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- Christal Richard
- 6 years ago
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7 ت و ط ي ة مازال عالم أجهزة الهواتف المحمولة يشهد الكثير من التحولات مع قدوم الهواتف الذكية فمنذ مجيي ها أصبحت هذه الهواتف جزءا أساسيا من حياتنا اليومية مما دفع العديد من مزودي الخدمات لتوفير تطبيقات هاتفية من أجل جلب أكبر عدد من الز ي ن. ولكن وعلى الرغم من هذا النجاح الباهر فقد ظل استخدام الهواتف الذكية في مجال التطبيقات الحساسة (كتطبيقات الدفع عن طريق المحمول) محدود الانتشار فمعظم مستخدمي هذه الهواتف يرفضون اي تمان أجهز م على معلوما م الحساسة بسبب ما يصل إلى مسامعهم عن ثغرات أمنية يتم اكتشافها بشكل دوري. الهدف من بحث الدكتوراه هذا هو تعزيز ثقة المستخدمين واتفهم الذكية وذلك تباع طريقتين مختلفتين تتمم الا ولى منهما الثانية ها ن الطريقتان هما الحماية الم بره نة security) (provable وتقنية الا جهزة العتادية primitives) (hardware المخصصة للحماية. تنقسم رسالة الدكتوراه إلى قسمين أساسيين الهدف من القسم الا ول هو إثبات محدودية التقنيات المستخدمة حاليا لحماية الهواتف الذكية لذلك يتناول هذا القسم دراسة تحليلية لنظام ي حماية واسع ي الانتشار النظام الا ول هو مستودع المفاتيح KeyStore) (Android والذي يعمل كدرع واق لمفاتيح التشفير السرية في نظام التشغيل أندرويد أما الثاني فهي مجموعة بروتوكولات التشفير Protocols) (Secure Channel المستخدمة في عالم البطاقات الذكية. تستعير دراستنا التحليلية أطرها من مبادئ الحماية المبره نة والتي قمنا لاعتماد عليها من أجل العثور على ثغرات أمنية نجاحنا في هذا ا ال يبين الدور الهام الذي يمكن أن تلعبه الحماية المبره نة لا نظمة الهواتف الذكية على الرغم من كو ا ت نظر في كثير من الا حيان على أ ا نظرية ومجردة. أما القسم الثاني فهو مخصص لبحث الا نظمة المعقدة والتي لا يمكن التا كد من شمولية أو تمامية نظم حمايتها بطريقة فعالة يبدأ هذا القسم بدراسة تفصيلية للمنهج ثناي ي البيي ة الح وسبية (Dual-execution-environment) يليها تركيز معمق على الحالة الخاصة التي يتم فيها بناء هذا المنهج ستخدام أجهزة عتادية معينة تسمى ARM TrustZone وذلك لا نشاء ما ي طلق عليه لبيي ة الح وسبية الموثوقة Environment).(TEE: Trusted Execution الهدف من هذا التركيز هو التعرف الجيد على هذه البيي ة الحوسبية من أجل إعادة التفكير بتصاميمها الحالية وذلك للتغلب على بعض القيود التي تحد من انتشار هذه التقنية. وفي هذا الا طار قمنا نشاء حل ين جديد ي ن: الا ول هو عبارة عن بيي ة حوسبية موثوقة يمكنها حماية بيا ا الحساسة حتى وإن تم اختراق الجزء الا كبر من نظام حمايتها أما الحل الثاني فهو عبارة عن تعزيز مجال استخدام هذه البيي ة ليشمل ليس فقط توفير الحماية للتطبيقات الحساسة ولكن أيضا مين خصاي ص حماي ية أكثر دقة (مثل الدفاع والعلاج الذاتي) لا نظمة حوسبية معقدة كنواة أنظمة التشغيل في الا جهزة المدمجة.
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11 إلى من ج ل د ت كتبي أ م المدرسة لا ا بذلك عل متني ح ب الع لم واحترام حامله... وإلى من عل مني أ ن كل إسراف قبيح ماعدا الا سراف في طلب العلم لا ن قيمة الا نسان ع ل م ه... إهداء إلى أبي وأمي مع التحية.
