Лекция 10 - Схемы разделения секрета. Жизненный цикл ключей

Ëåêöèÿ 10 Ñõåìû ðàçäåëåíèÿ ñåêðåòà. Æèçíåííûé öèêë êëþ÷åé

Ëåêöèÿ 10

Ñõåìû ðàçäåëåíèÿ ñåêðåòà.

Æèçíåííûé öèêë êëþ÷åé

Ìèõàèë Ëåîíèäîâè÷ Áóðÿêîâ

2012 ãîä


Ñõåìû ðàçäåëåíèÿ ñåêðåòà

Äîëÿ ñåêðåòà ó êàæäîãî ó÷àñòíèêà èíôîðìàöèîííîãî

âçàèìîäåéñòâèÿ ïîëó÷èòü ñåêðåò ìîæíî òîëüêî îáúåäèíèâøèñü.

Íàçíà÷åíèå:

I çàùèòà êëþ÷à îò ïîòåðè

I ðàçäåëåíèå îòâåòñòâåííîñòè � êîëëåãèàëüíûå ñîãëàøåíèÿ,

ñèñòåìû îðóæèÿ, êîðïîðàòèâíûå ðåøåíèÿ


























Ïîðîãîâûå (n, t)-ñõåìû ðàçäåëåíèÿ ñåêðåòà

n � êîëè÷åñòâî ó÷àñòíèêîâ

t � ìèíèìàëüíîå êîëè÷åñòâî ó÷àñòíèêîâ, íåîáõîäèìîå äëÿ

âîññòàíîâëåíèÿ ñåêðåòà

Âåêòîðíàÿ ñõåìà (t = n)

I s = (s1, . . . , sm) � äâîè÷íûé âåêòîð

I s1, s2, . . . , st � äîëè ñåêðåòà

I s = s1 + s2 + . . .+ st � ñåêðåò

Ïðîñòîå äåëåíèå s íà t ÷àñòåé ïîçâîëÿåò óçíàòü ÷àñòü ñåêðåòà.









I s = s1 + s2 + . . .+ st � ñåêðåò










I s = s1 + s2 + . . .+ st � ñåêðåò










I s = s1 + s2 + . . .+ st � ñåêðåò










I s = s1 + s2 + . . .+ st � ñåêðåò










I s = s1 + s2 + . . .+ st � ñåêðåò



Ñõåìà ðàçäåëåíèå ñåêðåòà ØàìèðàÈäåÿ: ×åðåç äâå òî÷êè ìîæíî ïðîâåñòè íåîãðàíè÷åííîå ÷èñëî

ïîëèíîìîâ ñòåïåíè 2. ×òîáû âûáðàòü èç íèõ åäèíñòâåííûé �

íóæíà òðåòüÿ òî÷êà.


Ñõåìà ðàçäåëåíèå ñåêðåòà Øàìèðà

p > M � ïðîñòîå ÷èñëî, îáùåèçâåñòíî

Çàäàäèì ïðîèçâîëüíûé ìíîãî÷ëåí ñòåïåíè k − 1

F (x) = (ak−1xk−1 + ak−2x

k−2 + . . .+ a1x +M) mod p,

M � ðàçäåëÿåìûé ñåêðåò

a1, a2, . . . , ak−1 � ñëó÷àéíûå ÷èñëà, íåèçâåñòíû



p > M � ïðîñòîå ÷èñëî, îáùåèçâåñòíî

Çàäàäèì ïðîèçâîëüíûé ìíîãî÷ëåí ñòåïåíè k − 1

F (x) = (ak−1xk−1 + ak−2x

k−2 + . . .+ a1x +M) mod p,

M � ðàçäåëÿåìûé ñåêðåò

a1, a2, . . . , ak−1 � ñëó÷àéíûå ÷èñëà, íåèçâåñòíû



Òåïåðü âû÷èñëÿåì êîîðäèíàòû n òî÷åê:

k1 = F (1) = (ak−1 · 1k−1 + ak−2 · 1k−2 + . . .+ a1 · 1+M) mod p

k2 = F (2) = (ak−1 · 2k−1 + ak−2 · 2k−2 + . . .+ a1 · 2+M) mod p

. . .

