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1 2008 (2008/09/30)

2

3 1 ISBN ISBN ISBN RSA (1) (2) (Solovay Strassen )

4

5 10,000 1,000,000 (code) (cryptography) 5

6

7 Chapter 1 ISBN ISBN (International Standard Book Number, ) ( ) ISBN ISBN ISBN 10 ( ) 9 04S1099Z ISBN 1.1 ISBN ISBN 10 9 ISBN ?? = ISBN ? ? ? X ISBN 10 X 7

8 8 CHAPTER 1. ISBN 1.2 ISBN ISBN a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 1,, a a ( X 10 ) ( 9 1=1 ia i ) 11 =? a 10? 9 ia i a 10 1= a ia i + 10 a 10 = ia i 1=1 11 n a, b n a b (mod n) a b n a b (mod n) l a = b + nl a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 ISBN 10 ia i 0 (mod 11) 1=1 4797a ( 5 ) a 5 5a a (mod 11) 1=1

9 a 5 4 (mod 11) 5 a a = 15 4 (mod 11) a 5 = ISBN 123? ?890? 1.3 ISBN ax b (mod 11) x x x 1 (mod 10). x 0, 1, 2, 3, (mod 10) (mod 10) (mod 10) x = 7, 17, 27, l x = l 3 (7 + 10l) = l 1 (mod 10) ax b (mod n) x 0 x 1 x 0 (mod n) x 1. x 1 x 0 (mod n) l x 1 = x 0 + ln ax 1 = a(x 0 + nl) = ax 0 + anl ax 0 b (mod n)

10 10 CHAPTER 1. ISBN n 0 n 1 0 n 1 (n ) x 1 (mod 9) (9 ) x 4 (mod 6), 2x 1 (mod 6) (mod 6) (mod 6) (mod 6) (mod 6) (mod 6) (mod 6) 2x 4 (mod 6) x = 2, 5 2x 1 (mod 6) ISBN ax b (mod n) ax + ny = b x, y a, b d a b x, y ax + by = d ( x, y ) a b (greatest common divisor) gcd(a, b) m d = gcd(a, b) m = dl ax 0 + by 0 = d x 0, y 0 a(x 0 l) + b(y 0 l) = dl = m ax + by = m x = x 0 l, y = y 0 l m d ax + by = m d d x, y

11 a, b d = gcd(a, b) ax + by = m x, y d m ax b (mod n) gcd(a, n) b ax b (mod n) d = gcd(a, n) d 1 a = a o d, n = n 0 d 0 < n 0 < n x 0 ax b (mod n) a(x 0 + n 0 ) = ax 0 + a 0 dn 0 = ax 0 + a 0 n ax 0 b (mod n) x = x 0 + n 0 0 < n 0 < n x 0 + n 0 x 0 (mod n) d = gcd(a, n) = 1 x 0, x 1 ax b (mod n) ax 0 b (mod n), ax 1 b (mod n) a(x 0 x 1 ) 0 (mod n) a(x 0 x 1 ) n a n x 0 x 1 n x 0 x 1 (mod n) ax b (mod n) n ax b (mod n) gcd(a, n) = 1 n a 0 (mod n) ax b (mod n) x + 9y = 1 x, y x + 6y = 4 x, y x + 6y = 1 x, y ISBN 11 ISBN = = ax b (mod n) a, b, n

12 12 CHAPTER 1. ISBN 1.4 ISBN ISBN ISBN ISBN ISBN ISBN X ISBN ( ISBN ISBN X 10 1 ISBN ISBN ) ISBN ( ) ISBN

13 Chapter 2 ISBN ( ) (Hamming code) F 2 = {0, 1} 0 1 F 2 F = 0, = 1, = 1, = = 0, 0 1 = 0, 1 0 = 0, 1 1 = = =, + =, + =, + = =, =, =, = 2 = = 0 a + a = 2a = 0 a = 0 a = a

14 14 CHAPTER 2. F 2 2 n n {0, 1,, n 1} ax b (mod n) ( a ) n n ax 1 (mod n) c c 1 a c a p F p = {0, 1,, p 1} p F 2 p 2 ( ) p n n ( ), ( ) + ( ) = ( ) ( ) ( ) ( 1) = 7 u, v (u, v) (u, v) = (v, u) (u, v) = 0 u v O xy- A(a, b), B(c, d) OA OB ( ) (a b) (c d)