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50 {0, 1} X Y X Y XY X X N N N N 1 f y R f(x) y f(x) f x y A A = x A x E : {0, 1} l {0, 1} l E(,.) {0, 1} l E(,.) E (.) l {0, 1} l {0, 1} k E {E : K(k)} K(k) k k E (.) E E 1 (.) E (.) E 1 (.) K(k) E E
51 Ẽ : {0, 1}l {0, 1} l T Ẽ(, T,.) {0, 1}l l Ẽ(, T,.) Ẽ (T,.) Ẽ 1 (T,.) Ẽ (, T ) Ẽ (T 1,.) Ẽ (T 2,.) T 1, T 2 T 1 T 2 + SE = (K, E, D) K E D K k R K(k) E m M M c C C M C c E (m) D c C m M D (E )(m) = m m
52 m m 1 m 2... m n m m i c i 1 E E 1 c i c i 1 m i c 0 [ ] E l [E] = (K, E, D) iv {0, 1} l m {0, 1} ln n N E (iv, m) D (iv, c) m m 1... m n m i = l c 0 iv i = 1...n c i E (c i 1 m i ) c 1... c n c c 1... c n c i = l c 0 iv i = 1...n m i E 1 (c i) c i 1 m 1... m n D (.,.)
53 [ ] E l [E] = (K, E, D) m {0, 1} ln n N [E] = (K, E, D ) E (m) R c 0 {0, 1} l c E (c 0, m) c 0 c D (c) c c 0 c 1... c n c i = l m D (c 0, c 1... c n ) m F : {0, 1} l {0, 1} l F (.) = F (,.) l l K X {0, 1} l i X + i l i 2 l/2 X 2 l l m {0, 1} ln r i F ( + i) c c 1... c n c i m i r i i = 1 n F [ ] F {0, 1} l [F ] = (K, E, D ) m = m 1... m n {0, 1} ln
54 E (m) m m 1... m n m i = l R {0, 1} l/2 c 0 0 l/2 i = 1...n c i m i F (c 0 + i) c 0 c 1... c n D (c) c c 0 c 1... c n c i = l i = 1...n m i c i F (c 0 + i) m 1... m n 0 2 l/2 l F () c 1 = m 1 F () c 2 = m 2 F () c 1 c 2 = m 1 m 2 m 1 m 2 nse = (K, ne, nd) K ne nd K k ne nd ne m M M N N N ne c C C c ne (N, m) N M C nd N N c C m M nd (N, c) nd (N, ne (N, m)) = m N m 1 2
55 [ ] E l [E] = (K, E, D) N N m {0, 1} ln [E] = (K, ne, nd ) ne (N, m) c 0 E (N) c E (c 0, m) c nd (N, c) c 0 E (N) m D (c 0, c) m [ ] E l [E] = (K, E, D) N N m {0, 1} ln [E] = (K, ne, nd ) ne 1, 2 (N, m) nd 1, 2 (N, c) c 0 E 2 (N) c E 1 (c 0, m) c c 0 E 2 (N) m D 1 (c 0, c) m ES = (, )
56 m H(.) (m) = H(( ) H(( ) m)). E l m = m 1... m n m i = l [E] m τ n τ i = E (m i τ i 1 ) i = 1...n τ 0 = l MA = (K, T, V) K T V K k N T m τ V m τ b V (m, T (m)) = 1 m SE = (K e, E, D) MA = (K m, T, V) m c τ c E 1 (m) τ T 2 (m)
57 c m τ m V 2 (m, τ) = 1 m c E 1 (m τ) τ = T 2 (m) c m τ m V 2 (m, τ) = 1 m c τ c E 1 (m) τ T 2 (c) τ V 2 (c, τ) = 1 c
58 A O A O Π A, SM A Π SM [ ]
59 A A p x {0, 1} A(x) p( x ) A k [ ] f p n f(n) < 1/p(n) p f p, N N, n N, (n N f(n) < 1/p(n)). A A Π A Π A A A A Π A Π A Π A Π A Π A A A Π Π
