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1 mahoro/2011autumn/crypto/
2 , ( ) mahoro/2011autumn/crypto/ DES MISTY AES RSA ElGamal
3 Mathematica Mathematica Windows > > ( ) Mathematica Mathematica file>open( new) ( ) Shift Enter 1 (1) (2). 3 6 (3). 3 2
4 1.3. Mathematica 3 (4). sin π 4 In Out [ ] In[1]:= 2+3 Out[1]= 5 In[2]:= 3^6 Out[2]= 729 In[3]:= 3/2 3 Out[3]= - 2 In[4]:= Sin[Pi/4] 1 Out[4]= Sqrt[2] 2 Mathematica ( ) (1). 30 (2) ( 27 (mod 11) ) (3) (mod 23) (4). 23x (mod 7) = 1 (23x 1 (mod 7) ) (5) (6). 45 ( 1 ) 45 In[5]:= Prime[30]
5 4 1 Out[5]= 113 In[6]:= Mod[27,11] Out[6]= 5 In[7]:= PowerMod[13,19,23] Out[7]= 2 In[8]:= PowerMod[23,-1,7] Out[8]= 4 In[9]:= FactorInteger[5040] Out[9]= {{2, 4}, {3, 2}, {5, 1}, {7, 1}} In[10]:= EulerPhi[45] Out[10]= 24 ( ) PowerMod[13,19,23] Mod[13^19,7l] PowerMod[13,19,23] PowerMod[23,-1,7] = EulerPhi[45] φ(45) 1 (1). 200 (2) (3) (mod 631) (4). 163x 1 (mod 17) (5). φ(40320) 1.4 ( )
6 = % ( ) 69 ( ) (1). 4 (2) (3). (4) , (a) 4 (b) 4 [ ]
7 6 1 (c) 36 (5) = 5, = 1, = 8, = 2, = 7, = 2, = 0, = (6). ( , ) ( ) 4, 2
8 , mahoro/2010autumn/crypto/ ( 86 ) ( 3 ) A 1 B 2 C 25 Z ( ) 2 C C 2 A ( )
9 , 1, 4, 12, 14 ANGOU A 19 T N O G K O A U I TOKAI TOKAI ANGOU 5 TOKAI BOSEI TOKAI 18, 0, 18, 22, (1). (2). (3).
10 (4) (1). 4 (2). 29 (3). 4 (4). 4 (1) (2) = (3) (4) =
11 (1) = (2) = (3) = = =
12 , ( ) mahoro/2011autumn/crypto/ (1). (2). (3) ( ) m, a, b q 1, q 2 0 r 1, r 2 < m { a = mq1 + r 1, b = mq 2 + r 2. r 1 = r 2 a b (mod m).
13 12 2 a b m ( ) m a b (mod m) a b m 6 { 31 = , 34 = 13 ( 3) (mod 13). 31 ( 34) = 65 = Z/NZ 2 (Z/NZ) N Z/NZ = {0, 1, 2,, N 1}. 3 (Z/NZ ) Z/NZ x, y Z/NZ x + y x, y x + y N x, y Z/NZ xy x, y xy N 2.6 Z/NZ 7 Z/5Z 4 ( ) N φ(n) 1, 2,, N N ((a, b) = 1 a, b )
14 2.6. Z/NZ : p n φ(p n ) = p n p n 1. ( ) 1 p n p p φ(p) = p 1 2 a, b (a, b) = 1 φ(ab) = φ(a)φ(b). ( ) p p, q φ(pq) = (p 1)(q 1) 3 N a Z/NZ. ax = 1 a φ(n) 1 ( ) N a N Z/NZ a φ(n) = 1.
15 RSA (1). p, q: (2). m = pq (3). φ(m) = (p 1)(q 1) (4). e: φ(m) (5). (m, e): (6). d: (ed 1 (mod φ(m)) d ) (7). M: ( ) (8). C = M e : Mathematica (1). p = Prime[100], q = Prime[134] ( ) (2). m = pq (3). n = (p 1) (q 1) (4). e: n (5). (m, e): (6). PowerMod[e, 1, n]: ( d ) (7). M: ( ) (8). PowerMod[M, e, m]: ( C ) C d = (M e ) d = M ed = M 1 = M.
