katagaitai workshop winter
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- りえ こいたばし
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1 katagaitai workshop 2018 winter 0CTF Finals: Authentication & Secrecy Shiho Midorikawa Shiho Midorikawa katagaitai workshop winter March 18, / 142
2 Introduction Introduction Shiho Midorikawa katagaitai workshop winter March 18, / 142
3 Introduction Who (Shiho Midorikawa) Cybozu Labs, Inc. CTF, Math, Crypto,... CTF scryptos, binja, vuls, fuzzi3 Shiho Midorikawa katagaitai workshop winter March 18, / 142
4 Introduction. Medium Hard Hard Shiho Midorikawa katagaitai workshop winter March 18, / 142
5 Introduction : Howgrave-Graham s Lemma LLL, Shiho Midorikawa katagaitai workshop winter March 18, / 142
6 Introduction : Authentication & Secrecy (0CTF 2017 Finals) Dual RSA Workshop Shiho Midorikawa katagaitai workshop winter March 18, / 142
7 NRI.. Shiho Midorikawa katagaitai workshop winter March 18, / 142
8 Shiho Midorikawa katagaitai workshop winter March 18, / 142
9 RSA RSA [RSA78] Definition 1 (RSA / ) p, q n = pq, ϕ(n) = (p 1)(q 1) e d = e 1 mod ϕ(n)., (e, n) RSA, d RSA. Definition 2 (RSA ) RSA (e, n) m Z/nZ c c = m e mod n. Definition 3 (RSA ) RSA (e, n) RSA d, c m m = c d mod n. Shiho Midorikawa katagaitai workshop winter March 18, / 142
10 RSA - m = c d mod n, m m ed m m kϕ(n) m 1 m (mod n). where k Z -, Euler s Thm.(gcd(a, n) = 1 a a ϕ(n) 1 (mod n)). Shiho Midorikawa katagaitai workshop winter March 18, / 142
11 RSA RSA [Bon99]. Alexander May [Ale07], May [May03],. [KO08]... Shiho Midorikawa katagaitai workshop winter March 18, / 142
12 RSA M? Shiho Midorikawa katagaitai workshop winter March 18, / 142
13 RSA 1: Stereotyped-Message Attack c m, m C x0 m = C + x 0,. (C + x 0 ) e c 0 (mod N), x 0. Shiho Midorikawa katagaitai workshop winter March 18, / 142
14 RSA 2: Boneh-Durfee s Attack RSA e, d. k. ed = 1 + k(p 1)(q 1) (, / ). ed = 1 + k(p 1)(q 1) = 1 + k(pq + 1 (p + q)) x = k, y = (p + q), A = N + 1, e x(a + y) (mod e). mod e d. Shiho Midorikawa katagaitai workshop winter March 18, / 142
15 RSA, x, y., RSA. [BD99], ROCA[NSS + 17],. Shiho Midorikawa katagaitai workshop winter March 18, / 142
16 Shiho Midorikawa katagaitai workshop winter March 18, / 142
17 ,,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
18 1996 Don Coppersmith EUROCRYPT Low-Exponent RSA with Related Messages [CFPR96] Finding a Small Root of a Univariate Modular Equation [Cop96b] Finding a Small Root of a Bivariate Integer Equation; Factoring with High Bits Known [Cop96a] [Cop96b]., 1997 Nicholas Howgrave-Graham [HG97]. Shiho Midorikawa katagaitai workshop winter March 18, / 142
19 [HG97] ( ).,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
20 Theorem 4 (Coppersmith, [HG97]) N, f(x) a 1 δ., f(x 0 ) 0 (mod N) x 0 : x 0 N 1 δ. a 1 [HG97],. Howgrave-Graham s Lemma,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
21 Theorem 5 (Coppersmith, [HG01]) N, 0 < β 1, f(x) 1 1., b N β, b N f(x 0 ) 0 (mod b), x 0 : x 0 N β2. [HG01] Thm. 4. b, N = k b (k N). Shiho Midorikawa katagaitai workshop winter March 18, / 142
22 Theorem 6 (Coppersmith, [May03]) N, 0 < β 1, f(x) 1 δ., b N β, b N f(x 0 ) 0 (mod b), x 0 log N, δ : x 0 N β2 δ. Shiho Midorikawa katagaitai workshop winter March 18, / 142
