CryptoGame201712
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- えりか なつ
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1 (CRISMATH 2017)
2 n l l 2
3 n A B n l à l à l à A \ B (, ) (, ) (, ) (, ) 3
4 n n n l n l A \ B ( -1, -1 ) ( -10, 0 ) ( 0, -10 ) ( -3, -3 ) 4
5 n l vs pk b m 0, m 1 b R {0,1} c c = Enc pk (m b ) n l l 5
6 (1/2) n l l [HT04, ADGH06, GK06, KN08a, KN08b, MS09, OPRV09, FKN10, NS12, KOTY17, etc.] l [Gra10, BCZ12, ADH13, AGLS14, HV16] l [Y16, YY17] l l [GKTZ12] l [GKMTZ13, GKTZ15] l [FK17] 6
7 (2/2) n l [DHR00, LMS05, ILM05, ILM08] l [BPR15, GPS16, RSS17] l [BC+10,HPS14a, HPS14b] n l [HP10, GLV10, PS11, HPS16] l [ACH11, GK12, HTYY12] n l [Nak08, Ros11, LJG15, CK+16, SB+16, FPS17] l [AM13, GHRV14, CG15, GHRV16, IY17] l [AD+14, BK14, KB14, KK+16, KB16] 7
8 n n n 8
9 n n [Y16] Yasunaga. Public-key encryption with lazy parties. IEICE Trans. Fund. (2016) [YY17] Yasunaga, Yuzawa. Repeated games for generating randomness in encryption. Cryptology eprint Archive: 2017/218 [IY17] Inasawa, Yasunaga. Rational proofs against rational verifiers. IEICE Trans. Fund. (2017)
10 IoT n à l l Enc( pk, m ; r ) 10
11 IoT n Enc( pk, m ; r ) 11
12 n S R (1) M S, M R (2) l 1. Good 2. Bad n S, R, A l S R l m M S M R 12
13 m 0, m 1 3. b ß R {0,1} 2. (m 0, m 1, st) ß A 1 (pk) m b c pk 5. b ß A 2 (st, c) 4. c ß Enc pk (m b ; r S ) pk c 1. (pk, sk) ß Gen(1 k ; r R ) 13
14 m 0, m 1 4. b ß {0,1} 3. (m 0, m 1, st) ß A 1 (pk) m b 6. G / B r S ß {0,1} k (if G) r S = 0 k (if B) 7. c ß Enc pk (m b ; r S ) c 5. m b ( m b M R? ) pk c pk 8. b ß A 2 (st, c) 1. G / B r R ß {0,1} k (if G) r R = 0 k (if B) 2. (pk, sk) ß Gen(1 k ; r R ) 14
15 n l Gen l Enc l m b n Gen, Enc G/B n Out = (Win, Val S, Val R, Num S, Num R ) l Win {0,1}, Win = 1 ó b = b l Val w {0,1}, Val w = 1 ó m M w l Num w : w {S, R} Good 15
16 n Out = (Win, Val S, Val R, Num S, Num R ) u w (Out) = u w sec ( Win) Val w + ( c w rand ) Num w l u sec w, c rand w > 0 l u sec w /2 > q w c rand w q w : Num w l Good u w sec /2 n (σ S, σ R ) U w (σ S, σ R ) = min E[u w (Out)] l min, M S, M R 16
17 n R m M R R l R S n 3 Π 3 l l Good Good l l Good 17
18 3 Π 3 (pk S, sk S ) ß Gen(1 k ; r 1S ) pk R pk S (pk R, sk R ) ß Gen(1 k ; r 1R ) r 2R ß Dec(sk S, c 1 ) c 1 r 2S ß R U r = r R 2 r 2S (= r L r R ) c 2, c 3 c 2 ß Enc(pk R, r 2S ; r 3S ) r 2R ß R U c 1 ß Enc(pk S, r 2R ; r 3R ) r 2S ß Dec(sk R, c 2 ) r = r R 2 r 2S (= r L r R ) c 3 ß Enc(pk R, m; r L ) m ß Dec(sk R, c 3 ) c 4 c 4 ß Enc(pk S, m; r R ) 18
19 Π 3 1 (1) Π 3 m M S M R (2) Π 3 n (1) (2) (3) (4) S R S R Good Good Good - Good Good - Good - Bad - - x Bad x l (1), (2) l (3) m M R \ M S c 2, c 3 l (4) m M S \ M R c 4 19
20 n l l n A \ B ( -1, -1 ) ( -10, 0 ) ( 0, -10 ) ( -3, -3 ) 20