15 (u, v + w) = (u, v) + (u, w) ( (u, v) = (u, w) = 0 (u, v + w) = 0 ) m n m n 2 3 ( 1 2 ) i j (i, j)- 0 0 m n A A t A t A n m (i, j)- A (j, i)- m n l m M m n N MN l n (i, j) M i N j ( ) ( ) ( ) = ( ( ) n 1 n n n 1 m n n (n 1 ) m (m 1 ) F 2 )

16 16 CHAPTER F 2 ( ) : (AB)C = A(BC) : (A + B)C = AC + BC, A(B + C) = AB + AC n n 1, 0 n I n l n M, n m N MI n = M, I n N = N 2.2 ( ) 1 4 ( ) 7 7 F 2 G, H G = H = G H F ( R. W. Hamming, ,

17 ) v = ( ) vg ( ) = ( ) ( 4 ) 3 ( ) ( ) H = = = 3 3 ( ) 3 ( ) ( 4 ) ( ) v vg G, H G, H 1. H 1 7 2

18 18 CHAPTER 2. H H = G ( ) vg 4 v 4 H G H G G 1 (1000abc) H a + b + c = 0 b + c = a + c = 0 (a, b, c) = (0, 1, 1) G G H v ( ) = ( ) G 1 v = ( ) vg G 1 3 ( ) = ( ) = ( ) + ( ) vg H H G 0

19 = = w (7 ) i e i i w + e i (F ) w Hw = 0 H(w + e i ) = Hw + He i = He i H e i H i F 2 x + y + z = 6 (1) 2x + 3y + 4z = 20 (2) 3x + 2y + 3z = 16 (3) x = y = z =

20 20 CHAPTER 2. x + y + z = 6 (1) = (1) y + 2z = 8 (2) = (2) (1) 2 y = 2 (3) = (3) (1) 3 x + y + z = 6 (1) = (1) y + 2z = 8 (2) = (2) + 2z = 6 (3) = (3) + (2) x + y + z = 6 (1) = (1) y + 2z = 8 (2) = (2) z = 3 (3) = (3) (1/2) x = 1 (1) (2) (3) y = 2 (2) (3) 3 z = 3 x = 1, y = 2, z = 3 (1) (0 ) (2) (3) ( )

21 (1) (0 ) (2) (3) ( ) { x z = 2 y + 2z = 8 (x, y, z) = ( 2, 8, 0), ( 3, 10, 1) = (1)

22 22 CHAPTER (2) (3) ( ) Step (1) Step 2. a 1/a 1 Step 3. (i, j) (i, j) a 0 i i a (i, j) 0

23 Step 1, 2, 3 (1), (2), (3) ( ) x 1,, x n b b b n (x 1,, x n ) = (b 1,, b n ) ( ) 0 = x 3 x 5 x 3 = s x 5 = t 1 x 1 + x 3 + x 5 = 2 x 3 = s x 5 = t x 1 = s t + 2

24 24 CHAPTER 2. x 1 = s t + 2 x 2 = 2s 2t x 3 = s (s, t ) x 4 = 3t + 3 x 5 = t s, t ( ). x + y + z = 3 x y + z = 1 x + z = x y z = s x = 2 s y = 1 z = s x y = z s ( ) x + y + z = 0 x y + z = 0 x + z = 0 x y z = 0 0 0,, x = s y = 0 z = s x y z = s 1 0 1

25 x + y + z = 4 (1) 2x + y + z = 6 x y + 2z = 3 (3) (5) x + y + z = 3 x + y + 2z = 6 x + y + z = 3 x + y + 2z + 2u = 2 2x + y + z + 2u = 4 x + 2y + 5z + 5u = 6 (2) (4) (6) x + y + z = 3 x + y + 2z = 6 x y + z = 2 { x + 2y + 3z = 4 2x + 3y + 4z = 5 x + 2y + z = 0 2x 3y 3z = 0 3x + y + 2z = 0 4x + y z = F 2 x + y + z = 1 x + y = 1 (1) x + y = 0 (2) x + z = 1 x + z = 0 y + z = 1 R n ( ) n R n F 2 F 2 n F 2 n F 2 R n ( ) a v = (v 1 v n ) av = (av 1 av n ) R n V R n (1) v, w V v + w V (2) v V a av V v 1,, v r n r i=1 a iv i (a i ) R n v 1,, v r r i=1 a iv i (a i ) R n v 1,, v r V v r = 0 V v 1,, v r 1 V V V V V V = R n R n e i i 1 0 e 1,, e n V V n

