TMPA-2013 Keynote: Zakharov Obfuscation

Ìàòåìàòè÷åñêèå
àñïåêòû çàäà÷è
îáôóñêàöèè ïðîãðàìì
Â.À. Çàõàðîâ
ô-ò ÂÌèÊ ÌÃÓ èì. Ì.Â. Ëîìîíîñîâà

ÎÁÔÓÑÊÀÖÈß ÏÐÎÃÐÀÌÌ
ýòî òàêàÿ ðàçíîâèäíîñòü ýêâèâàëåíòíûõ
ïðåîáðàçîâàíèé ïðîãðàìì, êîòîðàÿ ïðåäíàçíà÷åíà
äëÿ çàòðóäíåíèÿ ïîíèìàíèÿ ïðîãðàìì è
èçâëå÷åíèÿ èç íèõ ïîëåçíîé èíôîðìàöèè îá
àëãîðèòìàõ, ñòðóêòóðàõ äàííûõ, ñåêðåòíûõ
êëþ÷àõ, ñîäåðæàùèõñÿ â ïðîãðàììàõ.

ÎÁÔÓÑÊÀÖÈß ÏÐÎÃÐÀÌÌ
ýòî òàêàÿ ðàçíîâèäíîñòü ýêâèâàëåíòíûõ
ïðåîáðàçîâàíèé ïðîãðàìì, êîòîðàÿ ïðåäíàçíà÷åíà
äëÿ çàòðóäíåíèÿ ïîíèìàíèÿ ïðîãðàìì è
èçâëå÷åíèÿ èç íèõ ïîëåçíîé èíôîðìàöèè îá
àëãîðèòìàõ, ñòðóêòóðàõ äàííûõ, ñåêðåòíûõ
êëþ÷àõ, ñîäåðæàùèõñÿ â ïðîãðàììàõ.
Îñíîâíûå çàäà÷è

Êàê ïîñòðîèòü îáôóñêàòîð ïðîãðàìì?
Êàê îöåíèòü ñòîéêîñòü îáôóñêàöèè?

ÄÂÀ ÍÀÏÐÀÂËÅÍÈß ÈÑÑËÅÄÎÂÀÍÈÉ
Îáôóñêàöèÿ äëÿ íóæä êðèïòîãðàôèè
Die W., Hellman M. New directions in cryptography. IEEE
Transactions in Information Theory, 1976.
Îáôóñêàöèÿ ïîçâîëÿåò ïðåîáðàçîâûâàòü êðèïòîñèñòåìû ñ
ñåêðåòíûì êëþ÷îì â êðèïòîñèñòåìû ñ îòêðûòûì êëþ÷îì.
Äëÿ ýòîãî äîñòàòî÷íî ïîäâåðãíóòü îáôóñêàöèè ïðîãðàììó,
ðåàëèçóþùóþ àëãîðèòì øèôðîâàíèÿ ñ âñòàâëåííûì â íåå
ñåêðåòíûì êëþ÷îì. Ïðåîáðàçîâàííóþ òàêèì îáðàçîì
ïðîãðàììó ìîæíî èñïîëüçîâàòü â êà÷åñòâå ïðîãðàììû
øèôðîâàíèÿ êðèïòîñèñòåìû ñ îòêðûòûì êëþ÷îì.

Îáôóñêàöèÿ ïðîãðàìì ïîçâîëÿåò
ïðåâðàùàòü êðèïòîñèòåìû ñ ñåêðåòíûì êëþ÷îì â
êðèïòîñèñòåìû ñ îòêðûòûì êëþ÷îì,
ñòðîèòü ñèñòåìû âû÷èñëåíèé íàä çàøèôðîâàííûìè
äàííûìè (êðèïòîñèñòåìû ãîìîìîðôíûõ âû÷èñëåíèé),
èçáàâèòüñÿ îò ìîäåëè ñëó÷àéíîãî îðàêóëà â
êðèïòîãðàôè÷åñêèõ ïðîòîêîëàõ,
ñîçäàâàòü âåðèôèöèðóåìûå ñèñòåìû òàéíîãî ãîëîñîâàíèÿ,
îáåñïå÷èòü êîíôèäåíöèàëüíîñòü â ïîèñêîâûõ ñèñòåìàõ è
áàçàõ äàííûõ.

Íî äëÿ ýòîãî îáôóñêàöèÿ äîëæíà óäîâëåòâîðÿòü î÷åíü ñòðîãèì
òðåáîâàíèÿì ñòîéêîñòè, ïðèíÿòûì â êðèïòîãðàôèè.

Ñîâðåìåííîå ñîñòîÿíèå äåë â ýòîì íàïðàâëåíèè
èññëåäîâàíèé òàêîâî:
î÷åíü ìíîãî îòðèöàòåëüíûõ ðåçóëüòàòîâ,
è î÷åíü ìàëî ïîëîæèòåëüíûõ äîñòèæåíèé.

