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1 ( 9 1 )

2 RSA RSA RSA Fiat-Shamir RSA Solovay-Strassen Miller-Rabin Pollard p F p F p n ( F p version )

3 Information Analysis by Ken-ichi Shiota Z a, b Z (a, b) a, b Z (a, b) b 0 a b r (a, b) = (b, r) (a, b) = d d = ax + by x, y d, x, y ( ) (1) b = 0 d := a, 1 ( a 0 ) x :=, 1 ( a < 0 ) y := 0 (2) b 0 a) { r n }, { x n }, { y n } r 0 := a, r 1 := b, x 0 := 1, x 1 := 0, y 0 := 0, y 1 := 1, q := r n 2 r n 1 r n := r n 2 q r n 1 = r n 2 r n 1 (n = 2, 3, ). x n := x n 2 q x n 1 y n := y n 2 q y n 1 b) r n = 0 n d := r n 1, x := x n 1, y := y n 1 ( r n 1 > 0 ) d := r n 1, x := x n 1, y := y n 1 ( r n 1 < 0 )

4 Information Analysis by Ken-ichi Shiota r n r n = 0 (a, b) = (r 0, r 1 ) = (r 1, r 2 ) = = (r n 1, r n ) = r n 1 r n = ax n + by n a n ax + ny = 1 x, y G (1) G ( ) (2) : (ab) c = a (bc) for a, b, c G. (3) e : ae = ea = a for a G. (4) a G a 1 : a G, a 1 G s.t. aa 1 = a 1 a = e G ( ) : ab = ba for a, b G G G G G G a, b, c G ab = ac ba = ca = b = c G N a N = e for a G G a a n = e a G a 0 m, n a m = a n = e m, n d = (m, n) a d = e.

5 Information Analysis by Ken-ichi Shiota d = mx + ny x, y a d = a (mx+ny) = (a m ) x (a n ) y = e (1) G a m a m = e m a (2) G G G = {, a 2, a 1, e, a, a 2, } a G = a G a a G n ( ) a, b a b n a b n a b ( mod n) a b n ( mod n ) n Z/nZ a Z ā a mod n Z/nZ = { 0, 1,, n 1} Z/nZ : ā + b := a + b, ā b := ab Z/nZ 0 n Z/nZ n a Z (Z/nZ) ( ) (Z/nZ) ( ) (Z/nZ) 1

6 Information Analysis by Ken-ichi Shiota ax + ny = 1 x, y x ā 1 = ax + ny ax ( mod n ) (Z/nZ) 0 n 1 n ϕ(n) a n a ϕ(n) 1 ( mod n ) p (Z/pZ) = { 1, 2,, p 1 }, ϕ(p) = p p e (Z/p e Z) = { x 0 x < p e, x p }, ϕ(p e ) = p e p e p a Z p a p 1 1 ( mod p ). a Z a p a ( mod p ) (1) p (Z/pZ) p 1 g (Z/pZ) (Z/pZ) = g = { 1 = g 0, g = g 1, g 2,, g p 2 } g ( ) mod p (2) mod p ( mod p ) ϕ(p 1) g mod p g j ( 1 j p 2, (j, p 1) = 1 )

7 Information Analysis by Ken-ichi Shiota (Z/3Z) = { 2 0 = 1, 2 } (Z/5Z) = { 2 0 = 1, 2, 2 2 = 4, 2 3 = 3 } (Z/7Z) = { 3 0 = 1, 3, 3 2 = 2, 3 3 = 6, 3 4 = 4, 3 5 = 5 } (Z/11Z) = { 2 0 = 1, 2, 2 2 = 4, 2 3 = 8, 2 4 = 5, 2 5 = 10, 2 6 = 9, 2 7 = 7, 2 8 = 3, 2 9 = 6 } { } (Z/13Z) 2 0 = 1, 2, 2 2 = 4, 2 3 = 8, 2 4 = 3, 2 5 = 6, 2 6 = 12, 2 7 = 11, 2 8 = 9, 2 9 = 5, = 2 10 = 10, 2 11 = p e (Z/p e Z) p e 1 (p 1) mod p e ( ) m, n a, b x a ( mod m ), x b ( mod n ) mn x m n u, v mu + nv = 1 x := bmu + anv m n x anv = a(1 mu) a ( mod m ) m, n x a ( mod mn ) x a ( mod m ), x a ( mod n )

