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Protecting Obfuscation Against Algebraic Attacks

Protecting Obfuscation Against Algebraic Attacks. Boaz Barak Sanjam Garg Yael Tauman Kalai Omer Paneth Amit Sahai. Program Obfuscation . Obfuscation. Public Key. Virtual Black-Box (VBB). [ Barak- Goldreich - Impagliazzo - Rudich - Sahai - Vadhan -Yang 01].

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Protecting Obfuscation Against Algebraic Attacks

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  1. Protecting Obfuscation Against Algebraic Attacks Boaz Barak SanjamGarg Yael Tauman Kalai Omer Paneth Amit Sahai

  2. Program Obfuscation Obfuscation Public Key

  3. Virtual Black-Box (VBB) [Barak-Goldreich-Impagliazzo-Rudich-Sahai-Vadhan-Yang 01] Algorithm is an obfuscator for a class if: For every PPT adversary there exists a PPT simulator such that for every :

  4. VBB Impossibility [Barak-Goldreich-Impagliazzo-Rudich-Sahai-Vadhan-Yang 01] There exists contrived “unobfuscatable” programs. Code of a program equivalent to Secret Execute on itself Secret

  5. First Candidate Obfuscation [Garg-Gentry-Halevi-Raykova-Sahai-Waters 13] What is the security of the candidate? Assumption: The [GGHRSW13] obfuscator is an Indistingushability Obfuscator. No known attacks except [BGIRSVY01]. Indistinguishability Obfuscation(): For every pair of equivalent circuits :

  6. This Work A variant of the [GGHRSW13] obfuscator is VBB for all circuits in a generic model (underlying algebra is idealized)

  7. Multilinear Maps [Boneh-Silverberg 03, Garg-Gentry-Halevi 13] Encoding of under a set . • iff Idealy: any other operation is hard.

  8. The Generic MM Model Add Multiply ZT ?

  9. Our Result Virtual Black-Box obfuscation in the generic MM model: For . For assuming LWE.

  10. Avoiding VBB Impossibility In the Generic MM Model Code of a program equivalent to Secret Add Mul ZT Execute on itself Secret

  11. Interpretation Secure obfuscation against “algebraic attacks”. Warning:Non-algebraic attacks do exist [BGIRSVY01].

  12. Interpretation II This Work: VBB with Generic MultilinearMaps Multi-Message Semantically-Secure Multilinear Maps [Pass-Seth-Telang 13] for P/Poly (assuming LWE) [Pass-Seth-Telang 13] Virtual gray-box obfuscation for [Bitansky-Canetti-Kalai-P 14].

  13. Previous Works [Canetti-Vaikuntanathan13] [GGHRSW13] VBB from Black-Box Pseudo-Free Groups in the Generic Colored Matrix Model [Brakerski-Rothblum13] This Work in the Generic MM Model VBB in the Generic MM Model [Brakerski-Rothblum13] Assuming BSH

  14. The Construction • Construction for via branching programs • Bootstrap to P/Poly assuming LWE (leveled-FHE with decryption in )

  15. Branching Programs Program: Input:

  16. BP Evaluation Program: or Input: Output:

  17. Obfuscating BP • Randomizing [Kilian 88] • Encoding

  18. Step 1: Randomizing Program: or Input: Output:

  19. Step 1: Randomizing Program: or Input: Output:

  20. Step 2: Encoding Program: Obfuscation includes the encodings:

  21. Proof of Security

  22. Simulation Outline Test every monomial separately: By querying

  23. Problems 1. Inconsistent monomials: 2. Too many monomials:

  24. Changing the Sets

  25. Changing the Sets

  26. Changing the Sets

  27. Straddling Set System -matrices -matrices

  28. Straddling Set System

  29. Straddling Set System

  30. Too Many Monomials

  31. Pairing Level Together

  32. From Two Levels to One

  33. From Two Levels to One

  34. Dual-Input BP Input:

  35. Too Many Monomials

  36. Thank You!

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