Îáôóñêàöèÿ äëÿ îáåñïå÷åíèÿ êîìïüþòåðíîé
áåçîïàñíîñòè

Collberg C., Thomborson C., Low D. A taxonomy of obfuscating
transformations, Tech. Report, N 148, Dept. of Computer Science,
University of Auckland, 1997.
Îáôóñêèðóþùèå ïðåîáðàçîâàíèÿ ìîæíî èñïîëüçîâàòü äëÿ
çàùèòû èíòåëëåêòóàëüíîé ñîáñòâåííîñòè íà ïðîãðàììíîå
îáåñïå÷åíèå,
èíôîðìàöèîííîé çàùèòû ìîáèëüíûõ àãåíòîâ è
ìèêðîýëåêòðîííûõ ñõåì íà ýòàïå ïðîåêòèðîâàíèÿ,
à òàêæå äëÿ


Collberg C., Thomborson C., Low D. A taxonomy of obfuscating
transformations, Tech. Report, N 148, Dept. of Computer Science,
University of Auckland, 1997.
Îáôóñêèðóþùèå ïðåîáðàçîâàíèÿ ìîæíî èñïîëüçîâàòü äëÿ
çàùèòû èíòåëëåêòóàëüíîé ñîáñòâåííîñòè íà ïðîãðàììíîå
îáåñïå÷åíèå,
èíôîðìàöèîííîé çàùèòû ìîáèëüíûõ àãåíòîâ è
ìèêðîýëåêòðîííûõ ñõåì íà ýòàïå ïðîåêòèðîâàíèÿ,
à òàêæå äëÿ
ñîêðûòèÿ èñêóññòâåííûõ óÿçâèìîñòåé â ïðîãðàììàõ ,
ìàñêèðîâêè êîìïüþòåðíûõ ¾âèðóñîâ¿ ,
óäàëåíèÿ ¾âîäÿíûõ çíàêîâ¿ èç ïðîãðàìì .

Öåëü îáôóñêàöèè îêàçàòü ïðîòèâîäåéñòâèå ìåòîäàì
îáðàòíîé èíæåíåðèè è àëãîðèòìàì ñòàòè÷åñêîãî è
äèíàìè÷åñêîãî àíàëèçà ïðîãðàìì.

Ñîâðåìåííîå ñîñòîÿíèå äåë â ýòîì íàïðàâëåíèè
èññëåäîâàíèé òàêîâî:
ìíîãî ¾ýâðèñòè÷åñêèõ¿ ìåòîäîâ îáôóñêàöèè,
è íèêàêèõ îöåíîê èõ ñòîéêîñòè.


C. Wang, ¾A Security Architecture for survivability Mechanisms¿,
PhD thesis, Dep. of Computer Science, University of Virginia, 2000.
G. Wroblewski, ¾General Method of Program Code Obfuscation¿,
PhD thesis, Wroclaw University, 2002.
À.Â. ×åðíîâ, ¾Èññëåäîâàíèå è ðàçðàáîòêà ìåòîäîëîãèè
ìàñêèðîâêè ïðîãðàìì¿, Äèññ. íà ñîèñêàíèå ó÷. ñò. ê.ô.-ì.í,
ÂÌÊ ÌÃÓ, 2003.
Y. T. Kalai, ¾Attacks on the Fiat-Shamir Paradigm and Program
Obfuscation¿, PhD thesis, MIT, 2006


S. Drape, ¾Obfuscation of Abstract Data-Types¿, PhD thesis,
University of Oxford, 2004.
Ä.À. Ùåëêóíîâ, ¾Ðàçðàáîòêà ìåòîäèê çàùèòû ïðîãðàìì îò
àíàëèçà è ìîäèôèêàöèè íà îñíîâå çàïóòûâàíèÿ êîäà è
äàííûõ¿, Äèññ. íà ñîèñêàíèå ó÷. ñò. ê.ò.íàóê, ÌÃÒÓ èì. Í.Ý.
Áàóìàíà, 2009.
Mila Dalla Preda, ¾Code Obfuscation and Malware Detection by
Abstract Interpretation¿, Ph.D. Thesis. Universita degli Studi di
Verona, 2007.


Í.À. Êîíîíîâ, ¾Ñòðóêòóðíàÿ îïòèìèçàöèÿ è îáôóñêàöèÿ
êîìáèíàöèîííûõ öèôðîâûõ ñõåì â áàçèñå ÏËÈÑ/ÑÁÌÊ¿,
Äèññ. íà ñîèñêàíèå ó÷. ñò. ê.ò.í., ÌÈÝÒ, 2011.
J. Cappaert, ¾Code Obfuscation Techniques for Software
Protection¿, PhD thesis, Katholieke Universiteit Leuven, B. Preneel
(promotor), 112+14 pages, 2012.
C. Collberg, J. Nagra. ¾Surreptitious Software: Obfuscation,
Watermarking, and Tamperproong for Program Protection.¿
Addison-Wesley Professional, 2009.