16 2.6. Z/NZ Mathematica (1). M = PowerMod[C, d, m] 3 p = 233, q = 239, e = 101 d RSA 4 [ ( )] N φ(n) 1, 2,, N N ((a, b) = 1 a, b ) 1 [ ] p n φ(p n ) = p n p n 1. 2 [ ] a, b (a, b) = 1 φ(ab) = φ(a)φ(b). ( ) p, q φ(pq) = (p 1)(q 1) 3 [ ] N a Z/NZ. ax = 1 (a, N) = 1 a φ(n) 1 [ ( )] N a N Z/NZ a φ(n) = 1. 4 p, q a Z/pqZ, a 0 n Z a nφ(pq)+1 = a
17 p, q M e φ(pq) = (p 1)(q 1) (m, e) = (pq, e) d x, y ex + φ(pq)y = 1 x (pq, e) d d 0 a Z/pqZ a ed = a ( y)φ(pq)+1 = a C = M e C d C d = (M e ) d = M ed = M 2.8 RSA RSA (m, e) = (pq, e) d m = pq pq m (m, e) d x, y ex + φ(pq)y = 1 φ(pq) φ(pq) = (p 1)(q 1) p, q m = pq m RSA
18 2.9. (Z/NZ) , ( ) mahoro/2011autumn/crypto/ 2.9 (Z/NZ) 1 N (Z/NZ) := {a Z/NZ (a, N) = 1} Z/NZ Z/NZ ( ) a, b Z/NZ (a, N) = 1 (b, N) = 1 (ab, N) = 1 1 Z/NZ (1, N) = 1 1 (Z/NZ). a Z/NZ ax + Ny = 1 a 1 (Z/NZ) Z/NZ 2 ( ) G n G x e G x n = e 1 p, q (Z/pqZ) x x φ(pq) = x (p 1)(q 1) = 1. ( ) (Z/pqZ) φ(pq), 1 2
19
20 , ( ) mahoro/2011autumn/crypto/
21 ( ) ( ) 9 ( ) ( ) (Word ) 10 ( ) 11 ( ) 3.4 ID
22 ( ) 1bit 12 ( ) ID ID MD bit
23 22 3 MD bit SHA bit 160 bit SHA bit 256 bit SHA bit 384 bit SHA bit 512 bit ( 1) (1). K (2). ( ) r (3). r K t = C K (r) (4). t r ( 2) (1). K (2). ( ) r r K X = C K (r) (3). t r r r t = C K (r ) (4). t r r
24 3.6. MD , ( ) mahoro/2011autumn/crypto/ 3.6 MD5 md5make MD MD5 (1). URL md5make (2). md5make108a.lha ( Lhaca ) (3). ReadMe.txt (4). mahoro/2011autumn/crypto/ crypto2008.pdf.lha (5). crypto2008.pdf.lha crypto2008 crypto2008.pdf, crypto2008.pdf.md5 2 (6). md5make.exe ( md5make ) (7). crypto2008.pdf *.md5 MD5 (8). crypto2008.pdf crypto2008.pdf.md5 crypto2008.pdf (9). crypto2008.pdf.md5 (16 32 ) Web
25 24 3 ( ) [MD5 ] MD5 MD5 SHA-1, SHA-256 ( ) MultiHasher MD5 SHA- 1 (1). URL MultiHasher_1.1_win32.zip DOWNLOAD(Zip version) Mirror 1 (2). MultiHasher_1.1_win32.zip (3). MultiHasher.exe (4). Help RSA (1). (e, m) d (2). M M d C (3). C e
26 3.8. man-in-the-middle 25 (4). M C ( ) RSA 6 RSA (e, m) = (34987, ) = 1113 = man-in-the-middle man-in-the-middle (1). A B K (2). C A B C B (3). C B A ( C ) (4). A A 3.9 man-in-the-middle
27 (1). A (e A, m A ) CA (2). CA A (e A, m A ) (3). CA (e CA, m CA ) d CA (4). CA (e A, m A ) d CA P KC A (5). B CA P KC A e CA (e A, m A ) ( )
28 , ( ) mahoro/2011autumn/crypto/ 4.1 ( ) ( ) 13 (RSA ) RSA p, q m = pq e p, q, e p, q 4.2 ( ) ( ) ( )
29 28 4 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, (1). a, c, m Z (2). r 0 := a + c (mod m) (3). r n+1 := a r n + c (mod m) (1). (2). (3). 1 (4). (2)
30 (1). (2). (3). (4). 1 (5). (2)
31
32 , ( ) mahoro/2011autumn/crypto/ 5.1 ( ) ( ) 5.2 ( ) ( )
33 32 5 ( )
34 IC X
35
36 , ( ) mahoro/2011autumn/crypto/ 6.1 n n n n ρ (1). f(x) (2). x 1 (3). x i+1 = f(x i ) (mod n) (4). GCD(n, x 2i x i ) n 14 n = 2201 ρ f(x) = x x 1 = 1 x 2 = 2, x 3 = 5, x 4 = 26, x 5 = 677, x 6 = 522, x 7 = 1762, x 8 = GCD(n, x 8 x 4 ) = GCD(2201, ) = 31.
