Protecting Obfuscation Against Algebraic Attacks

    EUROCRYPT, pp. 221-238, 2013.

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    Keywords:
    microsoft researchnc1 circuitVirtual Black Boxmultilinear mapnsf grantMore(6+)
    Wei bo:
    Work done while the author was an intern at Microsoft Research New England

    Abstract:

    Recently, Garg, Gentry, Halevi, Raykova, Sahai, and Waters (FOCS 2013) constructed a general-purpose obfuscating compiler for NC1 circuits. We describe a simplified variant of this compiler, and prove that it is a virtual black box obfuscator in a generic multilinear map model. This improves on Brakerski and Rothblum (eprint 2013) who gav...More

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    Introduction
    • The goal of general-purpose program obfuscation is to make an arbitrary computer program “unintelligible” while preserving its functionality.
    • The work of Goyal et al [10] shows that there exists an oracle that can be implemented with trusted hardware of size that is only a fixed polynomial in the security parameter, with respect to which virtual black-box obfuscation is possible.
    • Once again, the focus of the paper is to consider oracles that abstract the natural algebraic functionality underlying actual plain-model candidates for general-purpose obfuscation.
    Highlights
    • The goal of general-purpose program obfuscation is to make an arbitrary computer program “unintelligible” while preserving its functionality
    • Work done while the author was an intern at Microsoft Research New England
    • Work done in part while visiting Microsoft Research, New England
    • Research supported in part from a DARPA/ONR PROCEED award, NSF grants 1228984, 1136174, 1118096, and 1065276, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant
    • We prove that the obfuscator O described in Section 5 is a good Virtual Black Box obfuscator for NC1 in the ideal graded encoding model
    Results
    • A branching program of width w and length n for -bit inputs is given by a permutation matrix Preject ∈ {0, 1}w×w such that Preject = Iw×w, and by a sequence: BP =
    • Note that by the way the authors defined the set ind(j) for input bit j ∈ [ ], and by the way the elements of Sj are indexed, Siin,bp11(i) ∈ Sinp1(i) and Siin,bp22(i) ∈ Sinp2(i).
    • To show that the zero testing call to the oracle M does not fail the authors need to show that the index set of the elements corresponding to h and h is the entire universe.
    • It follows from Theorem 2.1 that there exist polynomial functions n and w such that on input circuit C ∈ C , the branching program BP computed by O is of size n(|C|), width w(|C|), and computes on (|C|)-bit inputs.
    • To prove that O satisfies the virtual black-box property, the authors construct a simulator Sim that is given 1|C|, the description of an adversary A, and oracle access to the circuit C.
    • Instead the authors show how Sim can efficiently simulate the zero-test queries given oracle access to the circuit C.
    • Each single-input element has a value that depends on a subset of the formal variables that correspond to a specific input to the branching program.
    Conclusion
    • Since the values of the α variables are chosen at random by the obfuscation, it is unlikely that the adversary makes a query where the value of two single-input elements “cancel each other” and result in a zero.
    • If the authors think of e as an intermediate element in the evaluation of the branching program on some input x, the input-profile prof(e) represents the partial information that can be inferred about x based on the formal variables that appear in the value of e.
    • Given an input element e, D outputs a set of single-input elements with distinct input-profiles such that e = s∈D(e) s, where the equality between the elements means that their values compute the same function.
    Summary
    • The goal of general-purpose program obfuscation is to make an arbitrary computer program “unintelligible” while preserving its functionality.
    • The work of Goyal et al [10] shows that there exists an oracle that can be implemented with trusted hardware of size that is only a fixed polynomial in the security parameter, with respect to which virtual black-box obfuscation is possible.
    • Once again, the focus of the paper is to consider oracles that abstract the natural algebraic functionality underlying actual plain-model candidates for general-purpose obfuscation.
    • A branching program of width w and length n for -bit inputs is given by a permutation matrix Preject ∈ {0, 1}w×w such that Preject = Iw×w, and by a sequence: BP =
    • Note that by the way the authors defined the set ind(j) for input bit j ∈ [ ], and by the way the elements of Sj are indexed, Siin,bp11(i) ∈ Sinp1(i) and Siin,bp22(i) ∈ Sinp2(i).
    • To show that the zero testing call to the oracle M does not fail the authors need to show that the index set of the elements corresponding to h and h is the entire universe.
    • It follows from Theorem 2.1 that there exist polynomial functions n and w such that on input circuit C ∈ C , the branching program BP computed by O is of size n(|C|), width w(|C|), and computes on (|C|)-bit inputs.
    • To prove that O satisfies the virtual black-box property, the authors construct a simulator Sim that is given 1|C|, the description of an adversary A, and oracle access to the circuit C.
    • Instead the authors show how Sim can efficiently simulate the zero-test queries given oracle access to the circuit C.
    • Each single-input element has a value that depends on a subset of the formal variables that correspond to a specific input to the branching program.
    • Since the values of the α variables are chosen at random by the obfuscation, it is unlikely that the adversary makes a query where the value of two single-input elements “cancel each other” and result in a zero.
    • If the authors think of e as an intermediate element in the evaluation of the branching program on some input x, the input-profile prof(e) represents the partial information that can be inferred about x based on the formal variables that appear in the value of e.
    • Given an input element e, D outputs a set of single-input elements with distinct input-profiles such that e = s∈D(e) s, where the equality between the elements means that their values compute the same function.
    Related work
    • Our work deals with analyzing candidate generalpurpose obfuscators in an idealized mathematical model (the generic multilinear model). There has also been recent work suggesting general-purpose obfuscators in idealized mathematical models which currently do not have candidate instantiations in the standard model: the work of [5] describes a general-purpose obfuscator for NC1 in a generic group setting with a group G = G1×G2×G3×G4, where G1 is a pseudo-free Abelian group, G2 and G3 are pseudo-free non-Abelian groups, and G4 is a group supporting Barrington’s theorem, such as S5. In this generic setting, obfuscator described by [5] achieves Virtual Black-Box security. However, no candidate methods for heuristically implementing such a group G are known, and therefore, the work of [5] does not describe a candidate generalpurpose obfuscator at this time, though this may change with future work10.

      We note that question of whether there exists any oracle with respect to which virtual black-box obfuscation for general circuits is possible is a trivial question: one can consider a universal oracle that (1) provides secure encryptions eC for any circuit C to be obfuscated, and (2) given an encrypted circuit

      10 Indeed, one way to obtain a heuristic generic group G is by building it using a general-purpose obfuscator, but this would not be useful for the work of [5], since their goal is a general-purpose obfuscator.

      eC and an input x outputs C(x). The only way we can see this “solution” as being interesting is if one considers implementing this oracle with trusted hardware. The work of Goyal et al [10] shows that there exists an oracle that can be implemented with trusted hardware of size that is only a fixed polynomial in the security parameter, with respect to which virtual black-box obfuscation is possible. However, once again, the focus of our paper is to consider oracles that abstract the natural algebraic functionality underlying actual plain-model candidates for general-purpose obfuscation.
    Funding
    • Research conducted while at the IBM Research, T.J.Watson funded by NSF Grant No.1017660
    • This material is based upon work supported by the Defense Advanced Research Projects Agency through the U.S Office of Naval Research under Contract N00014-11-1-0389
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