60 A A [ ] F = {F : K(k)} F l l K(k) l l {0, 1} l {0, 1} l b {0, 1} f b (.) b F, (k) R K(k) R R f 0 F f 1 l l b f b(.) b
61 [, F (k) = 0 F, (k) = 1] [ 1 F, (k) = 1] F, F (k) (k) 0 F, (k) F (.) (k) 1 F, (k) f(.) F (.) K(k) f(.) f l l F [, F (k) = F (.) (k) = 1] [ f(.) (k) = 1] (k) F f(.) [ ] F = {F : K(k)} F {0, 1} l K(k) l {0, 1} l F [, F (k) = F (.) (k) = 1] [ f(.) (k) = 1] (k) F (.) K(k) f(.) f l [ ] F = {F : K(k)} F {0, 1} l K(k) F q, F (k), F (k) + q2 /2 l 1
62 E (LR(.,., b)) b {0, 1} E m 0 m 1 c E (m b ) E (LR(.,., b)) (m 0, m 1 ) m 0 = m 1 b {, } [ ] SE = (K, E, D) {, } SE [, SE (k) = E (LR(.,.,0)), O (k) = 1] [ E (LR(.,.,1)), O (k) = 1] (k) O O = D (.) = O = ϵ(.) ϵ(.) =
63 E (LR(.,., b)) D (.) = [ ] SE = (K, E, D) A E (.) k N SE, A (k) R K(k) c A E (.), D (.) D (c) c / SE A (k) = [ (k) = 1] (k) A, SE SE, A [ ] SE = (K, E, D) A SE B C A, SE (k) (k) + 2 B, SE C, SE (k)
64 [ ] SE = (K, E, D) A E (.) k N SE, A (k) R K(k) c A E (.) D (c) c / SE A (k) = [ (k) = 1] (k) A, SE SE, A [ ] SE = (K, E, D) A SE q d B A, SE (k) q d B, SE (k) m x m 1 x [ ] SE = (K, E, D) SE, SE (k) = [D E (.) (k) = 1] [D R(.) (k) = 1] (k)
65 R(.) D D m E k SE c E (m) R(.) c MA = (K, T, V) T (.) m V MA (m, τ) τ m V (m, τ) = 1 τ T (.) m [ ] MA = (K, T, V) F T (.) V (.,.) T (.) k N MA, F (k) R K(k) (m, τ) F T (.), V (.,.) V (m, τ) = 1 (m, τ) / MA F (k) = [ (k) = 1] (k) F, MA MA, F
66 AE[SE, MA] = (K, E AE, D AE) SE = (K e, E, D) MA = (K m, T, V) [ ] SE MA AE[SE, MA] [ ] MA SE AE[SE, MA] [ ] MA SE AE[SE, MA] [ ] SE MA AE[SE, MA]
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73 h m σ = h(m) m σ m σ σ = h(m) m h m σ σ M m c c E (m) c c c m
74 [ ] SE = (K, E, D) h SE E l h l m {0, 1} nl n N [h, SE] = (K, E, D ) E (m) σ h(m) c E (σ M) c D (c) D (c) σ m σ h(m)
75 [ ] h SE = (K, E, D) [h, SE] = (K, E, D ) A [h, SE] A, (k) (k) B SE A B A (m 0, m 1 ) E (LR(.,., b)) B A (m 0, m 1 ) B E (LR(.,.,b)) m 0 h(m 0) m 0 m 1 h(m 1) m 1 c E k (LR(m 0, m 1, b)) c A A b b B A B A A b B A, (k) = B, SE (k) SE B, SE (k) (k)
76 [ ] h [E] = (K, E, D ) E l [h, ] = (K, E, D) A A, (k) = 1 A E (LR(.,., b)) D (.) E l h l n N A A E (LR(.,.,b)), D (.) m 0 m 1 {0, 1} ln m 0 h(m 0) m 0 m 1 h(m 0) m 1 c E (LR(m 0, m 1, b)) c c 0 c 1 c 2 c 3 c c 1 c 2 c 3 x D (c ) b = 0 b = 1 A, (k) = 1 E (LR(.,., b)) c = E (m b = h(m 0) m b ) c E (h(h(m 0 ) m b ) h(m 0 ) m b ) c = c 0 c 1 {}}{ E (c 0 h(h(m 0 ) m b )) c 2 {}}{ E (c 1 h(m 0 )) c 3 {}}{ E (c 2 m b ) c c 0 c 1 c h(m 0 ) m b ) D (c ) m 0 b = 0 h(m 0 ) = h(m 1 ) m