8 Information Analysis by Ken-ichi Shiota m, n ϕ(mn) = ϕ(m) ϕ(n) n n = p e q f r g ϕ(n) = (p e p e 1 )(q f q f 1 ) (r g r g 1 ) = n ( 1 1 ) ( 1 1 ) ( 1 1 ) p q r ( ) s m 1, m 2,, m s 2 s a 1, a 2,, a s x a 1 ( mod m 1 ), x a 2 ( mod m 2 ),, x a s ( mod m s ) m 1 m 2 m s x ( ) x (1) m 1 M 1 = m 2 m s m 1 u 1 + M 1 v 1 = 1 u 1, v 1 w 1 := M 1 v 1 (2) j w 1 1 ( mod m 1 ), w 1 0 ( mod m 2 ),, w 1 0 ( mod m s ) w j 1 ( mod m j ), w j 0 ( mod m k ) ( k j ) w j (3) x := a 1 w 1 + a 2 w a s w s p p a x 2 a ( mod p ) p

9 Information Analysis by Ken-ichi Shiota p a 1 a p ( ) a := p 1 a p 0 a p ( ) = 3 2 ( mod 7 ) = 1. 7 ( ) 3 x 2 3 ( mod 7 ) x = ( ) ( ) a a Z a p 1 2 ( mod p ) p ( ) ( ) = 2 3 = 8 1 ( mod 7 ) = ( ) ( ) = 3 3 = 27 1 ( mod 7 ) = q q = p e1 1 per r q a (a ) q ( a := p 1 ) e1 ( ) er a p r ( q ) a, b q, r ( ) ( ) a b (1) a b ( mod q ) =. q q ( ) ( ) a 2 1 (2) a q = 1. = 1. q q (3) (4) ( ) ab = q ( a q ( ) { 1 = q )( ) b. q 1 q 1 ( mod 4 ) 1 1 q 3 ( mod 4 )

10 Information Analysis by Ken-ichi Shiota 9 (5) (6) ( ) { 2 = q ( ) r = q 1 q 1, 7 ( mod 8 ) 2 1 q 3, 5 ( mod 8 ) ( q r ) ( q ) r q 1 ( mod 4 ) r 1 ( mod 4 ) q r 3 ( mod 4 ) (1 6) (2) 1 2 ( ) ( ) ( ) ( ) ( ) = = = = = ( 1) = Pascal function jacobi_symbol(a,b:integer):integer; var c,j:integer; begin j:=1; if a<0 then begin a:=-a; if (b mod 4)=3 then j:=-j end; a:=a mod b; while a>1 do begin while (a mod 2)=0 do begin a:=a div 2; if ((b mod 8)=3) or ((b mod 8)=5) then j:=-j end; if a<>1 then begin if ((a mod 4)=3) and ((b mod 4)=3) then j:=-j; c:=b; b:=a; a:=c mod b end end; if a=0 then j:=0; jacobi_symbol:=j end;

11 Information Analysis by Ken-ichi Shiota P = ( plain text ) C = ( cipher text ) E : P C D = E 1 : C P P E C D P E E ( encryption ) D ( decryption ) K = A K P E A C D A P { } P E A C D A P A K P = C = {, A, B,, Z } = 27 Z/27Z = { 0, 1, 2,, 26 } K = Z/27Z { 0 } n K E n (x) = x + n ( in Z/27Z ) E n : P C E n n INFORMATION E 3 LQIRUPDWLRQ D n (y) = y n n 2.1.3