Ðàçðûâ ïðîëåãàåò ìåæäó

ôîðìàëüíîé ïîñòàíîâêîé çàäà÷è îáôóñêàöèè è
ïðèëîæåíèÿìè :
Îáëàñòü ïðèìåíåíèÿ îáôóñêàöèè îáøèðíà, íî ëèøü â
ðåäêèõ ñëó÷àÿõ óäàâàëîñü äîáèòüñÿ ñòðîãîé
ìàòåìàòè÷åñêîé ïîñòàíîâêè çàäà÷è îáôóñêàöèè ñ
ïîäõîäÿùèì îïðåäåëåíèåì ñòîéêîñòè îáôóñêàöèè.


ïîëîæèòåëüíûìè è îòðèöàòåëüíûìè ðåçóëüòàòàìè :
Åñòü ìíîãî ðåçóëüòàòîâ î íåâîçìîæíîñòè ïîñòðîåíèÿ
óíèâåðñàëüíûõ îáôóñêàòîðîâ, íî ìàëî ÷òî èçâåñòíî î
âîçìîæíîñòè ñòîéêîé îáôóñêàöèè äëÿ îòäåëüíûõ
ñïåöèàëüíûõ êëàññîâ ïðîãðàìì.


ïîëîæèòåëüíûìè è îòðèöàòåëüíûìè ðåçóëüòàòàìè :
Åñòü ìíîãî ðåçóëüòàòîâ î íåâîçìîæíîñòè ïîñòðîåíèÿ
óíèâåðñàëüíûõ îáôóñêàòîðîâ, íî ìàëî ÷òî èçâåñòíî î
âîçìîæíîñòè ñòîéêîé îáôóñêàöèè äëÿ îòäåëüíûõ
ñïåöèàëüíûõ êëàññîâ ïðîãðàìì.
òåîðèåé è ïðàêòèêîé îáôóñêàöèè :
Èçâåñòíî ìíîãî ïðàêòè÷åñêèõ ìåòîäîâ îáôóñêàöèè
ïðîãðàìì, îäíàêî, íà íèõ íå îêàçàëè íèêàêîãî âëèÿíèÿ
îñíîâîïîëàãàþùèå ðåçóëüòàòû èç îáëàñòè êðèïòîãðàôèè.

Äàëüíåéøèé ïðîãðåññ
áóäåò âîçìîæåí, åñëè óäàñòñÿ ñáëèçèòü îáà ýòèõ
íàïðàâëåíèÿ èññëåäîâàíèé çà ñ÷åò ñîçäàíèÿ
ñîãëàñîâàííîé ñèñòåìû òðåáîâàíèé ñòîéêîñòè, êîòîðûå
ìîæíî áóäåò ïðèìåíÿòü äëÿ ðàçðàáîòêè ðàçíûõ ìåòîäîâ
îáôóñêàöèè ïðîãðàìì â ðàçíûõ ïðèëîæåíèÿõ.

Äàëüíåéøèé ïðîãðåññ
áóäåò âîçìîæåí, åñëè óäàñòñÿ ñáëèçèòü îáà ýòèõ
íàïðàâëåíèÿ èññëåäîâàíèé çà ñ÷åò ñîçäàíèÿ
ñîãëàñîâàííîé ñèñòåìû òðåáîâàíèé ñòîéêîñòè, êîòîðûå
ìîæíî áóäåò ïðèìåíÿòü äëÿ ðàçðàáîòêè ðàçíûõ ìåòîäîâ
îáôóñêàöèè ïðîãðàìì â ðàçíûõ ïðèëîæåíèÿõ.

Áëàãîäàðÿ ýòîìó ìîæíî áóäåò
ïîíÿòü, êàêèì òðåáîâàíèÿì ñòîéêîñòè äîëæíû
óäîâëåòâîðÿòü òå èëè èíûå ðàçíîâèäíîñòè îáôóñêàöèè
ïðîãðàìì;
îöåíèòü, êàêèìè äîñòîèíñòâàìè è íåäîñòàòêàìè îáëàäàþò
ðàçíûå ìåòîäû îáôóñêàöèè,
ïðèñïîñîáèòü ôîðìàëüíûå ìåòîäû òåîðèè âû÷èñëåíèé è
êðèïòîãðàôèè äëÿ íóæä îáôóñêàöèè ïðîãðàìì.