77 [ ] h [E] = (K, E, D ) E l [h, ] = (K, E, D) A A, (k) = 1 A E (LR(.,., b)) D (.) E l h l A m m {0, 1} ln n N A A E (m) m h(m) m c E (m ) c c 0 c 1 c 2... c n+2 c c 1 c 2... c n+2 c A c c E (.) D (.) c c E (h(m) m) D (c ) = m( ). [ ] h [F ] = (K, E, D ) F {0, 1} l [h, ] = (K, E, D) A A, (k) = 1 A E (LR(.,., b)) D (.) A
78 A E (LR(.,.,b)), D m 0 m 1 m 2 {0, 1} l c E (LR(m 0, m 1, b)) c c 0 c 1 c 2 c 1 h(m 2) h(m 0 ) c 1 c 2 m 2 m 0 c 2 c c 0 c 1 c 2 x D k (c ) b = 0 b = 1 c = E (h(m b ) m b ) c c = c 0 c 1 {}}{ h(m b ) F (c 0 + 1) c 2 {}}{ m b F (c 0 + 2) c c 1 c 2 c c = c 0 c 1 {}}{ h(m 2 ) h(m 0 ) h(m b ) F (c 0 + 1) c 2 {}}{ m 2 m 0 m b F (c 0 + 2) b = 0 c c 0 h(m 2 ) F (c 0 + 1) m 2 F (c 0 + 2) c = E (h(m 2 ) m 2 ) D (c ) = m 2 ( ) b = 1 D (c ) = h(m 0 m 1 m 2 ) = h(m 0 ) h(m 1 ) h(m 2 )
79 E []( ) [, [AES]] = (K, E, D)
80 5.0 e e (K, E, D) e = (K e, E e, D e ) m E e (m) = E ((m) m), (.) e
81 e l l = 16 m A A e (m) = E k e (m) A e m m A A (.) (m) m (m) m m (m ) m (.) l c (m ) c c 0 c 1 c 2 c 3 c c c 2 c 3 c c m (m ) = (m) + 2 (.) l (= 16) (.) l c E ((m ) m ) e c E E (.,.) c = c 0 c 1 {}}{ E (c 0 ((m ) m )) c 3 {}}{ E (c 2 ((m) m)) c 2 {}}{ E (c 1 (m ) ) c {}}{ E (c 3, (m) m) D e (c ) m A m 32 (m ) = (m) + 2 A m 36 32
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94 m m E l = (K, E, D) SE = (K, E, D) E (m) = E ( l, m) D (c) = E ( l, c) m c A A A E (LR(.,.,b)) m 0 m 1 {0, 1} l c E (LR(m 0, m 0, b)) c E (LR(m 0, m 1, b)) c c 1 0 A, SE (k) = 1 (m, m) A A c = E (m) A m A m m N m N/2 A A E (.) c m (c) c E (m) = m c c
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97 4 10, 000 ( ) ( ) ( ) ( ) 2 15 = 32, 768 [ ] E l 1[E] = (K 1, ne 1, nd 1) MA = (K m, T, V) m {0, 1} ln n N SE = (K, E, D)
98 E (m) c ne 1 1(, m) + 1 c (c) c τ 1 τ 2 T 2 ( c ) τ 1 τ 2 c τ 1 D (c) c (c ) c τ c (c ) c τ 1 τ 2 T 2 (c ) τ 1 τ τ 1 τ 2 nd 1 1(, c ) + 1 T 2 V 2 c 8