26 26 CHAPTER 2. v 1,, v r V v 1,, v r r n M M M M V M M 0 V V M ( 0 r ) v 1 = (1 1 0), v 2 = (1 0 1), v 3 = (2 1 1) (1 0 1), (0 1 1) 2 ( v 1, v 2 ) v 1 = (1 2 3), v 2 = (4 5 6), v 3 = (7 8 9) F 2 v 1 = (1 1 0), v 2 = (1 0 1), v 3 = (0 1 1) ( ). x 1 x 2. = s 1 + s s m a 11 a 12. a 21 a 22. a m1 a m2. x n a 1n a 2n a mn s 1, s 2,, s m R n r m n r m = n r V R n V V V V R n

27 V ( ) V ( ) V r V r n M w V w n Mw = 0 V M x 1. x n = 0 n r (1 1 1) R 3 V V ( ) F 2 4 V V 2.5 k n (n, k) (n, k) H 1. H (n k) n n k H n k H V H n k V F 2 n n k C = V ( C ) C k 2. G k n C (k ) v vg G vg C H vg H 0 0 (vg ) w i 1 0

28 28 CHAPTER 2. n e i w + e i Hw = 0 H(w + e i ) = He i H i H H H H n 1 2 n 1 2 H H n (2 n 1) n H 0 H (2 n 1, 2 n n 1) ( ) F t w w + e e t H(w + e) = He t e He He e t H t e He (23, 12) G G = G C

29 G m n m n m K GK = I m (I m m ). x = x 1. x n e i i 1 n G m Gx = e i y i K = (y 1 y n ) GK = I m K vgk = v K G = ( ) GK = I K YES NO m 50 cm

30 30 CHAPTER cm (n, k) k n 2 k 2 k 2 n F 2 n u, v 2 k d d t = d/2 1 t = (d 1)/2 t F 2 n t n k n F 2 n t ( ) (Singleton ) n k + 1 d n k d (7, 4) =

31 (23, 12) 3 2

32

33 Chapter SHINSHUUNIVERSITY ABCDEFGHIJKLMNOPQRSTUVWXYZ DEFGHIJKLMNOPQRSTUVWXYZABC VKLQVKXXQLYHUVLWB QDJDQRNHQPDWVXPRWRVKL n SHINSHUUNIVERSITY 10 1 J. Caesar, BC100 BC44, CRSXCREEXSFOBCSDI 33

34 34 CHAPTER 3. n 25 (26 ) 26 A Z, B H, C T e z NAGANO A 1, 2, 3 14, 1, 7, 1, 14, ?? ( ). 3.2 RSA m, n mn

35 m , 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 RSA p 1 < p 2 < < p r n = p 1 p 2 p r + 1 p r n m n p i ( ). n 2 n n 1 ( ) n. x m 1 = (x 1)(x m 1 + x m x + 1) n n = lm, l 2, m 2 2 n 1 = (2 l ) m 1 = (2 l 1)((2 l ) m 1 + (2 l ) m (2 l ) + 1) 1 < 2 l 1 < 2 n 1 2 n 1 n 2 n = 2047 = , 2, = Mersenne, ,

36 36 CHAPTER 3. 2 n 1 2 n 1 (2 n 1) ( ) ( ). n 2 n n = 1, 2, 4, 8, 16 ( 2 n + 1 = 3, 5, 17, 257, 65537) n n n n + 1 ( ) n 2 (2 e ). m x m + 1 = (x + 1)(x m 1 x m 2 + x + 1) n m n = lm 2 n + 1 = (2 l ) m + 1 = (2 l + 1)((2 l ) m 1 (2 l ) m l + 1) 1 < 2 l + 1 < 2 n n + 1 n 2 2 n = = , = = ( ). n (n 1)! 1 (mod n) ((n 1)! = 1 2 (n 2) (n 1) ) (3 1)! = 1 2 = 2 1 (mod 3) (5 1)! = = 24 1 (mod 5) (7 1)! = = (mod 7) (11 1)! = = (mod 11) 1 Fermat, ,

37 (4 1)! = = 6 1 (mod 4) (6 1)! = = (mod 6) n = 13, ( ). p a p a p 1 1 (mod p) p = 5, a = = 81 1 (mod 5) p p = 4, a = = 27 3 (mod 4) a = 1 p p n n n ϕ(n) ( ). n a n a ϕ(n) 1 (mod n) p ϕ(p) = p 1 p p ϕ(4), ϕ(6), ϕ(8), ϕ(9), ϕ(10) 3 ϕ(8) 1 (mod 8) p, q n = pq ( ) ( ). m, n a, b x a (mod m), x b (mod n) x mn 1 Euler, ,