ÎÁÔÓÑÊÀÖÈß ×ÀÑÒÈ×ÍÎ ÇÀÙÈÙÅÍÍÛÕ
ÏÐÎÃÐÀÌÌ
R. Ostrovsky, Ecient computation on oblivious RAM, Proc. of
22nd ACM Symposium on Theory of Computing (STOC-90)
Çàùèùåííûé ïðîöåññîð P èìååò îòêðûòóþ ïàìÿòü M :
M ⇐⇒ P

ÎÁÔÓÑÊÀÖÈß ×ÀÑÒÈ×ÍÎ ÇÀÙÈÙÅÍÍÛÕ
ÏÐÎÃÐÀÌÌ
R. Ostrovsky, Ecient computation on oblivious RAM, Proc. of
22nd ACM Symposium on Theory of Computing (STOC-90)
Çàùèùåííûé ïðîöåññîð P èìååò îòêðûòóþ ïàìÿòü M :

Òåîðåìà

M ⇐⇒ P

Åñëè ñóùåñòâóþò îäíîñòîðîííèå ôóíêöèè, òî ëþáóþ
ïðîãðàììó π ìîæíî ïðåîáðàçîâàòü â ýêâèâàëåíòíóþ ïðîãðàììó
O(π) òàê, ÷òî:
1. Time(O(π)) = Time(π) × log3(Time(π));
2. Ïðè âûïîëíåíèè O(π) íà âû÷èñëèòåëüíîì óñòðîéñòâå ñ
çàêðûòûì ïðîöåññîðîì P è îòêðûòîé ïàìÿòüþ M íèêàêîé
ïðîòèâíèê, îãðàíè÷åííûé ïîëèíîìèàëüíûì âðåìåíåì, íå
ñïîñîáåí ðàñïîçíàòü ïðîãðàììó O(π) ïî
ïîñëåäîâàòåëüíîñòè åå îáðàùåíèé ê ïàìÿòè.

ÑÒÎÉÊÎÑÒÜ ÎÁÔÓÑÊÀÖÈÈ Â ÌÎÄÅËÈ
ÂÈÐÒÓÀËÜÍÎÃÎ ¾×ÅÐÍÎÃÎ ßÙÈÊÀ¿
[Barak B., Goldreich O., Impagliazzo R., et al., 2001]
Âåðîÿòíîñòíûé àëãîðèòì O íàçûâàåòñÿ îáôóñêàòîðîì,
ñòîéêèì â ìîäåëè ¾÷åðíîãî ÿùèêà¿, åñëè îí

óäîâëåòâîðÿåò ñëåäóþùèì òðåáîâàíèÿì:
1. (ôóíêöèîíàëüíîñòü) äëÿ ëþáîé ìàøèíû Òüþðèíãà M
M ≈ O(M).

2. (ïîëèíîìèàëüíîå çàìåäëåíèå) Ñóùåñòâóåò òàêîé ïîëèíîì
p(·), ÷òî äëÿ ëþáîé ìàøèíû Òüþðèíãà M
size(O(M)) ≤ p(size(M)), time(O(M)) ≤ p(time(M)).
A
S
ν

3. (ñòîéêîñòü) Äëÿ ëþáîé PPT (ïðîòèâíèêà ) ñóùåñòâóåò
PPT (ñèìóëÿòîð ) è ïðåíåáðåæèìî ìàëàÿ ôóíêöèÿ ,
òàêèå ÷òî íåðàâåíñòâî
|Pr{A(O(M)) = 1} − Pr{SM (1size(M) ) = 1}| ≤ ν(size(M))

âûïîëíÿåòñÿ äëÿ ëþáîé ìàøèíû Òüþðèíãà M .

Òåîðåìà [Barak B., Goldreich O.,
Impagliazzo R., et al., 2001]
Îáôóñêàòîðîâ, ñòîéêèõ â ìîäåëè
¾÷åðíîãî ÿùèêà¿, íå ñóùåñòâóåò .

Äîêàçàòåëüñòâî.

Ñóùåñòâóþò òàêèå âû÷èñëèìûå ôóíêöèè, ÷òî ëþáóþ èõ
ïðîãðàììíóþ ðåàëèçàöèþ íåâîçìîæíî îáôóñêèðîâàòü.


Ñóùåñòâóþò òàêèå âû÷èñëèìûå ôóíêöèè, ÷òî ëþáóþ èõ
ïðîãðàììíóþ ðåàëèçàöèþ íåâîçìîæíî îáôóñêèðîâàòü.
β, åñëè x = α ,
Fα,β (x) =
0 â îñòàëüíûõ ñëó÷àÿõ .
Gγ,δ (x) =

1, åñëè x(γ) = δ ,
0 â îñòàëüíûõ ñëó÷àÿõ .

Hα,β,γ,δ (x, y ) =

Fα,β (x),
Gγ,δ (x),

åñëè y = 0 ,
åñëè y = 0.