99 255 8 [ ] F = {F : K(k)} F l l K(k) R S R S m {0, 1} l R S (m) l F R S F, F (k) = [ F = 1] [ R S = 1] (k) [ ] F = {F : K(k)} F (.,.) (t, l) l K(k) R {0, 1} t {0, 1} l {0, 1} l F, F (k) = [ F = 1] [ R = 1] (k) [ ] SE = (K, E, D) A E (LR(.,., b)) D (.) A (m 0, m 1 ) E (LR(.,., b)) m 0 m 1 m 0 = m 1 A D (.) i j k N
100 SE, A (k) i 1 j 1 A (m 0, m 1 ) i i + 1 c i E k (LR(m 0, m 1, b)) c i A A c j j + 1 m D k (c) (j > i) (c c i ) = m A A b b D A E (LR(.,., b)) A D SE A A, SE (k) = [ SE, A (k) = 1] [ SE, A (k) = 1] (k) [ ] SE = (K, E, D) A E (.) D (.) E (.) i j k N
101 SE, A (k) i 1 j 1 A m i i + 1 c i E k (m) c i A A c j j + 1 m D k (c) (j > i) (c c i ) = m A A c D (c ) c / 1 0 SE A A, SE (k) = [ SE, A (k) = 1] (k) [ ] B E Ẽ B B E Ẽ B E Ẽ B SE B [B B, SE (k) = E = 1] [BẼ = 1] (k) [ ] SE = (K, E, D) m τ C SE (, m, τ) D τ (c)
102 D (c τ ) m SE C SE (, m, τ) m τ [ ] SE = (K, E, D) B B SE D (.) [ nse ] nse = (K, ne, nd) m nse = (K, ne, nd) SE ne (m) c ne (m) + 1 c SE nd (c) c nd (c) + 1 m [ T C MA ] F : {0, 1} t {0, 1} {0, 1} t T C MA = ( K, T, Ṽ) T C MA T (m) τ 1 τ 2 F (, m) τ 1 τ 2 τ 1 T C MA Ṽ (m, τ) τ 1 τ 2 F (, m) b [τ 1 = τ] τ 1 τ 2 b [ ] nse = (K, ne, nd) (, ) T C MA = ( K, T, Ṽ)
103 m = = ( K, Ẽ, D) Ẽ (m) c ne 1 (m) c (c) τ T 2 (c ) c τ D (c) c c τ c (c ) Ṽ 2(c, τ) 1 nd 1 (c ) T ( c) (t m) T ( c) = F (, c) F T C MA [ (k) ] A, T C MA F : {0, 1} n {0, 1} {0, 1} n T C MA A T C MA q F A, T C MA (k), F (k) + q2 2 n n/2 + [ q] [ q ] F q
104 [ A, (k) ] nse = (K, ne, nd) T C MA = ( K, T, Ṽ) (, ) F : {0, 1} n {0, 1} {0, 1} n T C MA A q B T C MA A, (k) (k) + [ ] MA B, T C m q F [ ] F (m) (T, m) = (T m) F (.,.) T m [ [ 2 ] ] E l [E ] m l m /l m µ µ = m /l m m [ 2 ] = [(m, m )] (µ + µ ) 2 2 l [ [ q ] ] E l [E ] m l m /l m µ µ = m /l Q q Q [ q ] = [(Q)] ( q i=1 µ i) 2 2 l
105 [ ] E n [E ](.) (.,.) (T, m) = [E ](T m) T m (, ) nse = (K, ne, nd) T C MA = ( K, T, Ṽ) A q A, (k), (k) + (k) A, (k) (k) + q2, 2 n n/2 + [ q] + [ ] [ q ] q [ ] m q (m) n N q 2 2 n n/2 + [ q] + [ ] [ q ] [ ] [ q ] [ ] [ q ] [ ] m [ ] [ i 2] m m i m i m µ [ ] q [(m, m i )] i=1 q (µ + µ i ) 2 i=1 2 n q 2 2 n n/2 + [ q] + [ ] n/2 + q2 n/2 + q2 1 2 n/2 + q 2 n + ( q i=1 µ i) 2 2 n + 2 n + ( q i=1 µ i) 2 q i=1 (µ + µ i ) 2 2 n + 3qµ2 + ( q + (2µ ) 2) 2 n + ( 2 n q i=1 µ2 i 2 n q i=1 µ i) 2 + q i=1 µ2 i 2 n