38 38 CHAPTER p, q ϕ(pq) = (p 1)(q 1). pq pq p q q p p q pq pq p + q 1 ϕ(pq) = pq p q + 1 = (p 1)(q 1) p, q a m a m(p 1)(q 1)+1 a (mod pq).. a pq a (p 1)(q 1) 1 (mod pq) a m(p 1)(q 1)+1 = (a (p 1)(q 1) ) m a a (mod pq) a pq (1) a p q (2) a q p (3) a p q 3 (3) a m(p 1)(q 1)+1 0 a (mod pq) (1) a q 1 1 (mod q) a m(p 1)(q 1)+1 = (a q 1 ) m(p 1) a a (mod q). a 0 (mod p) a m(p 1)(q 1)+1 0 (mod p) pq a a m(p 1)(q 1)+1 a (mod pq) (2) 3.3 RSA RSA (Rivest, Shamir, Adleman, 1977) ( ) A A

39 3.3. RSA 39 (1) p, q. ( p, q 200 ) (2) N = pq. (3) L = (p 1)(q 1). (4) L e. (5) ed 1 (mod L) d. N e p, q, L, d M N M M e (mod N) e, N C = M e d C C d (mod N) C d M ed M (mod N) M p = 7, q = 11 N = 7 11 = 77, L = (7 1)(11 1) = 60 e = 7 7d 1 (mod 60) d = 43 ( ) M = 50 M = 50 C = M e = (mod 77) C d = (mod 77) p = 3, q = 11, e = 7 N, L, d M = 15 RSA C d M ed M (mod N) ed 1 (mod L), L = (p 1)(q 1) m ed = m(p 1)(q 1) a a ed a (mod pq) d d ed 1 (mod L) L = (p 1)(q 1)

40 40 CHAPTER 3. d L N = pq p, q pq p, q 200 ( ) RSA 3 RSA RSA p, q 200 N, L, e, d p, q N, L e ed 1 (mod L) d 3. M 400 M e (mod N) ( ) 3.4 (1) ( ). a, b a = qb + r 0 q 0 r < b r ( ). a, b r 0 = a, r 1 = b i > 1 r i 1 = q i 1 r i + r i+1 (0 r i+1 < r i ) 1 Euclid, BC365 BC275,

41 3.4. (1) 41 r i+1 0 r i+1 < r i {r i } r i > 0 n r n+1 = 0 gcd(a, b) = r n gcd(200, 144). gcd(200, 144) = gcd(240, 252) 200 = = = = = a, b a = qb + r (0 b < r, q ) gcd(a, b) = gcd(b, r). d = gcd(a, b) d b r = a qb b r d gcd(b, r) d = gcd(b, r) d b a = qb + r a b d gcd(a, b) d = d gcd(a, b) = gcd(r 0, r 1 ) = gcd(r 1, r 2 ) = = gcd(r n 1, r n ) r n+1 = 0 r n 1 r n gcd(r n 1, r n ) = r n ( 1.3.5). a, b d a b x, y ax + by = d

42 42 CHAPTER 3. x, y x, y ( ) ( ). r 0 = a, r 1 = b, r i 1 = q i 1 r i + r i+1 (0 r i+1 < r i ), r n+1 = 0 r i = x i a + y i b x i, y i x n, y n s, t x i, y i r i+1 = r i 1 q i 1 r i = (x i 1 a + y i 1 b) q i 1 (x i a + y i b) = (x i 1 q i 1 x i )a + (y i 1 q i 1 y i )b x i+1 = x i 1 q i 1 x i, y i+1 = y i 1 q i 1 y i r 0 = a = 1 a + 0 b, r 1 = b = 0 a + 1 b x 0 = 1, y 0 = 0, x 1 = 0, y 1 = 1 x i, y i s + 144t = gcd(200, 144) = 8 s, t x 0 = 1, y 0 = 0 x 1 = 0, y 1 = 1 x 2 = = 1, y 2 = = 1 x 3 = = 2, y 3 = 1 2 ( 1) = 3 x 4 = 1 1 ( 2) = 3, y 4 = = 4 x 5 = = 5, y 5 = 3 1 ( 4) = ( 5) = gcd(220, 252) 220s + 252t = gcd(220, 252) s, t ax b (mod n) gcd(a, n) b ( 1.3.7) gcd(a, n) b b = b 0 gcd(a, n) x, q ax + nq = b ax 0 + nq 0 = gcd(a, n) x 0, q 0 x = b 0 x 0, q = b 0 q 0 ax + nq = b ( ) ( ) x 36 (mod 252) RSA p = 7, q = 11 e = 13 d (ed 1 (mod (p 1)(q 1)) )