Ïðåäïîëîæèì, ÷òî π ïðîãðàììà, âû÷èñëÿþùàÿ
ôóíêöèþ Hα,β,γ,δ , è O(π) ýòî îáôóñêàöèÿ ïðîãðàììû π.
Ðàñïîëàãàÿ ïðîãðàììîé O(π), òðåáóåòñÿ âûÿñíèòü, ïðàâäà
ëè, ÷òî α = γ è β = δ.


Åñëè òåêñò ïðîãðàììû O(π) íåäîñòóïåí, òî ýòî ìîæíî
ñäåëàòü òîëüêî ïîëíûì ïåðåáîðîì.


Åñëè òåêñò ïðîãðàììû O(π) íåäîñòóïåí, òî ýòî ìîæíî
ñäåëàòü òîëüêî ïîëíûì ïåðåáîðîì.
Åñëè òåêñò ïðîãðàììû O(π) äîñòóïåí, òî äîñòàòî÷íî
âû÷èñëèòü
O(π)[O(π)[·, 0], 1] .

ÂÈÐÒÓÀËÜÍÎÃÎ ¾ÑÅÐÎÃÎ ßÙÈÊÀ¿
ñòîéêèì â ìîäåëè ¾ñåðîãî ÿùèêà¿, åñëè îí óäîâëåòâîðÿåò
ñëåäóþùèì òðåáîâàíèÿì:
1. (ôóíêöèîíàëüíîñòü)
2. (ïîëèíîìèàëüíîå çàìåäëåíèå)
3. (ñòîéêîñòü) Äëÿ ëþáîé PPT A (ïðîòèâíèêà) ñóùåñòâóåò
PPT S (ñèìóëÿòîð) è ïðåíåáðåæèìî ìàëàÿ ôóíêöèÿ ν ,
|Pr{A(O(M)) = 1} − Pr{STr(M) (1size(M) ) = 1}| ≤ ν(size(M))

âûïîëíÿåòñÿ äëÿ ëþáîé ìàøèíû Òüþðèíãà M .
Îðàêóë Tr(M) â îòâåò íà çàïðîñ x âûäàåò ïàðó (y , trM (x)),
ñîñòîÿùóþ èç
ðåçóëüòàòà âû÷èñëåíèÿ y = M(x)
òðàññû trM (x) âûïîëíåíèÿ ÌÒ M íà âõîäå x .

Ðàññìîòðèì ñåìåéñòâî ðåàãèðóþùèõ ÌÒ (RMT), íà âõîä
êîòîðûõ ïîäàåòñÿ áåñêîíå÷íûé ïîòîê äàííûõ (çàïðîñîâ)
x1 , x2 , . . . , xn , . . . . RMT âû÷èñëÿåò áåñêîíå÷íûé ïîòîê âûõîäíûõ
äàííûõ (îòêëèêîâ) y1, y2, . . . , yn , . . . :
yn = Fn (x1 , x2 , . . . , xn ).

Òåîðåìà[Âàðíîâñêèé Í.Ï., 2002]
Åñëè ñóùåñòâóþò îäíîñòîðîííèå
ôóíêöèè, òî îáôóñêàòîðîâ, ñòîéêèõ â
ìîäåëè âèðòóàëüíîãî ¾ñåðîãî ÿùèêà ¿,
äëÿ ðåàãèðóþùèõ ÌÒ íå ñóùåñòâóåò .

Îòêðûòàÿ ïðîáëåìà
À ñóùåñòâóþò ëè îáôóñêàòîðû,
ñòîéêèå â ìîäåëè âèðòóàëüíîãî
¾ñåðîãî ÿùèêà ¿, äëÿ îáû÷íûõ
ìàøèí Òüþðèíãà?

ÎÁÔÓÑÊÀÖÈß ÄËß ÇÀÙÈÒÛ
ÀËÃÎÐÈÒÌÎÂ
ñòîéêî çàùèùàþùèì àëãîðèòìû, åñëè îí óäîâëåòâîðÿåò
|Pr{A(O(M),N) = 1} − Pr{SM (1size(M) ,N) = 1}| ≤ ν(size(M))

âûïîëíÿåòñÿ äëÿ ëþáîé òàêîé ïàðû ÌÒ (M, N), êîòîðàÿ
óäîâëåòâîðÿåò óñëîâèÿì
M ≈ N,
size(N) = poly (size(M)).

Òåîðåìà
Ñóùåñòâóåò îáôóñêàòîð, ñòîéêî
çàùèùàþùèé àëãîðèòìû ,
ïðåäñòàâëåííûå äåòåðìèíèðîâàííûìè
êîíå÷íûìè àâòîìàòàìè.