106 [ ] E [E ](.) (.,.) (T, m) = [E ](T m) T m (, ) nse = (K, ne, nd) T C MA = ( K, T, Ṽ) A B nse F T C MA A, (k) B, nse (k) + (k) F, T C MA nse [ ] nse = (K, ne, nd) T C MA = ( K, T, Ṽ) c i m i c i = σ i τ i = ne 1 (m) T 2 (σ i ) ( =, m, i) C (, m, i) > 1 c i (1) c i c i (2) D (c i ) = m m = m (σ i σi). c [ ] = 0 nse = (K, ne, nd) n ne 1 (n, m 1 ) ne 1 (n, m 2 ) m 1 m 2 C nse = (K, ne, nd) nse i σ i σ i σ i = ne 1 (m ) ne 1 (m) = σ i m m C m = m (σ i = σ i). c i c i τ i τ i c
107 [ ] = 0 2 t 2 (t, σ 1 ) 2 (t, σ 2 ) σ 1 σ 2 C T C MA = ( K, T, Ṽ) 2 i τ i τ i τ i = T 2 (σ i ) T 2 (σ i ) = τ i σ i σ i σ i = σ i q µ i ϵ + 1/2 n 2 + (1/2 n )(q 2 + 4qµ 2 + ( q i=1 µ i) 2 + q i=1 µ2 i ) n µ µ = m /n ϵ q n = 128 µ = q µ n q = 2 n/4 µ = 2 n/8 1 2 n/2 + q ( q + (2µ ) 2) 2 n + ( q i=1 µ i) 2 + q i=1 µ2 i 2 n 1 ( 2n/4 2 n/ n/4) ( 2 n/4 2 n/8) n/4 ( 2 n/8) 2 + 2n/2 2 n + 2 n 1 5 2n/2 + 2n/2 2 n + 23n/4 + 2 n/2 2 n 1 2 n/ n/ n/4 2 n 7 2 n/ n/4
108 MA[F ] = (K, T, V) F F m τ T V T (m) τ F (m) τ V (m, τ) F (m) = τ 1 0 T : {0, 1} n {0, 1} n T MA[F ] = (K, T, V) F T oma = (K, T ot, T ov) T of o T (.) l (.) l T C MA2 F : {0, 1} n {0, 1} {0, 1} n T C MA T C MA2 T C MA T C MA2 F (.,.) T C MA T C MA2 F 2 F 2 (m) τ F (, m) τ τ n F : {0, 1} n {0, 1} {0, 1} n T C MA2[F 2] T C MA = ( K, T, Ṽ) T C MA2[F 2] T (.) = n/2 (.) T C MA T of 2 A T C MA A A q A B F 2 T C MA2[F 2]
109 F A B (k) = A, T C MA B, F 2 (k) + 1/2n/2 B, F 2 (k) [] q 2 /2 n, F (k) + [ q ] + [] B O O F 2 R S T C MA T of 2 B A (m) τ T oo(m) τ A A (m, τ) b [τ = T oo(m)] b A A A 1 0 B O A B 1 O F 2 A [A T oo ] = [B O 1] A T C MA A T of 2 A, T C MA (k) = [AT of 2 ] = [A T of 2 ] + ( [A T or S ] [A T or S ] ) = ( [A T of 2 ] [A T or S ] ) + [A T or S ] = ( [B F 2 1] [B R S 1] ) + [A T or S ] ( ) = B, F 2 (k) + [AT or S ]
110 [A T or S ] A T or S R S (M, τ) A τ = T (R S (m)) [A T or S ] = [x R {0, 1} n, T (x) = τ] = 1 2 n/2 ( T (.) = n/2 (.) ) B F 2 F : {0, 1} n {0, 1} {0, 1} n q O O F (.,.) R(.,.) R(.,.) (T, m) n t n O {t} (m) t O(t, m) {t} t B 1 B b b B F (.,.) R(.,.) (T, m) (= (T m)) F (.,.) B B n m 1 = n τ 1 T T B m 2 = n τ 2 m 2 = n τ 2 n τ 3 τ 1 = τ 3 O = F (.,.) ( R(.,.)) τ 1 = E (E ( )) τ 2 = E (E (T )) ( ( τ 3 = E E ) E (E (T )) τ 2 )