43 3.5. (2) (2) RSA (400 ) a, e, n a e (mod n) (mod 123) = = = ( ) (mod 123) = = 102 ( ) = 102 (26) 102 (25) 102 (22 ) 102 (2i) 102 (2i) = (102 (2i 1) ) (20 ) 102 (21 ) 102 (22 ) 102 (23 ) 102 (24 ) 102 (25 ) 102 (26 ) = = = = = = = (mod 123) a e (mod n) 2 log 2 e e

44 44 CHAPTER mod (mod 71), (mod 127) a e (mod N) e a (2i) (mod N) ( ) 1300 (1) a, e, N (2) ans = 1 (3) e = 0 ans (4) e 1 (mod 2) ans ans a (mod N) (5) e e/2 (6) a a 2 (mod N) (7) (3) a, e, N, ans 4 e (5) (3) e = (mod 123) [1] N = 123 ( ) e = 100, a = 102, ans = 1 ( ) [2] e 0 (mod 2) : ans = 1, e = 50, a = (mod N) = 72 [3] e 0 (mod 2) : ans = 1, e = 25, a = 72 2 (mod N) = 18 [4] e 1 (mod 2) : ans = 1 18 (mod N) = 18, e = 12, a = 18 2 (mod N) = 78 [5] e 0 (mod 2) : ans = 18, e = 6, a = 78 2 (mod N) = 57 [6] e 0 (mod 2) : ans = 18, e = 3, a = 57 2 (mod N) = 51 [7] e 1 (mod 2) : ans = (mod N) = 57, e = 1, a = 51 2 (mod N) = 18 [8] e 1 (mod 2) : ans = (mod N) = 42, e = 0, a = 18 2 (mod N) = 78 [9] e = 0 ans = 42

45 ans e a = = = RSA p = 13, q = 17 N = pq e = 11 (1) L = ϕ(pq) (2) d (d ed 1 (mod L) ) (3) m = 3 (4) (3) (m = 3) 3.6 A A B A A B M N B N A M N A B A A B C C B ( ) B C

46

47 Chapter 4 RSA RSA N 2 N N 2 N N N 3 N 1 N N = mn 2 m, n m, n N N 47

48 48 CHAPTER N 2 N N N 3 N N N 1 N N = mn, m m n N 2 N N N 3 N N N ( ). N (1) 1 N ( ) (2) 1 (3) (N N ) (4) N (3) (5) N N ( ) N N p p a a p 1 1 (mod p) p a a = 2, 3 a p 1 (mod p) 1 p a p p 1 Eratosthenes, BC275 BC194,

49 p a p x x 2 a (mod p) a p ( ) a 1 p ( ) a = p 0 (a p ) 1 (a p ) 1 (a p ) ( ) p = , 2 2 4, 3 2 2, 4 2 2, 5 2 4, , 2, 4 3, 5, 6 ( ) 1 = 7 ( ) 2 = 7 ( ) 4 = 1, 7 ( ) 3 = 7 ( ) 5 = 7 ( ) 6 = ( a 11), a = 1, 2,, 10, p a, b ( ) ( ) a b (1) a b (mod p) = p p (2) ( ) ab = p ( a p ) ( ) b p ( ) ab 2 (3) b p = p (4) ( ) 1 = 1, p ( ) 1 = ( 1) (p 1)/2, p 1 Legendre, , ( ) a p ( ) 2 = ( 1) (p2 1)/8 p

50 50 CHAPTER 4. ( p (5) [ ] q p ( ) ( ) q p q = ( 1) (p 1)(q 1)/4 q p ) ( ) q = ( 1) (p 1)(q 1)/4 p ( ) ( ) 1 2. p 1 (mod 4) 1 p 3 (mod 4) 1 p ( ) ( ) p p q p 1, 7 (mod 8) 1 p 3, 5 (mod 8) 1 p 3 q p (mod 4) q 3 (mod 4) 1 1 ( ) (2) ( ) 24 = 31 ( ) 3 ( ) ( ) ( ) = (4) ( ) 2 = ( 1) (312 1)/8 = ( 1) 120 = 1 31 (5), (1) ( ) ( ) ( ) = ( 1) (3 1)(31 1)/4 = ( 1) 15 = ( 1) 1 = ( ) 24 = 1 31 ( ) n a n n = p e 1 1 p e 2 2 p er r ( ) e1 ( a a J(a, n) = p 1 p 2 ) e2 ( ) er a p r 1 n Jacobi, ,