Òåîðåìà
Ñóùåñòâóåò îáôóñêàòîð, ñòîéêî
çàùèùàþùèé àëãîðèòìû ,
ïðåäñòàâëåííûå äåòåðìèíèðîâàííûìè
êîíå÷íûìè àâòîìàòàìè.
Îáôóñêàòîð äåòåðìèíèðîâàííûõ êîíå÷íûõ àâòîìàòîâ ýòî
ïðîñòî àëãîðèòì ìèíèìèçàöèè êîíå÷íûõ àâòîìàòîâ.
Ýòî òèïè÷íûé ïðèìåð òðèâèàëüíîé îáôóñêàöèè àëãîðèòìîâ
ïóòåì ýôôåêòèâíîãî ïðèâåäåíèÿ ïðîãðàìì ê åäèíñòâåííîé
íîðìàëüíîé ôîðìå (ñòðîãàÿ íîðìàëèçóåìîñòü).

S. Goldwasser, G. N. Rothblum, On Best Possible Obfuscation,
TCC 2007.
Âåðîÿòíîñòíûé àëãîðèòì O íàçûâàåòñÿ íàèëó÷øèì
âîçìîæíûì îáôóñêàòîðîì, åñëè îí óäîâëåòâîðÿåò
3. (ñòîéêîñòü) Äëÿ ëþáîé PPT L (âûâåäûâàòåëü) ñóùåñòâóåò
òàêàÿ PPT S (ñèìóëÿòîð), ÷òî äëÿ äîñòàòî÷íî áîëüøèõ n
è äëÿ ïðîèçâîëüíîé ïàðû ÌÒ M1, M2, âû÷èñëÿþùèõ îäíó
è òó æå ôóíêöèþ è èìåþùèõ ðàçìåð n, ò. å. M1 ≈ M2,
size(M1 ) = size(M2 ) = n, äâà ðàñïðåäåëåíèÿ âåðîÿòíîñòåé
L(O(M1 )) è S(M2 )
âû÷èñëèòåëüíî íåîòëè÷èìû çà ïîëèíîìèàëüíîå âðåìÿ.

Òåîðåìà [S. Goldwasser, G. N. Rothblum, 2007]
Ñóùåñòâóåò íàèëó÷øèé âîçìîæíûé
îáôóñêàòîð äëÿ OBDD ïîëèíîìèàëüíîãî
ðàçìåðà.

Òåîðåìà [S. Goldwasser, G. N. Rothblum, 2007]
Ñóùåñòâóåò íàèëó÷øèé âîçìîæíûé
îáôóñêàòîð äëÿ OBDD ïîëèíîìèàëüíîãî
ðàçìåðà.
Òåîðåìà
Åñëè äëÿ ñåìåéñòâà 3-CNF ñóùåñòâóåò
íàèëó÷øèé âîçìîæíûé îáôóñêàòîð, òî
Σ
= PSPACE .
poly
2

[Barak B., Goldreich O., Impagliazzo R., et al., 2001]
Âåðîÿòíîñòíûé àëãîðèòì O îáëàäàåò ñâîéñòâîì

íåîòëè÷èìîãî îáôóñêàòîðà, åñëè îí óäîâëåòâîðÿåò

1. (ôóíêöèîíàëüíîñòü) äëÿ ëþáîé ìàøèíû Òüþðèíãà M
M ≈ O(M).

2. (ïîëèíîìèàëüíîå çàìåäëåíèå) Ñóùåñòâóåò òàêîé ïîëèíîì
p(·), ÷òî äëÿ ëþáîé ìàøèíû Òüþðèíãà M
size(O(M)) ≤ p(size(M)), time(O(M)) ≤ p(time(M)).

3. (ñòîéêîñòü) Äëÿ ëþáîé PPT A (ïðîòèâíèêà ) ñóùåñòâóåò
òàêàÿ ïðåíåáðåæèìî ìàëàÿ ôóíêöèÿ ν , ÷òî äëÿ ëþáîé
ïàðû ìàøèí Òüþðèíãà M1, M2, åñëè M1 ∼ M2, òî

|Pr{A(O(M1 )) = 1}−Pr{A(O(M2 )) = 1}| ≤ ν(size(M1 ) + size(M2 )

Îòêðûòûå ïðîáëåìû
Ñóùåñòâóþò ëè êëàññû ïðîãðàìì, äîïóñêàþùèõ
íåòðèâèàëüíóþ ñòîéêóþ îáôóñêàöèþ, çàùèùàþùóþ
àëãîðèòìû ?
Ñóùåñòâóþò ëè ïðîãðàììû, íå èìåþùèå ñòîéêîé
îáôóñêàöèè, çàùèùàþùåé àëãîðèòìû ?
Êàê ñâÿçàíû äðóã ñ äðóãîì îáôóñêàöèÿ, çàùèùàþùàÿ
àëãîðèòìû è íàèëó÷øàÿ âîçìîæíàÿ îáôóñêàöèÿ?