23 ,., M, b < N β M. β δ [Cop96b], [HG97] 1 f(x) 0 (mod M) [HG01] 1 f(x) 0 (mod b) [May03] f(x) 0 (mod b) May., Sagemath small_roots Pari/GP zncoppersmith May., (1 Modular ) May [May03]. Shiho Midorikawa katagaitai workshop winter March 18, / 142
24 . Thm. 6 Thm. 4, Thm Shiho Midorikawa katagaitai workshop winter March 18, / 142
25 ,,,.,.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
26 , Modular Modular. Berlekamp,., Modular. Shiho Midorikawa katagaitai workshop winter March 18, / 142
27 O x y ( 2, 0) ( 2, 0) ( 5, 3) ( 5, 3) (2 2, 6) ( 2 2, 6), 2 y = x 2 2, mod 3,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
28 :,.., Modular x 0 M x 0 < M, Modular 0 x < 3. Shiho Midorikawa katagaitai workshop winter March 18, / 142
29 O x y ( 2, 0) ( 2, 0) ( 5, 3) ( 5, 3) (2 2, 6) ( 2 2, 6) 0 x < 3, 6. Shiho Midorikawa katagaitai workshop winter March 18, / 142
30 , Modular.,,., Modular? Shiho Midorikawa katagaitai workshop winter March 18, / 142
31 .,., x 0 Modular f(x) 0 (mod M), k Z, f(x 0 ) km = 0. k = Shiho Midorikawa katagaitai workshop winter March 18, / 142
32 f(x 0 ) = 0 M = 0. f(x) = 0. f(x0 ) = 0, k = 0 f(x 0 ) 0 (mod M) Modular., Modular k = 0 f(x) = 0., f(x), k = 0. Shiho Midorikawa katagaitai workshop winter March 18, / 142
33 Howgrave-Graham s Lemma k = 0 Howgrave-Graham s Lemma. Lemma 7 (Howgrave-Graham, [HG97]) M, g(x) Z[x], ω. g(x) X, g(x 0 ) 0 (mod M) x 0 Z x 0 X., g(xx) < M ω g(x 0 ) = 0 ( ). deg g(x) deg g(x), g(x) = g i = gi 2, deg g(x) g(x). i=0 i=0 Shiho Midorikawa katagaitai workshop winter March 18, / 142
34 Howgrave-Graham s Lemma g(xx) < M ω. g(x 0 ) = deg g(x 0 ) i=0 g i x i i g i x i 0 i g i X i Shiho Midorikawa katagaitai workshop winter March 18, / 142
35 Howgrave-Graham s Lemma Cauchy-Schwarz ( i x ) ( ) iy i i x2 i i y2 i g i X i = (1 g i X i ) i i 1 ( g i X i ) 2 i=0,g i 0 g i 0 g i ω, 1 ( g i X i ) 2 = ω g(xx) < M. i=0,g i 0 i i Shiho Midorikawa katagaitai workshop winter March 18, / 142
36 Howgrave-Graham s Lemma, g(x 0 ) 0 (mod M) k Z g(x 0 ) = km. k g(xx) < M 0. g(x 0 ) = 0, g(x) = 0 x 0 ( ) Shiho Midorikawa katagaitai workshop winter March 18, / 142
37 Howgrave-Graham s Lemma Lemma 7 Modular, x 0 k = 0.,.? Shiho Midorikawa katagaitai workshop winter March 18, / 142
38 ,,.,.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
39 Proposition 8 M, f(x) 0 (mod M), g(x) 0 (mod M) x 0 Modular., a, b af(x) + bg(x) 0 (mod M) x 0. x 0 af(x 0 ) + bg(x 0 ) a 0 + b 0 0 (mod M).,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
40 ,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
41 f(x) 0 (mod M). mod M, M. Lemma 9 M, f(x). m, l g i,j (x) := M m i x j f i (x) (0 i m, 0 j l)., f(x 0 ) 0 (mod M) x 0 Z, g i,j (x 0 ) 0 (mod M m ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
42 x0 k f(x 0 ) = km. f i (x 0 ) = k i M i. gi,j (x) g i,j (x 0 ) = M m i x j 0 f i (x 0 ) = M m i x 0 k i M i = k i x j 0 M m., gi,j (x 0 ) 0 (mod M m ) ( ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
43 Lem. 9,., Lem. 7., Lem. 7..,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
44 -,.,. Definition 10 ( ) n R n {b 1,..., b n }., b 1,..., b n, L(b 1,..., b n ). L(b 1,..., b n ) = { x 1 b x n b n x i Z } b 1,..., b n B = t (b 1,..., b n ). L(B) = { Bx x Z n } Shiho Midorikawa katagaitai workshop winter March 18, / 142
45 - b1,..., b n : B = t (b 1,..., b n ):. B L(B) B. L(B) : (L(B) ) L(B) : n Shiho Midorikawa katagaitai workshop winter March 18, / 142
