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OBFUSCATOR 2. Resulting code does not “leak info” about P’s implementation in eyes of computationally bounded distinguishers P Q = O( ) P
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Introduction to Obfuscation
Mohammad MahmoodyUniversity of Virginia
*some slides borrowed from abhi shelat
Code Obfuscation • A program’s code can reveal how the program works.
That might reveal secrets planted in the program.
• Obfuscation: the task of making programs ‘unintelligible’ while preserving their functionality.
[Hada00][BGIRSVY01]
OBFUSCATOR
2. Resulting code does not “leak info” about P’s implementation in eyes of computationally bounded
distinguishers
1. Preserve Program’s Functionality
P Q = O( )P
Why do we care?• Software Protection: hiding the exact technologies used
• Software Patching
• Making private-key schemes public-key!
• Getting secure computation protocols from OWFs
• ..
Typical Solutions are : “best-effort”
Variable renaming, anti-debugger provisions,nonsense instructions, encrypted code segments,ROT-13 encoding of strings and names...
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Find obfuscation schemes with formal security definitions which rely on formal assumptions instead of human ones.
Goal:
Defining Obfuscation: What is the Ideal? • Like P in a black-box!
OBFUSCATORP P
Input to P
Output
Defining Obfuscation: 1st Try• Whatever one can do with O(P) could be done with
P
Input to P
Output
vsO( )P
P
• Not possible: consider the goal of finding a code that does compute
Defining Obfuscation: 2st Try• Whatever one can do with O(P) could be `simulated’ using
• For every algorithm there is a simulator such that output of A(O(P)) is indistinguishable from output of
P
Pvs
O( )PA( ) SDistinguisher
Just one bit Just one bit
Virtual Black-Box Obfuscation• O(.) is a randomized algorithm
• O(P) computes same function as P does for all P
• |O(P)| |P|
• Time(O(P)(x)) poly(Time(P(x)))
• For every poly-time adv A there is a simul S and negligible :
Now that we have a good definition, lets design some secure obfuscation method!
Celebrated results show impossibility in general for VBB
[HADA00][BGIRSVY01]
[GK05]
[HMS07]
[WEE05]
Some programsnecessarily leaksecret information about how they work
Reason behind Impossibility
versus
black-box access
unbeatable advantage
for Q=O(P)Run Q(Q) 𝑆𝑃=𝑆𝑄
Proof Sketch (reminiscent of halting problem)• Consider that has random numbers planted in it
• Given input x, always outputs 0 unless:• If then output • If x is a program that on outputs then output the secret
• Easy to see: given any obfuscated code we can extract by running Q over Q. This is not possible through oracle access
Possible for point functions
If (x == input) {Output 1}else {Ouptut 0}
[C97, CMR98, LPS04, DS05, W05]
Hope: a “weaker” meaningful definition exists• [BGIRSVY01] also introduced anther obfuscation notion called “indistinguishability obfuscation” IO in their appendix!
•Spoiler: IO is the current champion.
Indistinguishability obfuscation implied by VBB• Suppose P and R are two programs computing same function
• IO: For all equivalent circuits of same size : is indistinguishable from
PO( )PA( ) SJust one bit
Just one bit
RSJust one bit
O( )RA( )Just one bit
≈ ≈ ≈
Why is IO considered big thing?• “Candidate Indistinguishability Obfuscation…” by
Garg, Gentry, Halevi, Raykova, Sahai Waters, in 2013proposed the first scheme conjectured to be a secure IO
• Sahai-Waters (2013) showed how to get “main” crypto primitives assuming IO and “”
• Since then, tens of paper getting cool stuff using IO alone!
Can we rule out IO like we did for VBB?• Answer: sort of … no
• Theorem: If then perfectly secure IO exists!
• Interpretation: We cannot just use IO, we need to assume “ too
• Proof sketch: polynomial hierarchy collapses we can efficiently find smallest circuit Q that computes P outputting Q is a perfect IO obfuscation!
Power of IO:IO + one-way functions• Suppose O is a secure IO and using random bits for circuits of size • Let be a trivial circuit of size that outputs all the time!• Theorem: for is one way unless “P = NP”• Proof sketch:
• Interpretation: From now on we assume IO+OWFs but it is just same as assuming IO + ( )
Recap
• VBB is the stronger type of obfuscation, but it cannot exist for all circuits
• VBB could be achieved (probably) for limited class of functions though
• IO is the weaker type of obfuscation, and it seems it probably exists
• Lot of interesting things can be done by only using IO + BPP != NP
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