 3, 5, 7, 11, 13, 17, 19.,, 2,.,.,.,?.,,. 1 , 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x
1 UTF Youtube ( ) / 30
2011 11 16 ( ) 2011 11 16 1 / 30 1 UTF 10 2 2 16 2 2 0 3 Youtube ( ) 2011 11 16 2 / 30 4 5 ad bc = 0 6 7 (a, b, a x + b y) (c, d, c x + d y) (1, x), (2, y) ( ) 2011 11 16 3 / 30 8 2 01001110 10100011 (
2012 A, N, Z, Q, R, C
2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1 1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2 N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
AI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id
1 1.1 1.1 R R (1) R = 1 2 Z = 2 n Z (2) R 1.2 R C Z R 1.3 Z 2 = {(a, b) a Z, b Z Z 2 a, b, c, d Z (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac, bd) (1) Z 2 (2) Z 2? (3) Z 2 1.4 C Q[ 1] = {a + bi
p-sylow :
p-sylow :15114075 30 2 20 1 2 1.1................................... 2 1.2.................................. 2 1.3.................................. 3 2 3 2.1................................... 3 2.2................................
mahoro/2011autumn/crypto/
http://www.ss.u-tokai.ac.jp/ mahoro/2011autumn/crypto/ 1 1 2011.9.29, ( ) http://www.ss.u-tokai.ac.jp/ mahoro/2011autumn/crypto/ 1.1 1.1.1 DES MISTY AES 1.1.2 RSA ElGamal 2 1 1.2 1.2.1 1.2.2 1.3 Mathematica
76 3 B m n AB P m n AP : PB = m : n A P B P AB m : n m < n n AB Q Q m A B AQ : QB = m : n (m n) m > n m n Q AB m : n A B Q P AB Q AB 3. 3 A(1) B(3) C(
3 3.1 3.1.1 1 1 A P a 1 a P a P P(a) a P(a) a P(a) a a 0 a = a a < 0 a = a a < b a > b A a b a B b B b a b A a 3.1 A() B(5) AB = 5 = 3 A(3) B(1) AB = 3 1 = A(a) B(b) AB AB = b a 3.1 (1) A(6) B(1) () A(
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240-8501 79-2 Email: nakamoto@ynu.ac.jp 1 3 1.1...................................... 3 1.2?................................. 6 1.3..................................... 8 1.4.......................................
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i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
A µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x A (1)
7 2 2.1 A µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x 1 2.1.1 A (1) A = R x y = xy + x + y (2) A = N x y = x y (3) A =
( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................
Block cipher
18 12 9 1 2 1.1............................... 2 1.2.................. 2 1.3................................. 4 1.4 Block cipher............................. 4 1.5 Stream cipher............................
ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
O E ( ) A a A A(a) O ( ) (1) O O () 467
1 1.0 16 1 ( 1 1 ) 1 466 1.1 1.1.1 4 O E ( ) A a A A(a) O ( ) (1) O O () 467 ( ) A(a) O A 0 a x ( ) A(3), B( ), C 1, D( 5) DB C A x 5 4 3 1 0 1 3 4 5 16 A(1), B( 3) A(a) B(b) d ( ) A(a) B(b) d AB d = d(a,
1 n =3, 2 n 3 x n + y n = z n x, y, z 3 a, b b = aq q a b a b b a b a a b a, b a 0 b 0 a, b 2
n =3, 200 2 10 1 1 n =3, 2 n 3 x n + y n = z n x, y, z 3 a, b b = aq q a b a b b a b a a b a, b a 0 b 0 a, b 2 a, b (a, b) =1a b 1 x 2 + y 2 = z 2, (x, y) =1, x 0 (mod 2) (1.1) x =2ab, y = a 2 b 2, z =
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ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 8 8 1 9 9 1 10 10 1 E-mail:hsuzuki@icu.ac.jp 0 0 1 1.1 G G1 G a, b,
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漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
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5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x
Chap9.dvi
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高校生の就職への数学II
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Part y mx + n mt + n m 1 mt n + n t m 2 t + mn 0 t m 0 n 18 y n n a 7 3 ; x α α 1 7α +t t 3 4α + 3t t x α x α y mx + n
Part2 47 Example 161 93 1 T a a 2 M 1 a 1 T a 2 a Point 1 T L L L T T L L T L L L T T L L T detm a 1 aa 2 a 1 2 + 1 > 0 11 T T x x M λ 12 y y x y λ 2 a + 1λ + a 2 2a + 2 0 13 D D a + 1 2 4a 2 2a + 2 a
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
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r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
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2011 Future University Hakodate 2011 System Information Science Practice Group Report Project Name RSA Group Name RSA Code Elliptic Curve Cryptograrhy Group /Project No. 13-B /Project Leader 1009087 Takahiro
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1729 1 2016 10 28 1 1729 1111 1111 1729 (1887 1920) (1877 1947) 1729 (, 2016:17) 12 3 1728 9 3 729 1729 = 12 3 + 1 3 = 10 3 + 9 3 (1) 1 1 2 1729 1729 19 13 7 = 1729 = 12 3 + 1 3 = 10 3 + 9 3 13 7 = 91
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Solutions to Quiz 1 (April 20, 2007) 1. P, Q, R (P Q) R Q (P R) P Q R (P Q) R Q (P R) X T T T T T T T T T T F T F F F T T F T F T T T T T F F F T T F
Quiz 1 Due at 10:00 a.m. on April 20, 2007 Division: ID#: Name: 1. P, Q, R (P Q) R Q (P R) P Q R (P Q) R Q (P R) X T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F T 2. 1.1 (1) (7) p.44 (1)-(4)
1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1
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I, A 25 8 24 1 1.1 ( 3 ) 3 9 10 3 9 : (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4), (3,3,3) 10 : (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) 6 3 9 10 3 9 : 6 3 + 3 2 + 1 = 25 25 10 : 6 3 + 3 3
kou05.dvi
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