21 n n δ (0,1) : u w (Out) = ( c w rand ) Num w Gen + i=1,2, δ i-1 u w [i] u w [i] = u w sec ( Win) Val w i + ( c w rand ) Num w i U w (σ S, σ R ) = min E[u w (Out)] 21
22 n 2 Π 2 Repeat l Π 3 : Enc S R Good l S, R Good à l Good Bad Bad l Bad 22
23 2 Π 2 Repeat S (pk S, sk S ) ß Gen(1 k ; r 1S ) pk R pk S R (pk R, sk R ) ß Gen(1 k ; r 1R ) r 2R ß Dec(sk S, c 1 ) r 2S ß R U r = r 2R r S 2 c 2 ß Enc(pk R, r 2S ; r 3S ) c 3 ß m r c 1 c 2, c 3 r 2R ß R U c 1 ß Enc(pk S, r 2R ; r 3R ) r 2S ß Dec(sk R, c 2 ) r = r 2R r S 2 m ß c 3 r 23
24 2 Π 2 Repeat 2 (a) Pr[ m M S ] > c Enc S / (δ u Sec S ) (b) Pr[ m M R ] > c Enc R / (δ u Sec R ) (1) Π Repeat 2 m M S M R (2) Π Repeat 2 c Enc S : S Good u Sec S : S δ :, c Enc R, u Sec R 24
25 n n 2 l (1) (2) n 25
26 n l Ω =, l P l Q l S P = 0.6 P = 0.4 Q = S(Q, ω) Brier S 8 Q, ω = 2Q ω - Q(ω ) < 1 ω P / > 1 = - T ω Q(ω ) < /C 1 Q P - P ω S(P, ω) > - P ω S(Q, ω) / 1 / 1 P Q T ω 1 if ωc = ω 0 otherwise 26
27 n f f f(x) x f(x) n è << f(x) 27
![(2014)] n](/docs-images/104/161712253/images/28-3.jpg "R(T, x) f")
28 [Azar, Micali (2013), Guo (2014)] n R(T, x) f T x f(x) n è 28
29 x n t l g x 1 x M + + x P t 0 x M + + x P < t n t C = {i: x Y = 1} r {1,2,, n} l Q l x T è P Pr P = 1 = {Y:[ \]M} P l Q = P à l O(log n) g x M x < x P = S 8 Q, x T Q 0,1 Pr Q = 1 = t n t t 1, 0 29
30 Guo (2014) y M = f(x) g M d g < g f y < y f y M = f(x) g < y < g i g f y f x M x < x Y x YbM x PcM x P x {0,1} P d, S f O( d polylog(s) ) 30
31 Guo [IY17] n l l t < n/2 à 1 t n/2 à 0 n l l polylog(n) 3 l 31
32 n n l d, S f O( d polylog(s) ) n n 32
33 [HTYY17] Higo, Tanaka, Yamada, Yasunaga. Game-theoretic security for two-party protocols. Cryptology eprint Archive: 2016/1072
34 ?? A B?? n n 34
35 A B x?? A B?? y n n n l x, y, output( (A(x), B(y)) ) = ( 1( x > y ), 1( x > y ) ) B A A A l x 0, x 1, y s.t. 1( x 0 > y ) = 1( x 1 > y ), PPT B*, D B, Pr[ D B ( view B* ( A(x a ), B*(y) ) ) = 1 ] 1/2 a R {0,1} A B B B l x, y 0, y 1 s.t. 1( x > y 0 ) = 1( x > y 1 ), PPT A*, D A, Pr[ D A ( view A* ( A*(x), B(y b ) ) ) = 1 ] 1/2 b R {0,1} 35
36 n l x 0, x 1, y 0, y 1 s.t. 1( x 0 > y 0 ) = 1( x 1 > y 0 ) = 1( x 0 > y 1 ) = 1( x 1 > x 1 ) l a, b {0,1} (x a, y b ) D A b D B a l (suc A, suc B, guess A, guess B ) l l l l suc A = 1 ó A 1( x a > y b ) or suc B = 1 ó B 1( x a > y b ) or guess A = 1 ó D A b guess B = 1 ó D B a 36
37 n l A (1) 1( x a > y b ) (2) B y b (3) x a B l B l u A ( (A, D A ), (B, D B ) ) = suc A + guess A guess B l u B ( (A, D A ), (B, D B ) ) = suc B + guess B guess A 37
38 3 (A, B) ó (A, B) ó PPT A*, D A, D B, x 0, x 1, y 0, y 1 E[ u A ( (A*, D A ), (B, D B ) ) ] E[ u A ( (A, D A ), (B, D B ) ) ] PPT B*, D A, D B, x 0, x 1, y 0, y 1 E[ u B ( (A, D A ), (B*, D B ) ) ] E[ u B ( (A, D A ), (B, D B ) ) ] (A, B) ó A*, E[ u A ( A*, B ) ] E[ u A ( A, B ) ] B*, E[ u B ( A, B* ) ] E[ u B ( A, B ) ] 38