46 -, B = ( ). O b 1 b 2 Shiho Midorikawa katagaitai workshop winter March 18, / 142
47 -, B = ( ). O b 1 b 2 Shiho Midorikawa katagaitai workshop winter March 18, / 142
48 -, B =. ( ) b 2 b 1 O. Shiho Midorikawa katagaitai workshop winter March 18, / 142
49 -, B B L(B) = L(B ). ( ). Theorem 11 ([Pei13] Lemma 2.5) B 1, B 2, B 1 = UB 2 U Z n n. a a, 1. Shiho Midorikawa katagaitai workshop winter March 18, / 142
50 - Thm. 11, B, U L(B) = L(UB). ( ) ( ) B =, B 2 1 =, 1 1 ( ) 1 0 U =. 1 1, L(B) = L(B ) det(b) = det(b ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
51 - LLL 1982, Lenstra, Lenstra, and Lovász [LLL82] LLL., 1.,,. 1 Shiho Midorikawa katagaitai workshop winter March 18, / 142
52 - LLL, Thm. 12., SageMath,. sage: M = Matrix(ZZ, [[30, 25], [58, 23]]) sage: M.LLL() [ -2-27] [ 28-2] Shiho Midorikawa katagaitai workshop winter March 18, / 142
53 - LLL Theorem 12 (LLL, [LLL82]) B L, B = t (b 1, b 2,..., b n) B LLL., : b 1 b 2 b i 2 n(n 1) 4(n+1 i) det(b) 1 n+1 i., b 1 b 1 2 n 1 4 det(b) 1 n., LLL. Shiho Midorikawa katagaitai workshop winter March 18, / 142
54 -,, LLL.,., Step. Shiho Midorikawa katagaitai workshop winter March 18, / 142
55 - : m, l m, l,, Heuristic. : M, δ f(x), X f(x). : x 0 X, f(x 0 ) 0 (mod M) x 0, Shiho Midorikawa katagaitai workshop winter March 18, / 142
56 - Step 1. mod M m f(x) g i,j (x) (0 i m, 0 j l) (Lem. 9). Step 2. g i,j (x) B. Step 3. LLL B Thm. 12 B. Step 4. B 1, Howgrave-Graham s Lemma (Lem. 7). Step Step 5. B 1 h(x). Step 6. h(x) = 0,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
57 - Modular. M = , f(x) = x x f(x) 0 (mod M). x = 123, X = 200, x = 123. Shiho Midorikawa katagaitai workshop winter March 18, / 142
58 - : Step 1/6 Step 1. mod M m f(x) g i,j (x) (0 i m, 0 j l) (Lem. 9)., m, l m = 2, l = 1. (m + 1)(l + 1) = 6.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
59 - : Step 1/6 g 0,0 (x) = g 0,1 (x) = x g 1,0 (x) = x x g 1,1 (x) = x x x g 2,0 (x) = x x x x g 2,1 (x) = x x x x x Shiho Midorikawa katagaitai workshop winter March 18, / 142
60 - : Step 2/6 Step 2. g i,j (x) B. Howgrave-Graham s Lemma, g i,j (x) g i,j (xx),., ax + by = s, cx + dy = t 2 ( ) ( ) ( ) a b x s = c d y t. Shiho Midorikawa katagaitai workshop winter March 18, / 142
61 - : Step 2/6 g i,j (xx). gi,j (x) k g (k) i,j., (δ + 1)(m + 1),. 1 x x 2 x 3 x 4 x 5 g 0,0 g (0) 0,0 g (1) 0,0 X g(2) 0,0 X2 g (3) 0,0 X3 g (4) 0,0 X4 g (5) 0,0 X5 g 0,1 g (0) 0,1 g (1) 0,1 X g(2) 0,1 X2 g (3) 0,1 X3 g (4) 0,1 X4 g (5) 0,1 X5 g 1,0 g (0) 1,0 g (1) 1,0 X g(2) 1,0 X2 g (3) 1,0 X3 g (4) 1,0 X4 g (5) 1,0 X5 g 1,1 g (0) 1,1 g (1) 1,1 X g(2) 1,1 X2 g (3) 1,1 X3 g (4) 1,1 X4 g (5) 1,1 X5 g 2,0 g (0) 2,0 g (1) 2,0 X g(2) 2,0 X2 g (3) 2,0 X3 g (4) 2,0 X4 g (5) 2,0 X5 g 2,1 g (0) 2,1 g (1) 2,1 X g(2) 2,1 X2 g (3) 2,1 X3 g (4) 2,1 X4 g (5) 2,1 X5 Shiho Midorikawa katagaitai workshop winter March 18, / 142
62 - : Step 2/6,., f(x) M m i x j.,.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
63 - : Step 2/6 B = M 2 M 2 X MX 2 MX 3 X 4 X 5, 0. Shiho Midorikawa katagaitai workshop winter March 18, / 142
64 - : Step 3/6 Step 3. LLL B Thm. 12 B. det(b) = M X = M 6 X 15., LLL B = t (b 1,..., b 6 ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
65 - : Step 4/6 Step 4. B 1, Howgrave-Graham s Lemma (Lem. 7). Thm. 11, B, B,, U B = UB., 1 b 1 B b 1,..., b 6. Prop. 8, b 1 h(x) g i,j(x)., Thm. 12,., LLL Howgrave-Graham s Lemma.. Shiho Midorikawa katagaitai workshop winter March 18, / 142