12 Information Analysis by Ken-ichi Shiota (1) A K : A = (A p, A s ) A p :, A s : (2) E A A p (3) D A A p A s A K (1) A = (A p, A s ) A p ( A s ) (2) A p x y = E A (x) (3) A s y x = D A (y) (1) N(N 1) (2) N 2 N ( ) P = C, P E A C D A P E A D A = id C ( ) ( RSA ) x, A = (A p, A s ) Step 1 : Step 2 : Step 3 : A s y = D A (x) x y (x, y) E A (y) x = E A (y)

13 Information Analysis by Ken-ichi Shiota A s x = E A (y) y name D A (name) (x, D A (name)) D A (name)

14 Information Analysis by Ken-ichi Shiota 13 3 RSA 3.1 RSA x Z (x, mod n) n x 0 (x, mod n) < n p, q : n = pq m = ϕ(n) = (p 1) (q 1) e : m d : ed 1 ( mod m ) p, q, e n, m, d ( d ) : P = C = { x Z 0 x < n } : A p = (n, e) : A s = (p, q, m, d) n, e x y = E A (x) := (x e, mod n) d y w = D A (y) := (y d, mod n) x = w x x p w x ed 0 x ( mod p ). x p ed 1 ( mod p 1 ) w x ed x ( mod p ). q w x ed x ( mod n ). w x 0 x, w < n n w = x.

15 Information Analysis by Ken-ichi Shiota RSA RSA (1) n, e y y = (x e, mod n) x ( ) (2) d d n p, q (1) p, q (2) m (3) d a 2 b 2 ( mod n ) a ±b ( mod n ) 2 a, b p, q (a + b)(a b) 0 ( mod n ), a ± b 0 ( mod n ) n p, q a + b a b (a + b, n) n a 2 1 ( mod n ) a ±1 ( mod n ) 2 a, b p, q (3) (1) ( ) ed 1 = 2 s t ( t ) n w w 2st 1 ( mod n ) w t, w 2t,, w 2s t 1 ( mod n ) w 2rt r = 0 w 2r 1t 1 ( mod n ) 50% ( ) w w 2r 1t 1 ( mod n ), w 2rt 1 ( mod n ) w a a = w 2r 1t

16 Information Analysis by Ken-ichi Shiota p q x = p + q 2, y = p q 2 x n, y 0, x 2 y 2 = n b = 1, 2, b 2 + n ( = a 2 ) b y a, b y p 1 q 1 p 1 q 1 l ed 1 ( mod l ) d ( ) p 1 q 1 l d ϕ(n) K =( ) ϕ(n) K ϕ(n) ed 1 ( mod K ) d d n 512 ( ) ) RSA RSA MIPSYear 94) RSA RSA MIPSYear 95) RSA RSA-130

17 Information Analysis by Ken-ichi Shiota (1) (2) a 2 3 a ( = 1 ) (a) a (b) 3 a 2 ( = 1 ) ( = 2 ) 2 ( = 45 ) (c) a ( = 1 ) 2 ( = 2 ) (3) 2 (a) (20 + b) b 245 b ( = 8 ) (b) b (c) 245 (20 + b) b ( = 224 ) ( = 21 ) 2 ( = 67 ) (d) 20 + b ( = 28 ) b ( = 8 ) 2 ( = 36 ) (4)

18 Information Analysis by Ken-ichi Shiota (100a) a 2 3 a = 1 2 ( b) (10 + b) b + b (20 + b) b = 245 b = 8 3 (180 + c) c + c (360 + c) c ( ) 80 = 2167 c = 5

19 Information Analysis by Ken-ichi Shiota 18 4 (185 + d/10) d + d ( d) d ( ) 800 ( ) 50 = d = ( ) a { x n } n=0,1,2, (1) x 0 := ( ) (2) x n+1 := 1 ( x n + a ) ( n = 0, 1, 2, ) 2 x n x n a ( n ) a a p 1 p (Z/pZ) g mod p { x (Z/pZ) x } = { g 0, g 2, g 4, } ( ) a (Z/pZ) { ±1 } ; a p { x (Z/pZ) x }