ÎÁÔÓÑÊÀÖÈß, ÑÊÐÛÂÀÞÙÀß
ÊÎÍÑÒÀÍÒÓ
Ïóñòü M ýòî ïðîãðàììà ñ ïàðàìåòðîì (ïåðåìåííîé) x .
Îáîçíà÷èì Mc ïðèìåð ïðîãðàììû M , â êîòîðîé âìåñòî
ïàðàìåòðà x ïîäñòàâëåíà êîíñòàíòà c ∈ {0, 1}n .
ñêðûâàþùèì êîíñòàíòó, äëÿ ïàðàìåòðèçîâàííîãî ñåìåéñòâà
ïðîãðàìì F = {Mc : c ∈ {0, 1}n , n ≥ 1}, åñëè îí
óäîâëåòâîðÿåò ñëåäóþùèì òðåáîâàíèÿì:
|Pr{A[O(Mc0 ), Mc ] = 1} − Pr{SMc0 [1size(Mc0 ) , Mc ] = 1} ≤ ν(n)

âåðíî äëÿ ëþáîé ïàðû êîíñòàíò c0 ∈ {0, 1}n è c ∈R {0, 1}n .

ÎÁÔÓÑÊÀÖÈß, ÑÊÐÛÂÀÞÙÀß
ÊÎÍÑÒÀÍÒÓ
ÃÈÏÎÒÅÇÀ
Ñòîéêàÿ îáôóñêàöèÿ, ñêðûâàþùàÿ
êîíñòàíòó,
íåâîçìîæíà , åñëè M ýòî
óíèâåðñàëüíàÿ ìàøèíà Òüþðèíãà;
âîçìîæíà , åñëè M = E (key (x), m)
ýòî ïðîãðàììà øèôðîâàíèÿ ñòîéêîé
êðèïòîñèñòåìû ñ îòêðûòûì êëþ÷îì
key (x) è ñåêðåòíûì êëþ÷îì x.
x

x

ÎÁÔÓÑÊÀÖÈß ÏÐÅÄÈÊÀÒÎÂ
Âåðîÿòíîñòíûé àëãîðèòì O íàçûâàåòñÿ îáôóñêàòîðîì
ïðåäèêàòà π, çàäàííîãî íà ñåìåéñòâå ìàøèí Òüþðèíãà F ,
åñëè îí óäîâëåòâîðÿåò ñëåäóþùèì òðåáîâàíèÿì:
|Pr{A[O(M)] = π(M)} − Pr{SM [1size(M) ] = π(M)}| ≤ neg(size(M))

âåðíî äëÿ êàæäîé ÌÒ M èç F è åå îáôóñêàöèè O(M).

Òî÷å÷íîé íàçûâàåòñÿ ôóíêöèÿ fa : {0, 1}n → {0, 1}, a ∈ {0, 1}n ,
óäîâëåòâîðÿþùàÿ óñëîâèþ
1, åñëè x = a,
fa (x) =
0, åñëè x = a.
Ðàññìîòðèì ñåìåéñòâî Fn , ñîñòîÿùåå èç òî÷å÷íûõ ôóíêöèé
{fu : u ∈ {0, 1}n } è ôóíêöèè, òîæäåñòâåííî ðàâíîé 0. Íà ýòîì
ñåìåéñòâå îïðåäåëåí ïðåäèêàò P(f ) = (f ≡ 0).

Òåîðåìà [Çàõàðîâ Â.À., Âàðíîâñêèé Í.Ï., 2003]
Åñëè ñóùåñòâóþò îäíîñòîðîííèå ïåðåñòàíîâêè, òî
ïðåäèêàò P , îïðåäåëåííûé íà ñåìåéñòâå ïðîãðàìì,
âû÷èñëÿþùèõ ôóíêöèè ñåìåéñòâà Fn , èìååò ñòîéêóþ
îáôóñêàöèþ.

Äîêàçàòåëüñòâî
Íóæíî ñäåëàòü íåîòëè÷èìûìè äðóã îò äðóãà äâå ïðîãðàììû
prog π0 ;
prog πa ;
var x : string y : bit;
input (x) ;
const a : string;
y = 0; output (y);
input (x) ;
end of prog
if x==a then y=1 else y=0;
output (y);
end of prog

Äîêàçàòåëüñòâî
Íóæíî ñäåëàòü íåîòëè÷èìûìè äðóã îò äðóãà äâå ïðîãðàììû
prog π0 ;
prog πa ;
input (x) ;
const a : string;
y = 0; output (y);
input (x) ;
end of prog
if x==a then y=1 else y=0;
output (y);
end of prog

Íàì ïîíàäîáèòñÿ îäíîñòîðîííÿÿ ïåðåñòàíîâêà ϕ íà ìíîæåñòâå
ñòðîê {0, 1}n è ãåíåðàòîð ñëó÷àéíûõ ñòðîê, êîòîðûé ìîæíî
ïîñòðîèòü íà îñíîâå îäíîñòðîííåé ïåðåñòàíîâêè.