kn = F (n) = (ak−1 · nk−1 + ak−2 · nk−2 + . . .+ a1 · n+M) mod p

(i , ki , p, k − 1) ðàçäàåì ó÷àñòíèêàì ñõåìû



Òåïåðü âû÷èñëÿåì êîîðäèíàòû n òî÷åê:

k1 = F (1) = (ak−1 · 1k−1 + ak−2 · 1k−2 + . . .+ a1 · 1+M) mod p

k2 = F (2) = (ak−1 · 2k−1 + ak−2 · 2k−2 + . . .+ a1 · 2+M) mod p

. . .

kn = F (n) = (ak−1 · nk−1 + ak−2 · nk−2 + . . .+ a1 · n+M) mod p

(i , ki , p, k − 1) ðàçäàåì ó÷àñòíèêàì ñõåìû



Âîññòàíîâëåíèå êîýôôèöèåíòîâ

(èíòåðïîëÿöèîííûé ìíîãî÷ëåí Ëàãðàíæà):

F (x) =∑i

li (x)yi mod p

li (x) =∏i 6=j

x − xjxi − xj

mod p

(xi , yi ) � êîîðäèíàòû òî÷åê ìíîãî÷ëåíà


Ñõåìà ðàçäåëåíèå ñåêðåòà Øàìèðà. ÏðèìåðÃåíåðàöèÿ çíà÷åíèé

1. Ïóñòü M = 11, n = 5, t = 3.

2. Âîçüìåì p = 13. Ïîñòðîèì ìíîãî÷ëåí ñòåïåíè k − 1 = 2:

F (x) = (7x2 + 8x + 11) mod 13

3.

k1 = F (1) = (7 · 12 + 8 · 1 + 11) mod 13 = 0

k2 = F (2) = (7 · 22 + 8 · 2 + 11) mod 13 = 3

k3 = F (3) = (7 · 32 + 8 · 3 + 11) mod 13 = 7

k4 = F (4) = (7 · 42 + 8 · 4 + 11) mod 13 = 12

k5 = F (5) = (7 · 52 + 8 · 5 + 11) mod 13 = 5

4. Ðàñïðåäåëåíèå (i , ki , 13, 2) ïî ó÷àñòíèêàì



1. Ïóñòü M = 11, n = 5, t = 3.


F (x) = (7x2 + 8x + 11) mod 13

3.

k1 = F (1) = (7 · 12 + 8 · 1 + 11) mod 13 = 0

k2 = F (2) = (7 · 22 + 8 · 2 + 11) mod 13 = 3

k3 = F (3) = (7 · 32 + 8 · 3 + 11) mod 13 = 7

k4 = F (4) = (7 · 42 + 8 · 4 + 11) mod 13 = 12

k5 = F (5) = (7 · 52 + 8 · 5 + 11) mod 13 = 5




1. Ïóñòü M = 11, n = 5, t = 3.


F (x) = (7x2 + 8x + 11) mod 13

3.

k1 = F (1) = (7 · 12 + 8 · 1 + 11) mod 13 = 0

k2 = F (2) = (7 · 22 + 8 · 2 + 11) mod 13 = 3

k3 = F (3) = (7 · 32 + 8 · 3 + 11) mod 13 = 7

k4 = F (4) = (7 · 42 + 8 · 4 + 11) mod 13 = 12

k5 = F (5) = (7 · 52 + 8 · 5 + 11) mod 13 = 5




1. Ïóñòü M = 11, n = 5, t = 3.