66 - : Step 4/6 Thm. 12, 1 b 1 ( ). b (M 6 X 15 ) 1 6 (1), B 1 b 1 6, Howgrave-Graham s Lemma b 1 M 2 6 (2) b 1 ( ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
67 - : Step 4/6 Thm. 12 Eq. 1 Howgrave-Graham s Lemma Eq. 2, (M 6 X 15 ) 1 6 M 2 6 (3) (M 6 X 15 ) 1 6 = M 2 6 = , Eq. 3. Shiho Midorikawa katagaitai workshop winter March 18, / 142
68 - : Step 5/6 Step 5. B 1 h(x). b 1 i xi. gi,j (xx), b 1 i Xi x i. b 1. h(x) = 6x x x x x Shiho Midorikawa katagaitai workshop winter March 18, / 142
69 - : Step 6/6 Step 6. h(x) = 0,., Howgrave-Graham s Lemma, x = 123. Thm. 12, 1, LLL. SageMath. Shiho Midorikawa katagaitai workshop winter March 18, / 142
70 - [HG97]. SageMath f28c8fdbbd01b8193aa4af6af Shiho Midorikawa katagaitai workshop winter March 18, / 142
71 -, 1 Modular, X. Thm. 4., Thm. 6. b, M. Shiho Midorikawa katagaitai workshop winter March 18, / 142
72 - Thm.6 f(x) 0 (mod b). b, M. mod b, b. Lemma 13 b, M, b M. f(x), m, l g i,j (x) := M m i x j f i (x) (0 i m, 0 j l)., f(x 0 ) 0 (mod b) x 0 Z, g i,j (x 0 ) 0 (mod b m ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
73 - Thm.6 t = M b. x0 k f(x 0 ) = kb. f i (x 0 ) = k i b i. gi,j (x) g i,j (x 0 ) = M m i x j 0 f i (x 0 ) = (b t) m i x 0 k i b i = t m i k i x j 0 bm., gi,j (x 0 ) 0 (mod b m ) ( ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
74 - Thm.6 Lem. 13, mod b m. LLL, Howgrave-Graham s Lemma., Thm. 6. Shiho Midorikawa katagaitai workshop winter March 18, / 142
75 - 1, 2 3, n. 2. n. Shiho Midorikawa katagaitai workshop winter March 18, / 142
76 -, Howgrave-Graham s Lemma 2. Lemma 14 (Howgrave-Graham for Bivariate Polynomial) M, g(x, y) Z[x, y], ω. g(x, y) X, Y, g(x 0, y 0 ) 0 (mod M) x 0, y 0 Z x 0 X, y 0 Y., g(xx, yy ) < M ω g(x 0, y 0 ) = 0 ( ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
77 - 1., Howgrave-Graham s Lemma n. Shiho Midorikawa katagaitai workshop winter March 18, / 142
78 -,. Lemma 15 b, M, b M. f(x, y), m, l, τ g i,j,k (x, y) := M m i x j y k f i (x, y) (0 i m, 0 j l, 0 k τ)., f(x 0, y 0 ) 0 (mod b) x 0, y 0 Z, g i,j,k (x 0, y 0 ) 0 (mod b m ).,,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
79 - LLL,, 2. n Heuristic [JM06],. Shiho Midorikawa katagaitai workshop winter March 18, / 142
80 - Q. 2,? A. (resultant) Gröbner. Gröbner., 2 f(x, y), g(x, y), x 0, y 0, y h = Res y (f, g) h 1, h(x 0 ) = 0. LLL 1, 2 2.,,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
81 - Q. β A. β Howgrave-Graham, Eq. 3. [May03]. Shiho Midorikawa katagaitai workshop winter March 18, / 142
82 Unravelled Linearization, Thm. 12.,,,., 2009 [HM09]. Shiho Midorikawa katagaitai workshop winter March 18, / 142
83 Unravelled Linearization Boneh-Durfee s Attack.,. x(a + y) (mod e). Shiho Midorikawa katagaitai workshop winter March 18, / 142
84 Unravelled Linearization f(x, y) = xy + xa + 1 mod e m, xy. y, x i y j e m k f k (x),. X, Y X i Y j e m k (XY ) k, X, Y i, j e m X i Y j e m k (XY ) k.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
85 Unravelled Linearization,. Boneh-Durfee[BD99],.,, d < n ,,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
86 Unravelled Linearization Herrmann, May [HM09] Unravelled Linearization,., u = xy + 1. xy + 1 +Ax 0 (mod e) }{{} u u+ax 0 (mod e) Shiho Midorikawa katagaitai workshop winter March 18, / 142