20 Information Analysis by Ken-ichi Shiota p a 2 b 2 ( mod p ) = a b ( mod p ) a b ( mod p ). a 2 1 ( mod p ) = a 1 ( mod p ) a 1 ( mod p ) ( ) p x 2 a ( mod p ) x (1) a (2) p = 2u + 1 ( u ) x := ±a u+1 2. (3) p = 4u + 1 ( u ) (3a) a u 1 ( mod p ) x := ±a u+1 2. (3b) a u 1 ( mod p ) x := ±a u u. (4) p = 8u + 1 ( u ) ( ) b (4a) b = 3, 5, 7, 11, = 1 b p (4b) a 2u 1 ( mod p ) (4b-1) a u 1 ( mod p ) x := ±a u+1 2. (4b-2) a u 1 ( mod p ) x := ±a u+1 2 b 2u. (4c) a 2u 1 ( mod p ) (4c-1) a u b 2u 1 ( mod p ) x := ±a u+1 2 b u. (4c-2) a u b 2u 1 ( mod p ) x := ±a u+1 2 b 3u. (5) p = 2 k u + 1 ( k 3, u ) ( ) b (5a) b = 3, 5, 7, 11, = 1 b p (5b) i = 1, 2,, k 1 e i = 0 1 : (5b-1) a 2k i 1u b (2k i e k 2 e i 1 )u 1 ( mod p ) e i := 0, (5b-2) a 2k i 1u b (2k i e k 2 e i 1)u 1 ( mod p ) e i := 1. (5c) x := ±a u+1 2 b (e 1+2e k 2 e k 1 )u (2) p = 23 = ( u = 11 ), a = 2 x = ±2 u+1 2 = ± ( ( 5) 2 = ) (3a) p = 29 = ( u = 7 ), a = 7 7 u = x = ±7 u+1 2 = ± ( ( 6) 2 = ) (3b) p = 29 = ( u = 7 ), a = 5 5 u = x = ±5 u ( ( 11) 2 = )

21 Information Analysis by Ken-ichi Shiota p (2) ( ) a u = a p 1 a 2 = 1. p ) 2 a a u+1 = (a u+1 2. (3) a 2u = a p 1 2 a u 1 ( ) a = 1. p a u 1 (3a) a u 1 a a u+1 = ) 2 (a u+1 2. (3b) a u 1 ( ) 2 2u = 2 p = 1. p (4) a u 2 2u 1 a a u+1 2 2u = (a u u) 2. a 4u = a p 1 2 a 2u 1 ( ) a = 1. p a 2u 1 (4a) a 2u a u 1 a u 1 (4a-1) a u 1 a a u+1 = ) 2 (a u+1 2. (4a-2) a u 1 ( ) b 4u = b p = 1. p a u b 4u 1 a a u+1 b 4u = (a u+1 2 b 2u) 2. (4b) a 2u 1 a 2u b 4u a u b 2u 1 a u b 2u 1 (4b-1) a u b 2u 1 a a u+1 b 2u = (a u+1 2 b u ) 2.

22 Information Analysis by Ken-ichi Shiota 21 (4b-2) a u b 2u 1 a u b 2u b 4u 1. a a u+1 b 6u = (a u+1 2 b 3u ) 2. (5) (4) ( ) p e x 2 a ( mod p e ) x ( ) a (1) = 1 p (2) (x 1 ) 2 a ( mod p ) x (3) (x e ) 2 a ( mod p e ) x e ( e = 2, 3, ) x e := x e 1 + (2x e 1 ) 1 (a (x e 1 ) 2 ). (2x e 1 ) 1 mod p 2x e 1 ( ) a (x e 1 ) 2 0 ( mod p e 1 ) (a (x e 1 ) 2 ) 2 0 ( mod p 2e 2 ). 2e 2 = e + (e 2) e (a (x e 1 ) 2 ) 2 0 ( mod p e ). (x e 1 + (2x e 1 ) 1 (a (x e 1 ) 2 )) 2 = (x e 1 ) 2 + (a (x e 1 ) 2 ) + ((2x e 1 ) 1 ) 2 (a (x e 1 ) 2 ) 2 (x e 1 ) 2 + (a (x e 1 ) 2 ) = a ( mod p e ) ( ) n x 2 a ( mod n ) x (1) n = p e 1 1 pe m ( ) a (2) p j = 1 p j (3) (x j ) 2 a ( mod p ej j ) x j (4) x x j ( mod p e j j ) ( j = 1, 2,, m )