Äëÿ ïðîãðàììû π0 : 1) âûáðàòün äâå ñëó÷àéíûå ñòðîêè w , u,
2) âû÷èñëèòü v = ϕ(w ) è σ = wi ui mod 2.
i=1


i=1
Äëÿ ïðîãðàììû πa : 1) âûáðàòü ñëó÷àéíóþ ñòðîêó u,
n
2) âû÷èñëèòü v = ϕ(a) è σ = 1 + ai ui mod 2.
i=1


i=1
Äëÿ ïðîãðàììû πa : 1) âûáðàòü ñëó÷àéíóþ ñòðîêó u,
n
2) âû÷èñëèòü v = ϕ(a) è σ = 1 + ai ui mod 2.
i=1
Òîãäà êàæäàÿ èç ïðîãðàìì π0 , πa , ãäå a ∈ {0, 1}n ïðèìåò âèä:
prog O(π);
const u,v : string, σ : bit;
input (x) ;
if ϕ(x)==v then
n
if σ == xi ∗ ui mod 2 then y=0 else y=1
i=1
else y=0;
output (y);

end of prog

Òåîðåìà

Ïóñòü O1, O2 îáôóñêàòîðû ôóíêöèîíàëüíûõ ñâîéñòâ
π1 , π2 ñîîòâåòñòâåííî, è ïðè ýòîì îáëàñòü çíà÷åíèé
îáôóñêàòîðà O2 ñîäåðæèòñÿ â îáëàñòè îïðåäåëåíèÿ
îáôóñêàòîðà O1.
Òîãäà êîìïîçèöèÿ O = O1O2 ÿâëÿåòñÿ îáôóñêàòîðîì
îáîèõ ïðåäèêàòîâ π1 è π2.

ÇÀÊËÞ×ÅÍÈÅ
Íóæíî ïðîäîëæàòü ýòîò ñïèñîê îïðåäåëåíèé,
ôîðìóëèðóÿ âñå áîëåå è áîëåå ñëàáûå òðåáîâàíèÿ
ñòîéêîñòè, ïðèãîäíûå äëÿ ðåøåíèÿ äðóãèõ
ïðèëîæåíèé îáôóñêàöèè.

ÇÀÊËÞ×ÅÍÈÅ
Íóæíî ïðîäîëæàòü ýòîò ñïèñîê îïðåäåëåíèé,
ôîðìóëèðóÿ âñå áîëåå è áîëåå ñëàáûå òðåáîâàíèÿ
ñòîéêîñòè, ïðèãîäíûå äëÿ ðåøåíèÿ äðóãèõ
ïðèëîæåíèé îáôóñêàöèè.
Íóæíî àêòèâíåå ïðèâëåêàòü äëÿ îáôóñêàöèè
äîñòèæåíèÿ êðèïòîãðàôèè è òåîðèè ñëîæíîñòè
ñèñòåìû ãîìîìîðôíîãî øèôðîâàíèÿ,
òðóäíîðåøàåìûå çàäà÷è.

ÄÎÑÒÈÆÅÍÈß ÏÎÑËÅÄÍÈÕ ËÅÒ
Â èþëå 2013 ã. áûëà îïóáëèêîâàíà ñòàòüÿ

Candidate Indistinguishability Obfuscation and Functional
Encryption for All Circuits
S. Garg, C. Gentry, S. Halevi, M. Raykova, A. Sahai, B. Waters
â êîòîðîé àâòîðû ïîêàçàëè, ÷òî âåðíà

Òåîðåìà [S. Carg, C. Gentry, et al, 2013]
Íåîòëè÷èìàÿ îáôóñêàöèÿ âîçìîæíà äëÿ
ïðîèçâîëüíûõ ïðîãðàìì
(ïðè íåêîòîðûõ ïðåäïîëîæåíèÿõ î òðóäíîñòè
ðåøåíèÿ çàäà÷ òåîðèè ãðóïï)

ÄÎÑÒÈÆÅÍÈß ÏÎÑËÅÄÍÈÕ ËÅÒ
30 ñåíòÿáðÿ 2013 ã. áûëà îïóáëèêîâàíà ñòàòüÿ

Virtual Black-Box Obfuscation for All Circuits via Generic
Graded Encoding.

Zvika Brakerski, Guy N. Rothblum
We present a new general-purpose obfuscator for all polynomial-size
circuits. The obfuscator uses graded encoding schemes, a
generalization of multilinear maps. We prove that the obfuscator
exposes no more information than the program's black-box
functionality, and achieves virtual black-box security, in the generic
graded encoded scheme model.

Áëàãîäàðþ çà
âíèìàíèå
Âàøè âîïðîñû?

TMPA-2013 Keynote: Zakharov Obfuscation

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TMPA-2013 Keynote: Zakharov Obfuscation