87 Unravelled Linearization f(u, x) = u + Ax,., g i,j,k (u, x, y) = x i y j e m k f k (u, x). u, x y. Shiho Midorikawa katagaitai workshop winter March 18, / 142
88 Unravelled Linearization,., x y., x, y (unhelpful )., xy + 1 u, x, y.,,,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
89 Unravelled Linearization LLL u = xy + 1 x, y. u + LLL Unravelled Linearization., Boneh-Durfee s Attack,. [HM09], Herrmann [Her11], Boneh-Durfee s Attack [HM10]. Shiho Midorikawa katagaitai workshop winter March 18, / 142
90 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy Shiho Midorikawa katagaitai workshop winter March 18, / 142
91 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, RSA,. alice_public_key.py bob_public.key.py communication.py keygen.sage send.bak Shiho Midorikawa katagaitai workshop winter March 18, / 142
92 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy Alice, Bob, (e, N 1, N 2 )., e., d Boneh-Durfee Small Secret Exponent Attack., keygen.sage. Shiho Midorikawa katagaitai workshop winter March 18, / 142
93 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy keygen.sage Shiho Midorikawa katagaitai workshop winter March 18, / 142
94 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy n, n d. nd n. tmp tmp = n/2 n d. Shiho Midorikawa katagaitai workshop winter March 18, / 142
95 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, x 1, x 2 n d, tmp, p 1 = x 1 x , p1 = x 1 x n/2. Shiho Midorikawa katagaitai workshop winter March 18, / 142
96 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, y 2 tmp, p 2 = x 1 y p2 n/2. Shiho Midorikawa katagaitai workshop winter March 18, / 142
97 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy y 1 n d, q 1 = y 1 y q1 n/2. Shiho Midorikawa katagaitai workshop winter March 18, / 142
98 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy,. x1, x 2, y 1, y 2, p 1 = x 1 x 2 + 1, p 2 = x 1 y 2 + 1, q 1 = y 1 y Shiho Midorikawa katagaitai workshop winter March 18, / 142
99 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, p 1, q 1, p 2, n/2. n-bit RSA. N 1 = p 1 q 1, ϕ(n 1 ) = (p 1 1)(q 1 1) = x 1 x 2 y 1 y 2.. Shiho Midorikawa katagaitai workshop winter March 18, / 142
100 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, d, q 2. d x1 x 2 y 1 y 2, ϕ(n 1 ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
101 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, e = d 1 mod (p 1 1)(q 1 1) e, ed 1 (p 1 1)(q 1 1). ed 1 (mod ϕ(n1 )), k 1, ed = 1 + k 1 ϕ(n 1 ), k 1. Shiho Midorikawa katagaitai workshop winter March 18, / 142
102 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, q 2 = x 2 k 1 + 1, q 2. k1 d, q 2 n/2. Shiho Midorikawa katagaitai workshop winter March 18, / 142
103 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy N 2 = p 2 q 2, ϕ(n 2 ) = (p 2 1)(q 2 1) = x 1 x 2 k 1 y 2. ed = 1 + k1 (x 1 x 2 y 1 y 2 ), ed = 1 + k 1 ϕ(n 1 ) = 1 + k 1 (x 1 x 2 y 1 y 2 ) = 1 + y 1 (x 1 x 2 k 1 y 2 ) = 1 + y 1 ϕ(n 2 ), N 1, N 2 e, d RSA. Shiho Midorikawa katagaitai workshop winter March 18, / 142
104 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy,. Definition 16 p 1, q 1, p 2, q 2, N 1 = p 1 q 1, N 2 = p 2 q 2. e, d ed 1 (mod ϕ(n 1 )), ed 1 (mod ϕ(n 2 )), (e, N 1 ), (e, N 2 ) RSA, d RSA., RSA Variant. Shiho Midorikawa katagaitai workshop winter March 18, / 142