23 Information Analysis by Ken-ichi Shiota n (1) n (2) 2 x 2 a ( mod n ) (1) (2) (2) (1) n p n = p e m ( (p, m) = 1 ) x 2 a ( mod n ) x 1 x 2 ( mod p e ), x 1 x 2 ( mod m ) 2 x 1, x p e = (x 1 + x 2, n)

24 Information Analysis by Ken-ichi Shiota Fiat-Shamir : A, ( ) P, ( ) V : : p, q : A, n = pq :, t : P ( ), a = (t 2, mod n) : P ID ( ) P t t V A P t A Step 1 : Step 2 : Step 3 : Step 4 : P r x := (r 2, mod n) V V b = 0 1 b P b P { r y := tr V b = 0 b = 1 V : { x y 2 ( mod n ) b = 0 ax y 2 ( mod n ) b = A (1) V b = 0 P t (2) V b = 1 P y Step 1 x := y 2 /a V ( t ) ax y 2 ( mod n ) (3) P V { b = 0 r b = 1 y

25 Information Analysis by Ken-ichi Shiota 24 (4) b m P V ) m 0 ( m ) ( RSA : ( RSA ) P, V : : p, q : P RSA, n = pq : P p, q V P mod n a t 2 a ( mod n ) t t Step 1 : Step 2 : Step 3 : V s a := (s 2, mod n) P P t 2 a ( mod n ) t t V V a t 2 ( mod n ) V s t s t ( mod p ), s t ( mod q ) 1/2 p = (s t, n) p V Step 1 : Step 2 : Step 3 : V s a := (s 2, mod n) P P t 2 a ( mod n ) t P A t V

26 Information Analysis by Ken-ichi Shiota V a P P ( ) B Step 1 : Step 2 : Step 3 : Step 4 : V s a := (s 2, mod n) P V A s P P t 2 a ( mod n ) t P A t V A B p, q p q 3 ( mod 4 ) n = pq Blum n = pq Blum a n mod n (1) x 2 a mod n x mod n 4 ±x 1, ±x 2 ( ) ( ) ±x1 ±x2 = 1, = 1 n n (2) (1) n C Step 1 : Step 2 : Step 3 : Step 4 : Step 5 : Step 6 : A p q 3 ( mod 4 ) p, q n = pq B B x a := (x 2, mod n) A A e = 1 1 e B B x A A x 2 a ( mod n ) ( x = e A B n)

27 Information Analysis by Ken-ichi Shiota C ( x (1) A x 2 a ( mod n ) (1) n) (2) (2) B x 2 a ( mod n ) ( ) x ( x = n n) ( ) x A e = e x n (1) C n mod 4 3 Blum ( x (2) n mod x 2 a ( mod n ) n) A B a x e = B ( x n) C A n Blum B p q B D Step 1 : Step 2 : Step 3 : Step 4 : A x a := (x 2, mod n) B B e = 1 1 e A ( ) x A (x ) 2 a ( mod n ) = e x x B n ( ) x B (x ) 2 a ( mod n ) = e n : P V : : G, H : ( ), σ : G H : ( P ) P σ σ V