F (x) = (7x2 + 8x + 11) mod 13

3.

k1 = F (1) = (7 · 12 + 8 · 1 + 11) mod 13 = 0

k2 = F (2) = (7 · 22 + 8 · 2 + 11) mod 13 = 3

k3 = F (3) = (7 · 32 + 8 · 3 + 11) mod 13 = 7

k4 = F (4) = (7 · 42 + 8 · 4 + 11) mod 13 = 12

k5 = F (5) = (7 · 52 + 8 · 5 + 11) mod 13 = 5



Ñõåìà ðàçäåëåíèå ñåêðåòà Øàìèðà. Ïðèìåð

Âîññòàíîâèì ìíîãî÷ëåí ïî k2,k3,k5

1. Çàïèñûâàåì ñèñòåìó èç òðåõ óðàâíåíèé:

(a2 · 22 + a1 · 2 +M) mod 13 = 3

(a2 · 32 + a1 · 3 +M) mod 13 = 7

(a2 · 52 + a1 · 5 +M) mod 13 = 5




1. Çàïèñûâàåì ñèñòåìó èç òðåõ óðàâíåíèé:

(a2 · 22 + a1 · 2 +M) mod 13 = 3

(a2 · 32 + a1 · 3 +M) mod 13 = 7

(a2 · 52 + a1 · 5 +M) mod 13 = 5




2. Ïîñòðîèì èíòåðïîëÿöèîííûé ìíîãî÷ëåí Ëàãðàíæà:

l1(x) =x − x2x1 − x2

· x − x3x1 − x3

= 9x2 + 6x + 5 mod 13

l2(x) =x − x1x2 − x1

· x − x3x2 − x3

= 6x2 + 10x + 8 mod 13

l3(x) =x − x1x3 − x1

· x − x2x3 − x2

= 11x2 + 10x + 1 mod 13




2. Ïîñòðîèì èíòåðïîëÿöèîííûé ìíîãî÷ëåí Ëàãðàíæà:

F (x) = 3 · l1(x) + 7 · l2(x) + 5 · l3(x) mod p

a2 = 3 · 9 + 7 · 6 + 5 · 11 = 7 mod 13

a1 = 3 · 6 + 7 · 10 + 5 · 10 = 8 mod 13

M = 3 · 5 + 7 · 8 + 5 · 1 = 11 mod 13

Îòñþäà F (x) = (7x2 + 8x + 11) mod 13 è M = 11


Ñõåìà Áëýêëè

I Ðàçäåëÿåìûé ñåêðåò � êîîðäèíàòû òî÷êè â t-ìåðíîìïðîñòðàíñòâå.

I Äîëè ñåêðåòà � óðàâíåíèÿ t − 1-ìåðíûõ ãèïåðïëîñêîñòåé.

I Äëÿ âîññòàíîâëåíèÿ íåîáõîäèìî çíàòü t óðàíåíèéãèïåðïëîñêîñòåé.

Ìåíåå ýôôåêòèâíà, ÷åì ñõåìà Øàìèðà:

I â ñõåìå Øàìèðà êàæäàÿ äîëÿ òàêîãî æå ðàçìåðà êàê è

ñåêðåò;

I â ñõåìå Áëýêëè êàæäàÿ äîëÿ â t ðàç áîëüøå.


Ñõåìà Áëýêëè â òðåõìåðíîì ñëó÷àå


Ñõåìû, îñíîâàííûå íà ÊÒÎ. Ñõåìà Ìèíüîòòà

Ïóñòü ïîñëåäîâàòåëüíîñòü d1 < d2 < . . . < dn, di ∈ N îáëàäàåò

ñâîéñòâàìè:

I ∀i 6= j : gcd(di , dj) = 1

I d1 · d2 · . . . · dt > dn−t+2 · . . . · dnÑåêðåò M � d1 · d2 · . . . · dt > M > dn−t+2 · . . . · dn.Äîëè ñåêðåòà � îñòàòêè îò äåëåíèÿ M íà d1, d2, . . . , dn


Ñõåìû, îñíîâàííûå íà ÊÒÎ. Ñõåìà Àñìóòà�Áëóìà

M � ñåêðåò

p � ïðîñòîå ÷èñëî, áîëüøåå M

d1, d2, . . . , dn � âçàèìíî ïðîñòûå, òàêèå, ÷òî:

I di > p

I di+1 > diI d1 · d2 · . . . · dt > p · dn−t+2 · . . . · dn

r � ñëó÷àéíîå

M ′ = M + rpÄîëè ñåêðåòà � {p, di , ki}, ãäå ki = M ′ mod di


Ñõåìû, îñíîâàííûå íà ðåøåíèè ñèñòåì óðàâíåíèé.