105 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy RSA Variant,, Variant. Variant Variant [Hin07]., Authentication & Secrecy Dual RSA Variant. Shiho Midorikawa katagaitai workshop winter March 18, / 142
106 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, genkey 1 N 1, N 2, d. 1024, 342. d = , d N = max(n1, N 2 ) N , d Boneh-Durfee s Attack d < N Shiho Midorikawa katagaitai workshop winter March 18, / 142
107 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, RSA Variant RSA.,., Variant RSA. Dual RSA, d N [PHL + 17]. 2014, Shiho Midorikawa katagaitai workshop winter March 18, / 142
108 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy Theorem 17 ([PHL + 17]) (e, N 1 ), (e, N 2 ) Dual RSA, d., d < N ϵ N 1, N 2., ϵ = ,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
109 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy., N1 = p 1 q 1, N 2 = p 2 q 2 k 1, k 2 2. ed = 1 + k 1 (N 1 (p 1 + q 1 1)) ed = 1 + k 2 (N 2 (p 2 + q 2 1)), N1, d. Shiho Midorikawa katagaitai workshop winter March 18, / 142
110 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy LLL. A, ( ) A 0 M = e N 1 L(M). v = t (d, k 1 ), ( ) Ad Mv = ed k 1 N 1., t = t (Ad, ed k 1 N 1 ) L(M)., t L(M). Shiho Midorikawa katagaitai workshop winter March 18, / 142
111 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy : 2., d., a1, x 1, a 2, x 2 4 (x 1, x 2 ) d = a 1 x 1 + a 2 x 2.. x 1 = 1, x 2 = 0, a 1 = d, a 2 = 0.,., x 1, x 2. 2 [PHL + 17]. Shiho Midorikawa katagaitai workshop winter March 18, / 142
112 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy., d,. d, ( ed ) k1 N 1 = k 1 (p 1 + q 1 1) 1 0 B = e N 1. L(B) ( ) ( ) d d B = k 1 ed k 1 N 1, d. Shiho Midorikawa katagaitai workshop winter March 18, / 142
113 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy LLL,.,. (1, 0)., LLL, x 1, x 2., A (A, 0) LLL, LLL. ( ) A 0. e N 1.,, LLL decoration( ). Shiho Midorikawa katagaitai workshop winter March 18, / 142
114 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy M LLL M = ( λ1 λ 2 )., a1, a 2 Z t = a 1 λ 1 + a 2 λ 2. λ1 = (l 11, l 12 ), λ 2 = (l 21, l 22 ) Ad = a 1 l 11 + a 2 l 21. A d = a1 l 11 A + a 2 l 21 A., d. Shiho Midorikawa katagaitai workshop winter March 18, / 142
115 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, d, N 2. ed = 1 + k2 (N 2 (p 2 + q 2 1))., l 11 = l 11 A, l 21 = l 21 A. ed = 1 + k 2 (N 2 (p 2 + q 2 1)) e(a 1 l 11 + a 2 l 21) = 1 + k 2 (N 2 (p 2 + q 2 1)) ea 1 l 11 = 1 + k 2 (N 2 (p 2 + q 2 1)) ea 2 l 21 x = k 2, y = (p 2 + q 2 1), z = a 2, el 11, x(n 2 + y) el 21z + 1 = ea 1 l 11 0 (mod el 11) Shiho Midorikawa katagaitai workshop winter March 18, / 142
116 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy 3.,., 3.,., u = xy + 1 Unravelled Linearization. f. f(u, x, z) = u + xn 2 el 21z Shiho Midorikawa katagaitai workshop winter March 18, / 142
117 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy,, LLL.. m, t 2, i, j, k, l 2 g i,j,k (u, x, y, z) = x i z j f k (u, x, z)(el 11) m k h j,k,l (u, x, y, z) = y j z k l f l (u, x, z)(el 11) m l., g i,j,k (u, x, y, z) 0 i m k, 0 j m k i, 0 k m, h j,k,l (u, x, y, z) 1 j t, m t k m, 0 l k. Shiho Midorikawa katagaitai workshop winter March 18, / 142
118 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy 3, 1.,, 3 1. Gröbner. Shiho Midorikawa katagaitai workshop winter March 18, / 142
119 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy Gröbner,. [ 08] 6.,.,,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
