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Structure-Preserving Compilers from New Notions of Obfuscations

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

The dream of software obfuscation is to take programs, as they are, and then generically compile them into obfuscated versions that hide their secret inner workings. In this work we investigate notions of obfuscations weaker than virtual black-box (\(\textsf{VBB} \)) but which still allow obfuscating cryptographic primitives preserving their original functionalities as much as possible.

In particular we propose two new notions of obfuscations, which we call oracle-differing-input obfuscation (\(\textsf{odiO} \)) and oracle-indistinguishability obfuscation (\(\textsf{oiO} \)). In a nutshell, \(\textsf{odiO} \) is a natural strengthening of differing-input obfuscation (\(\textsf{diO} \)) and allows obfuscating programs for which it is hard to find a differing-input when given only oracle access to the programs. An \(\textsf{oiO} \) obfuscator allows to obfuscate programs that are hard to distinguish when treated as oracles.

We then show applications of these notions, as well as positive and negative results around them. A few highlights include:

  • Our new notions are weaker than \(\textsf{VBB} \) and stronger than \(\textsf{diO} \).

  • As it is the case for \(\textsf{VBB} \), we show that there exist programs that cannot be obfuscated with \(\textsf{odiO} \) or \(\textsf{oiO} \).

  • Our new notions allow to generically compile several flavours of secret-key primitives (e.g., SKE, MAC, designated verifier NIZK) into their public-key equivalent (e.g., PKE, signatures, publicly verifiable NIZK) while preserving one of the algorithms of the original scheme (function-preserving), or the structure of their outputs (format-preserving).

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Notes

  1. 1.

    Note that a function-preserving transformation is also format-preserving. This is because the former does not modify the algorithms of the original primitive. Hence, the format of the output is preserved by definition.

  2. 2.

    As for straight-line knowledge soundness, we do not consider succinctness (i.e., we do not cover \(\textsf{dv}\text {-}\textsf{SNARG}\)/\(\textsf{pv}\text {-}\textsf{SNARG}\)) since, in order to have a straight-line extraction, the size of the proof is proportional to the size of the witness.

  3. 3.

    We will elaborate on this later, but intuitively this is because the obfuscated program will use the puncturable PRF to generate a fresh symmetric key for different input (e.g., messages, initialization vectors). Hence, on decryption/verification, the receiver needs to evaluate the same PRF in order to recompute the symmetric key used to decrypt/verify a particular ciphertext/signature.

  4. 4.

    Note that Barak et al. [8] demonstrates the impossibility of transforming a SKE into a PKE (through the obfuscation of its encryption algorithm \(\textsf{Enc}({\textsf{k}},\cdot )\)) by building a (contrived) secure SKE that, after applying the transformation, yields an insecure PKE. However, their contrived SKE is not key indistinguishable. For this reason, in order to prove the impossibility of our \(\textsf{oiO} \)-based transformation (5) (from key indistinguishable SKE to PKE) we need to rework their result.

  5. 5.

    This follows the same spirit of the UCE framework proposed by Bellare et al. [9] that allows to identify which property of the random oracle model (ROM) is needed to imply security of the (ROM-based) construction..

  6. 6.

    The rest of the input besides the key is irrelevant for this discussion.

  7. 7.

    In particular, soundness (of underlying designated-verifier non-interactive proof system) must hold even if the adversary has oracle access to the verification algorithm. The latter is essential during the reduction to simulate the input-output behavior of the two circuits (treated as oracles). Hence, our transformation does not apply to non-interactive proofs systems that suffer from the so called verifier rejection problem, i.e., giving oracle access to the verifier allows the adversary to break soundness..

  8. 8.

    Despite the construction is the same, the sampler required to prove knowledge soundness is different.

  9. 9.

    If, instead of generating \(\textsf{iv}\) using the first PRF, we allow the circuit to take directly in input \(\textsf{iv}\) then the PKE (output by the transformation) is trivially broken. This is because (following the syntax of the \(\textsf{IV}\)-based SKE) \(\textsf{iv}\) is included into the ciphertext. Hence, an adversary can break the selective IND-CPA security of the compiled PKE by simply re-encrypting a message using the \(\textsf{iv}\) that is included into the challenge ciphertext.

  10. 10.

    Recall that \(|C_0| = |C_1|\) by definition of sampler (Definition 3.1).

  11. 11.

    For instance, we can have that \(\textsf{S}_{b}\) only outputs circuits whose description starts with a bit b, and that \(\textsf{Obf}_b\) rejects any circuit whose description starts with the bit \(1-b\).

  12. 12.

    Otherwise, if \(\textsf{S}\in \mathcal {S}_{\textsf{odiO}}\), there exists a \((\{\textsf{S}\})\)-\(\textsf{odiO} \)-obfuscator that in turn is also a \((\{\textsf{S}\})\)-\(\textsf{diO} \)-obfuscator.

  13. 13.

    Indeed, any PPT obfuscator \(\textsf{Obf}\) that satisfies correctness and polynomial slowdown is a \((\{\textsf{S}\})\)-\(\textsf{odiO} \)-obfuscator (resp. \((\{\textsf{S}\})\)-\(\textsf{oiO} \)-obfuscator), e.g., \(\textsf{Obf}\) is the identity function or \(\textsf{Obf}\) is an \(\textsf{iO} \)-obfuscator.

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Acknowledgments

The authors would like to thank the anonymous reviewers for useful feedback. The research described in this paper received funding from: the Concordium Blockhain Research Center, Aarhus University, Denmark; the Carlsberg Foundation under the Semper Ardens Research Project CF18-112 (BCM); the European Research Council (ERC) under the European Unions’s Horizon 2020 research and innovation programme under grant agreement No 803096 (SPEC).

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Authors and Affiliations

  1. Protocol Labs, San Francisco, USA

    Matteo Campanelli

  2. Aarhus University, Aarhus, Denmark

    Danilo Francati & Claudio Orlandi

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  1. Matteo Campanelli
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Correspondence to Danilo Francati .

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Campanelli, M., Francati, D., Orlandi, C. (2023). Structure-Preserving Compilers from New Notions of Obfuscations. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_23

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