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Candidate Obfuscation via Oblivious LWE Sampling

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Advances in Cryptology – EUROCRYPT 2021 (EUROCRYPT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12698))

Abstract

We present a new, simple candidate construction of indistinguishability obfuscation (iO). Our scheme is inspired by lattices and learning-with-errors (LWE) techniques, but we are unable to prove security under a standard assumption. Instead, we formulate a new falsifiable assumption under which the scheme is secure. Furthermore, the scheme plausibly achieves post-quantum security.

Our construction is based on the recent “split FHE” framework of Brakerski, Döttling, Garg, and Malavolta (EUROCRYPT ’20), and we provide a new instantiation of this framework. As a first step, we construct an iO scheme that is provably secure assuming that LWE holds and that it is possible to obliviously generate LWE samples without knowing the corresponding secrets. We define a precise notion of oblivious LWE sampling that suffices for the construction. It is known how to obliviously sample from any distribution (in a very strong sense) using iO, and our result provides a converse, showing that the ability to obliviously sample from the specific LWE distribution (in a much weaker sense) already also implies iO. As a second step, we give a heuristic contraction of oblivious LWE sampling. On a very high level, we do this by homomorphically generating pseudorandom LWE samples using an encrypted pseudorandom function.

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Notes

  1. 1.

    We defer a comparison with the independent and concurrent works [GP20, BDGM20b] to Sect. 1.3.

  2. 2.

    In detail, assume we start with a functional encoding where the encoding size is \(O(m^a)\) and the opening size is \(O(m^{1-\delta })\) for some constants \(a, \delta >0\), ignoring any other polynomial factors in the security parameter or the input size. The size of the obfuscated circuit above is then bounded by \(O(m^a + Q m^{1-\delta })\). By choosing \(m = 2^{n/(a+ \delta )}\) and recalling \(Q = 2^n/m\), the bound becomes \( O(2^{n(1-\varepsilon )})\) for \(\varepsilon = \delta /(a+\delta )\).

  3. 3.

    Recall that previously we relied on a “packed” homomorphic evaluation, where we could evaluate a function \(f~:~{\{0,1\}}^\ell \rightarrow {\{0,1\}}^m\) on a commitment to \(\mathbf {x}\) to get a commitment \(\mathbf {c}_f = \mathbf {A}\cdot \mathbf {s}_f + \mathbf {e}_f + f(\mathbf {x}) \cdot \tfrac{q}{2}\). The above relies on a slight variant that’s even further packed and allows us to homomorphically evaluate a function \(g~:~{\{0,1\}}^\ell \rightarrow \mathbb {Z}_q^m\) over a commitment to \(\mathbf {x}\) and derive a commitment \(\mathbf {c}_{g} = \mathbf {A}\cdot \mathbf {s}_{g} + \mathbf {e}_{g}+ g(\mathbf {x})\).

  4. 4.

    We believe that this change could also be applied retroactively to remove the use of a random oracles in BDGM.

  5. 5.

    As stated in BDGM Sect. 4.4: “We stress that, in contrast with the instantiation based on the Damgard-Jurik encryption scheme (Sect. 4.3), this scheme does not satisfy the syntactical requirements to apply the generic transformations (described in Sect. 4.2) to lift the scheme to the plain model.”.

  6. 6.

    Interestingly, since decrypting random ciphertexts is a (weak-)PRF, the two approaches may be more similar than may appear.

  7. 7.

    Note that if we write \(\mathbf {C}= [\mathbf {C}_1 \mid \cdots \mid \mathbf {C}_\ell ]\) where \(\mathbf {C}_1,\ldots ,\mathbf {C}_\ell \in \mathbb {Z}_q^{m \times m \log q}\) and \(\mathbf {x}= (x_1,\ldots ,x_\ell )\), then

    $$\begin{aligned} \mathbf {C}- \mathbf {x}^\top \otimes \mathbf {G}= [ \mathbf {C}_1 - x_1 \mathbf {G}\mid \ldots \mid \mathbf {C}_\ell - x_\ell \mathbf {G}] \end{aligned}$$

    .

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Acknowledgments

We thank Yilei Chen and Vinod Vaikuntanathan for insightful discussions on cryptanalysis and bootstrapping.

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  1. NTT Research Inc., San Francisco, USA

    Hoeteck Wee & Daniel Wichs

  2. Northeastern University, Boston, USA

    Daniel Wichs

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Wee, H., Wichs, D. (2021). Candidate Obfuscation via Oblivious LWE Sampling. In: Canteaut, A., Standaert, FX. (eds) Advances in Cryptology – EUROCRYPT 2021. EUROCRYPT 2021. Lecture Notes in Computer Science(), vol 12698. Springer, Cham. https://doi.org/10.1007/978-3-030-77883-5_5

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