120 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, LLL Howgrave-Graham s Lemma, Gröbner,., Gröbner, 16. Shiho Midorikawa katagaitai workshop winter March 18, / 142
121 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy. ( ) A 0 Step 1. LLL, e N 1. Step 2. Step 1., a 1, a 2 d = a 1 l 11 + a 2l 21 l 11, l 21. Step 3. Step 2. l 11, l 21 mod (el 11 )m f(u, x, z) g i,j,k (u, x, y, z), h j,k,l (u, x, y, z). Step 4. g i,j,k (u, x, y, z), h j,k,l (u, x, y, z) B. Step 5. LLL B Thm. 12 B. Shiho Midorikawa katagaitai workshop winter March 18, / 142
122 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy Step 6. B, Howgrave-Graham s Lemma (Lem. 7). Step 7. Step 6., Unravelled Linearization u = xy + 1. Step 8. Step 7. Gröbner. Step 9. Step 8. x, y, z,.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
123 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy, N 1, N 2, e d, RSA. CTF,, 8 10., CTF.,.. Shiho Midorikawa katagaitai workshop winter March 18, / 142
124 0CTF 2017 Finals: Authentication&Secrecy 0CTF 2017 Finals: Authentication&Secrecy CTF Crypto,,.,.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
125 Workshop Workshop Shiho Midorikawa katagaitai workshop winter March 18, / 142
126 Workshop Workshop 2018/02/22 Springer.,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
127 Workshop Workshop, Howgrave-Graham s Lemma 10.,... Shiho Midorikawa katagaitai workshop winter March 18, / 142
128 Workshop Workshop, Heuristic.,. <., /,,. Shiho Midorikawa katagaitai workshop winter March 18, / 142
129 Workshop Workshop Python range/xrange. n i m range(n, m+1) (i = m, m 1,..., n) range(m, n-1, -1) Shiho Midorikawa katagaitai workshop winter March 18, / 142
130 Workshop Workshop SageMath Gröbner Tips, Gröbner CPU.., Gröbner. Tips, Singular giac. Shiho Midorikawa katagaitai workshop winter March 18, / 142
131 Workshop Workshop, 20 Gröbner, AWS 128GB 20,., 1GB. 2, giac. SageMath, 2. vs = map(str, f.parent().gens()) f = PolynomialRing(QQ, vs)(f) Shiho Midorikawa katagaitai workshop winter March 18, / 142
132 Workshop Workshop,.,.,., assertion. Shiho Midorikawa katagaitai workshop winter March 18, / 142
133 Workshop Workshop URL. gh-pages/content/ctf/2017/0ctf%20finals/ cr1000-authenticationsecrecy/solve.sage,. Gröbner giac,, giac. Shiho Midorikawa katagaitai workshop winter March 18, / 142
134 Workshop Workshop.. Shiho Midorikawa katagaitai workshop winter March 18, / 142
135 Workshop I [Ale07] [BD99] [Bon99] Alexander May. Using LLL-Reduction for Solving RSA and Factorization Problems: A Survey. LLL+25 Conference in honour of the 25th birthday of the LLL algorithm, Dan Boneh and Glenn Durfee. Cryptanalysis of RSA with Private Key d Less than N 0.292, pp Springer Berlin Heidelberg, Berlin, Heidelberg, Dan Boneh. Twenty years of attacks on the rsa cryptosystem. NOTICES OF THE AMS, Vol. 46, pp , Shiho Midorikawa katagaitai workshop winter March 18, / 142
136 Workshop II [CFPR96] Don Coppersmith, Matthew K. Franklin, Jacques Patarin, and Michael K. Reiter. Low-exponent RSA with related messages. In Advances in Cryptology - EUROCRYPT 96, International Conference on the Theory and Application of Cryptographic Techniques, Saragossa, Spain, May 12-16, 1996, Proceeding, pp. 1 9, [Cop96a] Don Coppersmith. Finding a small root of a bivariate integer equation; factoring with high bits known. In Advances in Cryptology - EUROCRYPT 96, International Conference on the Theory and Application of Cryptographic Techniques, Saragossa, Spain, May 12-16, 1996, Proceeding, pp , Shiho Midorikawa katagaitai workshop winter March 18, / 142