28 Information Analysis by Ken-ichi Shiota G ( = H ) n G, H A = (a ij ), B = (b ij ) σ n a ij = b σ(i)σ(j) for i, j E Step 1 : Step 2 : Step 3 : Step 4 : P n φ F = φ(h) C = (c ij ) = (b φ 1 (i)φ 1 (j)) V V e = 1 2 e P e P { φ τ := φ σ V e = 1 e = 2 V : { F = τ(h) c τ(i)τ(j) = b ij for i, j e = 1 F = τ(g) c τ(i)τ(j) = a ij for i, j e = E (1) V e = 1 P σ (2) V e = 2 P τ Step 1 τ(g) V ( σ ) F = τ(g) (3) P V { e = 1 φ e = 2 τ (4) e m P V ( 1 2) m 0 ( m )

29 Information Analysis by Ken-ichi Shiota RSA n n n ( ) ( ) a n 1 1 ( mod n ) for a Z s.t. (a, n) = a n 1 1 ( mod n ) a n ( ) n n = 561 = (a, n) = a 2 1 ( mod 3 ) a 10 1 ( mod 11 ) a 16 1 ( mod 17 ) 560 2, 10, 16 a ( mod ) ( ) n ( Carmichael ) n (1) n (2) n = p 1 p 2 p r n (a) p j (b) r 3 (c) j p j 1 n ( Alford, Graville and Pomerance (1992) ) n 1 ( )

30 Information Analysis by Ken-ichi Shiota n 1 = p e 1 1 pe 2 2 per r j a n 1 j 1 ( mod n ) a (n 1)/p j j 1 ( mod n ) a j n (Z/nZ) a j m j m j p ej (Z/nZ) ϕ(n) m j ϕ(n) ( m 1, m 2,, m r ) = p e 1 1 pe 2 2 per r = n 1 ϕ(n) = n 1 n n 1 j n 1 = p e 1 1 pe 2 2 pe r r m ( p j m < n ) j a n 1 j 1 ( mod n ) (a (n 1)/p j j 1, n) = 1 a j n n q n q a n 1 j 1 ( mod q ) a (n 1)/p j j 1 ( mod q ) a j (Z/qZ) p ej ϕ(q) = q 1 p e1 1 pe2 2 pe r r = (n 1)/m n 1 m > n 1 = n 1 > n 1 q 1 n n j 6.2 Solovay-Strassen n a Z ( ) ( ( ) a n 1 a 2 ( mod n ) n) n 50% a (Z/nZ) ( )

31 Information Analysis by Ken-ichi Shiota (1) ( ) a (a) n n p n = pm mod p a Z a 1 ( mod m ) ( ( ) a a ( a ) = = 1, a n) n ( mod m ). p m (b) n p 2 n n = p e m ( e 2, (p, m) = 1 ) a = 1 + n p = 1 + pe 1 m n q ( p ) a 1 ( mod q ) ( a = 1 n) a n n 1 p e 1 m 1 ( mod p e ). 2 (2) (1) a x (Z/nZ) ( ) ax ( ) ( ) (Z/nZ) ( ) Step 1 : Step 2 : Step 3 : Step 4 : a ( 1 < a < n ) (a, n) > 1 (a, n) n ( ) ( ) Step 1 ( ) n ( ) m ) m ( Miller-Rabin n 1 2 n 1 = 2 s t n a ( (a, n) = 1 ) : (i) a t 1 ( mod n ) ( ) (ii) r ( 0 r < s ) s.t. a 2rt 1 ( mod n ) n a 2st = a n 1 1 ( mod n ) 1 mod n ±1 a 2s 1t 1 ( mod n ) a 2s 1t ±1 ( mod n ) a 2s 2t ±1 ( mod n ) (ii) (i)

32 Information Analysis by Ken-ichi Shiota n 75% a (Z/nZ) ( ) ( ) a ( ) ( ) ) m n ( ) m ( Solovay-Strassen Miller-Rabin 0 Adleman-Rumery