39 n à l l u A A à A* l l l suc A (A, B) guess A A or A* à B guess B B or B* à A l u B n à l D A = D B = D rand A à A abort u A l B A D B D A = D rand A à A abort u A l B* B A B*, D B D A = D rand B B* abort u B l B 39
40 (A, B) ó PPT A*, D A, D B, x 0, x 1, y 0, y 1 s.t. E[ u A ( (A*, D A ), (B, D B ) ) ] > E[ u A ( (A, D A ), (B, D B ) ) ] PPT B*, D A, D B, x 0, x 1, y 0, y 1 s.t. E[ u B ( (A, D A ), (B*, D B ) ) ] > E[ u B ( (A, D A ), (B, D B ) ) ] n A D B è D B B n D A, D B è D rand 40
41 (A, B) ó PPT A*, D A, x 0, x 1, y 0, y 1 s.t. PPT D B E[ u A ( (A*, D A ), (B, D B ) ) ] > E[ u A ( (A, D rand ), (B, D B ) ) ] PPT B*, D B, x 0, x 1, y 0, y 1 s.t. PPT D A E[ u B ( (A, D A ), (B*, D B ) ) ] > E[ u B ( (A, D A ), (B, D rand ) ) ] (A, B) D B ó PPT A*, D A, x 0, x 1, y 0, y 1 PPT D B s.t. E[ u A ( (A*, D A ), (B, D B ) ) ] E[ u A ( (A, D rand ), (B, D B ) ) ] PPT B*, D B, x 0, x 1, y 0, y 1 PPT D A s.t. E[ u B ( (A, D A ), (B*, D B ) ) ] E[ u B ( (A, D A ), (B, D rand ) ) ] 41
42 (A, B) ó PPT A*, D A, x 0, x 1, y 0, y 1 PPT D B s.t. E[ u A ( (A*, D A ), (B, D B ) ) ] E[ u A ( (A, D rand ), (B, D B ) ) ] PPT B*, D B, x 0, x 1, y 0, y 1 PPT D A s.t. E[ u B ( (A, D A ), (B*, D B ) ) ] E[ u B ( (A, D A ), (B, D rand ) ) ] n n A D B à a {0,1} 42
![(1) PPT D A, Pr[ D A (view A (A(x a ), B*(y b ))) = 1 ] 1/2 (2) Pr[ output((a(x a ), B(x b )) = ( 1( x > y ), 1( x >](/docs-images/104/161712253/images/43-1.jpg "y ) ) abort] = Pr[ output(a(x a ), B*(x b )) = ( 1( x > y ), 1( x > y ) ) abort] Pr[ D B ( view B* ( A(x a ), B*(y b")
43 n n n n l x, y, output( (A(x), B(y)) ) = ( 1( x > y ), 1( x > y ) ) B A l x 0, x 1, y, PPT D B, Pr[ D B ( view B ( A(x a ), B(y) ) ) = 1 ] 1/2, a R {0,1} B A l x 0, x 1, y 0, y 1, PPT B*, D B s.t. (1) PPT D A, Pr[ D A (view A (A(x a ), B*(y b ))) = 1 ] 1/2 (2) Pr[ output((a(x a ), B(x b )) = ( 1( x > y ), 1( x > y ) ) abort] = Pr[ output(a(x a ), B*(x b )) = ( 1( x > y ), 1( x > y ) ) abort] Pr[ D B ( view B* ( A(x a ), B*(y b ) ) ) = 1 ] 1/2 vs A B b B* n A B 43
44 4 (A, B) (A, B) n l A à A abort D B u A l B A D B (B, D rand ) à (B, D B ) D A u B l B A B*, D B (B, D rand ) à (B*, D B ) D A u B l B 44
45 n n l n l l l 45
46 n l l l à à l l à n l l 46
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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
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(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/101/151465237.jpg "(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
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