137 Workshop III [Cop96b] Don Coppersmith. Finding a small root of a univariate modular equation. In Advances in Cryptology - EUROCRYPT 96, International Conference on the Theory and Application of Cryptographic Techniques, Saragossa, Spain, May 12-16, 1996, Proceeding, pp , [Her11] [HG97] Mathias Herrmann. Lattice-based Cryptanalysis using Unravelled Linearization. PhD thesis, Nicholas Howgrave-Graham. Finding small roots of univariate modular equations revisited, pp Springer Berlin Heidelberg, Berlin, Heidelberg, Shiho Midorikawa katagaitai workshop winter March 18, / 142
138 Workshop IV [HG01] [Hin07] [HM09] Nick Howgrave-Graham. Approximate integer common divisors. In Joseph H. Silverman, editor, Cryptography and Lattices, pp , Berlin, Heidelberg, Springer Berlin Heidelberg. Hinek, M. Jason. On the Security of Some Variants of RSA. PhD thesis, Mathias Herrmann and Alexander May. Attacking power generators using unravelled linearization: When do we output too much? In Mitsuru Matsui, editor, Advances in Cryptology ASIACRYPT 2009, pp , Berlin, Heidelberg, Springer Berlin Heidelberg. Shiho Midorikawa katagaitai workshop winter March 18, / 142
139 Workshop V [HM10] [JM06] Mathias Herrmann and Alexander May. Maximizing small root bounds by linearization and applications to small secret exponent rsa. In Phong Q. Nguyen and David Pointcheval, editors, Public Key Cryptography PKC 2010, pp , Berlin, Heidelberg, Springer Berlin Heidelberg. Ellen Jochemsz and Alexander May. A strategy for finding roots of multivariate polynomials with new applications in attacking rsa variants. In Xuejia Lai and Kefei Chen, editors, Advances in Cryptology ASIACRYPT 2006, pp , Berlin, Heidelberg, Springer Berlin Heidelberg. Shiho Midorikawa katagaitai workshop winter March 18, / 142
140 Workshop VI [KO08] [LLL82] [May03] Noboru Kunihiro and Kazuo Ohta. RSA. Bulletin of the Japan Society for Industrial and Applied Mathematics, A. K. Lenstra, H. W. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, Vol. 261, No. 4, pp , Dec Alexander May. New RSA Vulnerabilities Using Lattice Reduction Methods Shiho Midorikawa katagaitai workshop winter March 18, / 142
141 Workshop VII [NSS + 17] Matus Nemec, Marek Sys, Petr Svenda, Dusan Klinec, and Vashek Matyas. The Return of Coppersmith s Attack: Practical Factorization of Widely Used RSA Moduli. In Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, CCS 17, pp , New York, NY, USA, ACM. [Pei13] Chris Peikert. Lattices in Cryptography - Lecture 1: Mathematical Background, http: //web.eecs.umich.edu/~cpeikert/lic15/lec01.pdf. Shiho Midorikawa katagaitai workshop winter March 18, / 142
142 Workshop VIII [PHL + 17] Liqiang Peng, Lei Hu, Yao Lu, Jun Xu, and Zhangjie Huang. Cryptanalysis of dual rsa. Designs, Codes and Cryptography, Vol. 83, No. 1, pp. 1 21, Apr [RSA78] Ronald L Rivest, Adi Shamir, and Leonard Adleman. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, Vol. 21, No. 2, pp , [ 08],,,,,,. -., Shiho Midorikawa katagaitai workshop winter March 18, / 142
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