51 n a, b (1) a b (mod n) J(a, n) = J(b, n) (2) J(ab, n) = J(a, n)j(b, n) (3) gcd(a, n) > 1 J(a, n) = 0 (4) J(1, n) = 1, J( 1, n) = ( 1) (n 1)/2, J(2, n) = ( 1) (n2 1)/8 (5) a, b J(a, b)j(b, a) = ( 1) (a 1)(b 1)/4 J(a, b) = ( 1) (a 1)(b 1)/4 J(b, a) J(26, 45) (2) J(26, 45) = J(2, 45)J(13, 45) (4) J(2, 45) = 1 (5) J(2, 13) = 1 J(13, 45) = ( 1) (13 1)(45 1)/4 J(45, 13) = J(6, 13) = J(2, 13)J(3, 13) J(3, 13) = ( 1) (3 1)(13 1) J(13, 3) = J(1, 3) = 1 J(26, 45) = 1 J(a, b) a a = 2 e a 0 (a 0 ) J(a, b) = J(2, b) e J(a 0, b) (5) J(28, 45) a, b J(a, b) p a ( ) a a (p 1)/2 (mod p) p a p ( ) a 0 a = 1 p b a b 2 (mod p) a (p 1)/2 b p 1 1 (mod p) ( ) a a = 1 a (p 1)/2 p (mod p) 2 1 ±1 1

52 52 CHAPTER (Solovay Strassen ) Solovay Strassen (Solovay Strassen ). p (1) 1 < a < p 1 (2) gcd(a, p) > 1 p (3) j a (p 1)/2 (mod p) ( ) (4) J(a, p) (5) j J(a, p) (mod p) p (6) j J(a, p) (mod p) p 1/2 ( ) a p = J(a, p) j J(a, p) (mod p) p p 1/2 (200 p ) p Solovay Strassen a , , , 19, 30, , 14, 18, 47, 51, , 16, 38, 47, 69, , 10, 12, 16, 17, 22, 29, 38, 53, 62, 69, 74, 75, 79, 81, , 13, 41, 64, 92, , , 9, 27, 40, 81, 94, 112, , , 12, 27, 30, 31, 39, 58, 64, 69, 75, 94, 102, 103, 106, 121, , 17, 59, 86, 128, , 55, 64, 89, 98, , , 22, 23, 70, 80, 89, 99, 146, 147, , 51, 124, , 43, 68, 117, 142, 149 a p 3 (p 3)/2 (9, 15, 21, 27, 33, 35, 39, 51, 55, 57, 63, 69, 75, 77, 81, 87, 93, 95,

53 4.4. (SOLOVAY STRASSEN ) 53 99, 111, 115, 119, 123, 129, 135, 141, 143, 147, 155, 159, 161, 171, 177, 183, 187, 189, 195) a Solovay Strassen 1/2 Solovay Strassen a ( Solovay Strassen ) 10 1/2 10 = 1/1024 ( ) p Solovay Strassen a p a

54

55 [4] [3] [2] [1] ( ) 55

56

57 , 1, X , x 7 (mod 9) (x, y) = (4, 3) (x, y) = (2, 4) ( ) s (1) (x, y, z) = (2, 1, 1) (2) (3) (x, y, z) = (0, 0, 3) (4) (x, y, z) = ( 2 + s, 3 2s, s) (5) (x, y, z, u) = ( 6 + s, 16 3s, s, 4) (6) (x, y, z) = (3s, 5s, 7s) (1) (x, y, z) = (1, 1, 1) (2) (1 2 3), (0 1 2) (1 1 0), (0 1 1) (1 0 1), (0 1 1) ( ), ( ), ( ) C C 3 = NAGANOKENMATSUMOTOSHI , 103, ϕ(4) = 2, ϕ(6) = 2, ϕ(8) = 4, ϕ(9) = 6, ϕ(10) = N = 33, L = 30, d = (mod 33), (mod 33) = , = gcd(220, 252) = 4, (s, t) = (8, 7) x = d =

58 58 CHAPTER , (mod 71), (mod 127) (1) L = 192 (2) d = 35 (3) (mod 221) (4) (mod 221) , 1, 1, 1, 1, 1, 1, 1, 1,

59 [1] (ISBN ) [2] (ISBN ) [3] (ISBN ) [4] SGC 5, (ISSN ) 59

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