37 = n 1 n n n ρ f(x) n x 2i x i f(x) x 2i x i 6.2 Z/NZ 5 ( ) G. G H G H a G a := {a n n Z} a 5 p Z/pZ (Z/pZ) a (Z/pZ) (Z/pZ) = a a 15 p = 7 a = 3 3 = {3 0 = 1, 3 1 = 3, 3 2 = 9 = 2, 3 3 = 27 = 6, 3 4 = 81 = 4, 3 5 = 5, 3 6 = 1} (Z/7Z) = ( ) G G G e a G a G = e 16 ( ) p (Z/pZ) = p 1 (a, p) = 1 a p 1 1 (mod p).
38 p 1 (p 1 ) (1). n (2). n n = pq, p,q > 1, Q (3). M = (p 1)m (p 1) (4). (a, n) = 1 a (a, p) = 1 a p 1 1 (mod p) (5). a M = a (p 1)m = (a p 1 ) m 1 m = 1 (mod p) (6). a M 1 0 (mod p) a M 1 p (7). M (n, a M 1) p n ( ) 17 p 1 p 1 (1). n = 2201 (2). a = 2, M (3). (n, a M 1) (4). (n, a 2! 1) = (2201, 3) = 1, (n, a 3! 1) = (2201, 63) = 1, (n, a 4! 1) = (2201, ) = 1, (n, ) = (2201, 1023) = 31. (5) = = p = 31 p 1 = 31 1 = 30 = M = 2 5 = 10 p 1 = 30 M p 1 (n, a M 1) a M 1 (n, a M 1) = (n, a M 1 (mod n)) n p 1 a = 2, M 2!, 3!, 4!,
39 (1). m n (2). Q(x) := (x + m) 2 n (3). x = ±1, ±2, Q(x) (4). Q(x) x (5). Q(x) (6). t 2 Q(x) (7). s 2 (x + m) 2 (8). s 2 t 2 (mod n) (9). GCD(n, s ± t) n 18 n = n = m = 46. Q(x) := (x + 46) Q(1) := (1 + 46) = 8 = 2 3. Q(3) := (3 + 46) = 200 = Q(1)Q(3) = = (2 3 5) 2 = t 2. (1 + 46) 2 (3 + 46) 2 = = s 2. GCD(n, s t) = GCD(2201, 2263) = =
40 , ( ) mahoro/2011autumn/crypto/ mahoro@tokai-u.jp 24
41 ρ (1). p n (2). x 1, x 2,, x p, x p+1 mod p Z/pZ p x i, x j x i x j (mod p) i < j. (3). k = j i x i+1 x 2 i + 1 x 2 j + 1 x j+1 (mod p) x i+1 x j+1 (mod p) x i+t x j+t (mod p) (t = 1, 2, 3, ) (4). i kl l x kl x kl+k x kl+2k x kl+kl x 2kl (mod p) m = kl x 2m x m (mod p) (5). p m (6). x 2m x m 0 (mod p) GCD(n, x 2m x m ) p n n (7). x 2m x m n
42 n = [ ] m = = 172 Q(x) := (x + 172) [Q(x) ] x [Q(x) mod 2 ] x Q(x) mod 2 mod (mod 2) = = Q(1)Q( 1)Q( 2) = = = t 2 Q(1), Q( 1), Q( 2) (1 + m) 2 ( ) 2, ( ) 2, ( ) 2 s 2 = ( ) 2 ( ) 2 ( ) 2 = gcd(s t, n) = 131 gcd(s + t, n) = 227 n n =
43
44 , ( ) mahoro/2010autumn/crypto/ 7.1 g 2 x a = g x g a x x = log g a (a log g a ) x x ( ) g 2, p x a = g x (mod p) g x p p a p x a Ind x x = Ind g a (mod p) mod p g, a x
45 ElGamal p g (Z/pZ) g = (Z/pZ) a (Z/pZ) g x = a x x g, a x = Ind g a (Z/pZ) (mod p) g, a x [ ] (1). p, g, x, (0 < x < p) (2). (Z/pZ) (3). h = g x (4). (p, g, h) x [ ] (1). m m p (2). r, (0 < r < p) (3). c 1 = g r, c 2 = mh r (4). (c 1, c 2 ) [ ] (1). c 2 (c x 1) 1 m (2). c 2 (c x 1) 1 = mh r ((g r ) x ) 1 = mh r ((g x ) r ) 1 = mh r (h r ) 1 = m Elgamal g, h h = g x x
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
, = = 7 6 = 42, =