111 [ F 1] = [B F 2 1] + [ F ] [ R 1] = [B R S 1] + [ R] [ ] B, F 2 (k) =, F, F [] {}}{ (k) + [ R] [ q] {}}{ [ F ] (k) + [] + [ q ] [ q ] F q {x : x 0 = n, x i = R(x 0,.)} q B q q 1 q(q + 1) [] = 2 n+1 q(q + 1)/2 n+1 q 2 /2 n q = 1 1/2 n { n, y} y R( n,.) [] = [x R {0, 1} n, x = n ] 1/2 n q = m 1 x i i q = m [] = {}}{ [ m 1 ] + [x m ] m(m 1) = 2 n+1 + m 2 n m(m + 1) = 2 n+1 q = m A A i M i C i = σ i τ i A i C i = σ i τ i m i A F T C MA F nse F T 2 Ṽ 2
112 A M F σ (ne 1 (M)) σ T 2 τ C = σ τ A A C = σ τ F τ Ṽ 2 b b F F nd 1 ((σ)) A A C = σ τ F (σ, τ) A q d j A A j i q τ j = τ i i j q j τ j = τ j m j = M j A, (k) (k) + [ ] + [] MA [] = 0 F, T C A q Q j (C j = σ j τ j ) E Q E 1 E τ j / {τ 1,..., τ q } E 2 E τ j {τ 1,..., τ q } E 2,1 E 2 q < j τ j τ j E 2,2 E 2 q j τ j = τ j E 2,2,1 E 2,2 m j = M j E 2,2,2 E 2,2 m j M j Q A A, (k) = [E] [E] = [E 1 E 2,2,2 ] + [E 2,1 ] + [E 2,2,1 ] E 2,1 τ j j A j E 2,2,1 A, (k) = [E 1 E 2,2,2 ] + [] + [] [E 1 E 2,2,2 ] [] [E 1 E 2,2,2 ] E 1 E 2,2,2 j C j = σ j τ j F T C MA (m j, τ j ) T 2
113 E 1 τ j T 2 E 2,2,2 τ j T 2 σ j i ne(.) M i M i ne 1 (M i ) ne 1 (M i) τ j (= τ j) σ j = ne 1 (M j ) σ j σ j σ j ) σ j = ne 1(m j ) M j m j [E 1 E 2,2,2 ] = [Fforges] = (k) F, T C MA [] [] = [ i j τ j = τ i] T 2 ((ne 1 (m j ))) = T 2 ((ne 1 (M i ))) T C MA q T 2 (.) i( j) j T C MA [] F 2 (.,.) q [] = [ ] E 2,2,1 q j C j = σ j τ j C j C j σ j σ j τ j = τ j m j = M j 3 4 [] = 0 nse = (K, ne, nd) n ne 1 (n, m 1 ) ne 1 (n, m 2 ) m 1 m 2 j nse = (K, ne, nd) nse 3 σ j = ne 1(m j ) ne 1 (M j ) = σ j m j M j 4 m j = M j
114
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116
117
118
119
120
121 +
122 + + +
123 + [ ] S N N = 2 N = m m N = m+1 m = 2 M 3 (= m + 1) D D
124 D D D M D m = 4 3 (= m 1)
125 + + +
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128 + 12, 883
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130 + + <
131
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145 <
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158 (m 0, m 1 ) R m c c {0, 1} m 1 c S n S L n 2 R n n X i c i (m i 0, mi 1 ) c i i X (m i 0, mi 1 ) L f : {0, 1} n {0, 1} m (F, e, d) (f) e x {0, 1} n f X X = e(, x) d Y y {0, 1} m y = d(y ) F X Y Y = F (X) f x f (e, F, d) X e y = d(f (X)) f(x) = d(f (X)) (F, X, d) y = d(f (X)) + +
159 f x, x f(x, x ) f f (e, F, d) X = e(x) F d X = e(x ) y = d(f (X, X )) x f(x; x ) x
160
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