33 Information Analysis by Ken-ichi Shiota ( ) n < (a, n) < n a 1 < (a, n) < n (a, n) n (a, n) (a, n) 7.2 Pollard p 1 n p 1 p n p n p 1 p 1 = p e1 1 pe2 2 pe r r M ( M = 10 5 ) j p ej j M (1) M q k f k := log qk M K := ( q f k k M < qf k+1 k ) (2) a (3) (a, n) 1 < (a, n) < n q k M (4) (a, n) = 1 d := (a K 1, n) (5) d = n (2) d < n d p ( ) q f k k d p K j p e j K p 1 j a K 1 ( mod p ) 7.3 2

34 Information Analysis by Ken-ichi Shiota x n x 2 n x 2 n = ( ) x 2 ( ) ( mod n ) X 2 Y 2 ( mod n ) (X + Y ) (X Y ) 0 ( mod n ) (X ± Y, n) n R, B 1 x n R x n + R x 2 n B n = R = 1000, B 1 = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = 345 = n = = n = = n = = n = = n = = n = =

35 Information Analysis by Ken-ichi Shiota n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = n = = , 2, 20, 21 a := , b := (a + b, n) = ( , ) = 14533, (a b, n) = ( , ) = = p 900, = p n = x p x p x log p log n ( ) ( 1) F x 2 n 2 2 X 2 Y 2 ( mod n ) (d, n) > 1 d

36 Information Analysis by Ken-ichi Shiota ( ) R R := { x R y R s.t. xy = 1 } ( R ) R R F F = F { 0 } F p F p p Z/pZ = { 0, 1, 2,, p 1 } F p p F p F p : F p a + b := (a + b, mod p), (x, mod p) x p 0 (x, mod p) p 1 F p a b := (a b, mod p) a F p ( a 0 ) b ax + ny = 1 x a b F p = bx a

37 Information Analysis by Ken-ichi Shiota F F F ( 2 1 = 2, 2 2 = 4, 2 3 = 3, 2 4 = 1. ) F ( 3 1 = 3, 3 2 = 2, 3 3 = 6, 3 4 = 4, 3 5 = 5, 3 6 = 1. ) 8.3 F p n p F p = { 0, 1, 2,, p 1 } p G(t) F p n F F := { F p n 1 } G(t) F F p n F p n F a(t) + b(t) := ( a(t) + b(t) F p ) F a(t) b(t) := ( a(t) b(t) G(t) ) ( 0 )

38 Information Analysis by Ken-ichi Shiota F 4 p = 2, G(t) = t 2 + t F 4 α F 2 α 2 + α + 1 = α α α α α+1 α α α α α+1 α+1 α α α α α+1 α 0 α α+1 1 α+1 0 α+1 1 α ( α 1 = α, α 2 = α + 1, α 3 = 1. ) F 8 p = 2, G(t) = t 3 + t F 8 β F 2 β 3 + β + 1 = β β+1 β 2 β 2 +1 β 2 +β β 2 +β β β+1 β 2 β 2 +1 β 2 +β β 2 +β β+1 β β 2 +1 β 2 β 2 +β+1 β 2 +β β 0 1 β 2 +β β 2 +β+1 β 2 β 2 +1 β+1 0 β 2 +β+1 β 2 +β β 2 +1 β 2 β β β+1 β β+1 β β 2 +β 0 1 β 2 +β β β+1 β 2 β 2 +1 β 2 +β β 2 +β β β+1 β 2 β 2 +1 β 2 +β β 2 +β+1 β β 2 β 2 +β β+1 1 β 2 +β+1 β 2 +1 β+1 β 2 +1 β 2 +β+1 β 2 1 β β 2 β 2 +β β β β 2 +1 β 2 +β+1 β+1 β 2 +β β 2 +β β β 2 β 2 +β+1 β + 1 ( β 1 = β, β 2 = β 2, β 3 = β + 1, β 4 = β 2 + β + 1, β 5 = β 2 + β, β 6 = β 2 + 1, β 7 = 1. ) F 16 p = 2, G(t) = t 4 + t F 16 p = 2, G(t) = t 4 + t F 16 γ F 2 γ 4 + γ + 1 = 0 δ F 2 δ 4 + δ = 0 δ = γ F 16 F 16