http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1 1 2016.9.26, http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1.1 1 214 132 = 28258 2 + 1 + 4 1 + 3 + 2 = 7 6 = 42, 4 + 2 = 6 2 + 8
インターネット概論 第07回(2004/11/12) 「SFC-CNSの現状」
/ / / : AES 128bit) 196bit 256bit 128bit) 10 12 14 196bit) 12 12 14 256bit) 14 14 14 (n, e) (n, d) M M : 2 ( 101 5) [e ] [e ] n = p * q (p q ) (n) = (p-1)(q-1) gcd( (n), e) = 1; 1 < e < (n) d = e^-1
2001 Miller-Rabin Rabin-Solovay-Strassen self-contained RSA RSA RSA ( ) Shor RSA RSA 1 Solovay-Strassen Miller-Rabin [3, pp
200 Miller-Rabin 2002 3 Rabin-Solovay-Strassen self-contained RSA RSA RSA ( ) Shor 996 2 RSA RSA Solovay-Strassen Miller-Rabin [3, pp. 8 84] Rabin-Solovay-Strassen 2 Miller-Rabin 3 4 Miller-Rabin 5 Miller-Rabin
() 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)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................
セアラの暗号
1 Cayley-Purser 1 Sarah Flannery 16 1 [1] [1] [1]314 www.cayley-purser.ie http://cryptome.org/flannery-cp.htm [2] Cryptography: An Investigation of a New Algorithm vs. the RSA(1999 RSA 1999 9 11 2 (17
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18 12 9 1 2 1.1............................... 2 1.2.................. 2 1.3................................. 4 1.4 Block cipher............................. 4 1.5 Stream cipher............................
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2011 Future University Hakodate 2011 System Information Science Practice Group Report Project Name Visualization of Code-Breaking RSA Group Name RSA C
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
1 1 1 1 1 1 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z) xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q) 1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b
40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x,
9.. x + y + 0. x,y, x,y, x r cos θ y r sin θ xy x y x,y 0,0 4. x, y 0, 0, r 0. xy x + y r 0 r cos θ sin θ r cos θ sin θ θ 4 y mx x, y 0, 0 x 0. x,y 0,0 x x + y x 0 x x + mx + m m x r cos θ 5 x, y 0, 0,
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b)
CALCULUS II (Hiroshi SUZUKI ) 16 1 1 1.1 1.1 f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b) lim f(x, y) = lim f(x, y) = lim f(x, y) = c. x a, y b
IW2001-B2 1 Internet Week 2001 ( ) Copyright 2001 All Rights Reserved, by Seiji Kumagai IW2001-B2 2 CodeRed Copyright 2001 All Rights
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