Ñõåìà Êàðíèíà � Ãðèíà � Õåëëìàíà

I Çàäàí n + 1 âåêòîð ~v0, ~v1, . . . , ~vn ðàçìåðíîñòè m

I Ðàíã ëþáîé ìàòðèöû, ñîñòàâëåííîé èç m âåêòîðîâ, ðàâåí

m

I Âåêòîð ~v0 èçâåñòåí âñåì ó÷àñòíèêàì.

I Ñåêðåò � ïðîèçâåäåíèå 〈~u, ~v0〉I Äîëè ñåêðåòà � ñêàëÿðíûå ïðîèçâåäåíèÿ 〈~u, ~vi 〉 è âåêòîðû

~viI Äëÿ âîññòàíîâëåíèÿ ñåêðåòà ïî èçâåñòíûì äîëÿì (è

íàáîðó âåêòîðîâ ~v1, ~v2, . . . , ~vn) ðåøàåòñÿ ñèñòåìà èç móðàâíåíèé äëÿ íàõîæäåíèÿ âåêòîðà .


Ñõåìû ïðåäâàðèòåëüíîãî ðàñïðåäåëåíèÿ êëþ÷åé

I Åñëè n ó÷àñòíèêîâ èíôîðìàöèîííîãî îáìåíà ⇒ ∼ n2

êëþ÷åé (ìíîãî õðàíèòü)

I Ðàñïðåäåëåíèå íåêîòîðûõ âñïîìîãàòåëüíîãî êëþ÷åâîãî

ìàòåðèàëà ⇒ êàæäûé ó÷àñòíèê ñàì ãåíåðèðóåò êëþ÷è

I Ñõåìà Áëîìà


Ñõåìà Áëîìà

Ïðåäâîðèòåëüíûé ýòàï

I Ñèììåòðè÷åñêàÿ ìàòðèöà D ∈ F k×k � ëèáî ñåêðåòíà (åñëè

ïðåäïîëàãàåòñÿ äîáàâëåíèå ó÷àñòíèêîâ), ëèáî çàáûâàåòñÿ.

I IA � ñëó÷àéíûé âåêòîð äëèíû k � îòêðûòûé ¾êëþ÷¿ AgA = DIA � çàêðûòûé ¾êëþ÷¿ A

Îáìåí îòêðûòûìè êëþ÷àìè ìåæäó A è B

1. SA = gTA IB = (gT

A IB)T = (ITA DT IB)T = ITB DIA

2. SB = gTB IA = (DIB)

T IA = ITB DT IA = ITB DIA

3. SA = SB


Æèçíåííûé öèêë èñïîëüçîâàíèÿ êëþ÷åé

1. Ðåãèñòðàöèÿ ïîëüçîâàòåëÿ

2. Èíèöèàëèçàöèÿ (óñòàíîâêà ÏÀ ñðåäñòâ)

3. Ãåíåðàöèÿ êëþ÷åé

4. Óñòàíîâêà êëþ÷åé

5. Ðåãèñòðàöèÿ êëþ÷åé

6. Îáû÷íûé ðåæèì ðàáîòû

7. Õðàíåíèå êëþ÷åé

8. Çàìåíà êëþ÷åé

9. Àðõèâèðîâàíèå êëþ÷åé

10. Óíè÷òîæåíèå êëþ÷åé

11. Âîññòàíîâëåíèå êëþ÷åé

12. Îòìåíà êëþ÷åé










































































































































































Óñëóãè òðåòüåé ñòîðîíû

1. Ñåðâåð èì¼í àáîíåíòîâ

2. Ðåãèñòðàöèîííûé ñåðâåð

3. Ïðîèçâîäñòâî êëþ÷åé

4. Ñåðâåð õðàíåíèÿ êëþ÷åé

5. Öåíòð óïðàâëåíèÿ êëþ÷àìè

6. Ñåðòèôèêàöèîííûé ñåðâåð

7. Öåíòð óñòàíîâêè âðåìåííûõ ìåòîê

8. Öåíòð íîòàðèçàöèè

















































































Documents

Лекция 10 - Схемы разделения секрета. Жизненный цикл ключей