39 Information Analysis by Ken-ichi Shiota F 9 p = 3, G(t) = t 2 + t F 9 α F 3 α 2 + α + 2 = α α+1 α+2 2α 2α+1 2α α α+1 α+2 2α 2α+1 2α α+1 α+2 α 2α+1 2α+2 2α 2 1 α+2 α α+1 2α+2 2α 2α+1 α 2α 2α+1 2α α+1 2α+2 2α α+2 2α α α α+1 α+2 2α+1 α+2 α 2α+2 α α α+1 α+2 2α 2α+1 2α α α+1 α+2 2α 2α+1 2α α 2α+2 2α+1 α α+2 α+1 α 2α α+1 α+2 2α+2 2 α+1 α+2 2α 2 α 2α+1 α+2 2 2α+2 1 α 2α 2α+1 α+1 1 2α+1 2 2α 2α+2 α+2 ( α 1 = α, α 2 = 2α + 1, α 3 = 2α + 2, α 4 = 2, α 5 = 2α, α 6 = α + 2, α 7 = α + 1, α 8 = 1. ) 8.4 ( F p version ) ( F p version ) F p a(t), b(t), a(t), b(t) d(t), u(t), v(t) a(t)u(t) + b(t)v(t) = d(t) (1) b(t) = 0 d(t) := a(t), u(t) := 1 (2) b(t) 0 a) { r n (t) }, { u n (t) }, { v n (t) } r 0 (t) := a(t), r 1 (t) := b(t), u 0 (t) := 1, u 1 (t) := 0, v 0 (t) := 0, v 1 (t) := 1, q (t) := r n 2 (t) r n 1 (t) r n (t) := r n 2 (t) q(t) r n 1 (t) = r n 2 (t) r n 1 (t) (n = 2, 3, ). u n (t) := u n 2 (t) q(t) u n 1 (t) v n (t) := v n 2 (t) q(t) v n 1 (t)

40 Information Analysis by Ken-ichi Shiota 39 b) r n (t) = 0 n d(t) := r n 1 (t), u(t) := u n 1 (t), v := v n 1 (t) (3) c := (d(t) ), f := (c F p ) d(t) := f d(t), u(t) := f u(t), v(t) := f v(t) ( ) ( F 2 ) ( ) = 1 (3) F p n G(t) F p n F = { F p n 1 } G(t) a(t) F ( a(t) 0 ) ( version ) a(t), G(t) u(t) ( 8.3 G(t) F p n ) p n p n F p n F p - n F p n p n 1 g F F = { g, g 2,, g pn 1 = 1 } g F p n

41 Information Analysis by Ken-ichi Shiota : (1995) 2. : = 4 ( 1996 ) 3. : = 5 ( 1980 ) 4. : ( 1987 ) 5. D. M. Bressoud: Factorization and Primality Testing = Undergraduate Texts in Mathematics, Springer-Verlag ( 1989 ) 6. H. Cohen: A Course in Computational Algebraic Number Theory = Graduate Texts in Mathematics, 138, Springer-Verlag ( 1993 ) 7. N. Koblitz: A Course in Number Theory and Cryptography, 2nd Edition = Graduate Texts in Mathematics, 114, Springer-Verlag ( 1994 ) 8. A. K. Lenstra, H. W. Lenstra Jr. (Eds.): The developement of the number field sieve = Lecture Notes in Mathematics, 1554, Springer-Verlag ( 1993 ) 9. A. Salomaa: Public-Key Cryptgraphy = EATCS Monographs on Theoretical Computer Science, 23, Springer-Verlag ( 1990 ) ( )

2001 Miller-Rabin Rabin-Solovay-Strassen self-contained RSA RSA RSA ( ) Shor RSA RSA 1 Solovay-Strassen Miller-Rabin [3, pp

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

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