Abstract
We develop attacks on the security of variants of pseudo-random generators computed by quadratic polynomials. In particular we give a general condition for breaking the one-way property of mappings where every output is a quadratic polynomial (over the reals) of the input. As a corollary, we break the degree-2 candidates for security assumptions recently proposed for constructing indistinguishability obfuscation by Ananth, Jain and Sahai (ePrint 2018) and Agrawal (ePrint 2018). We present conjectures that would imply our attacks extend to a wider variety of instances, and in particular offer experimental evidence that they break assumption of Lin-Matt (ePrint 2018).
Our algorithms use semidefinite programming, and in particular, results on low-rank recovery (Recht, Fazel, Parrilo 2007) and matrix completion (Gross 2009).
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Notes
- 1.
The work of Ananth, Jain, and Sahai [2] also considered degree-3 polynomials. We do not have attacks on such degree-3 polynomials; we discuss this further below.
- 2.
This cubic version of their assumption was made explicit in an update to [2].
- 3.
Along the same lines, we note that if \(\mathcal Q\) is nice and \({\mathbb {E}}_{q \sim \mathcal Q} q = 0\) (as we observe later, the latter can be enforced without loss of generality) then \(\mathcal Q\) is also \(\varLambda (n)\)-bounded for \(\varLambda (n) \leqslant O(n)\). The reason is that if \(\mathcal Q\) is nice and has \({\mathbb {E}}\, q = 0\) then
$$ {\mathbb {E}}\Vert q\Vert ^2 = \sum _{i,j \leqslant n} {\mathbb {E}}\, Q_{ij}^2 = \sum _{i,j \leqslant n} Var(Q_{ij}) = n^2\,.$$For every i, j and every q in the support of \(\mathcal Q\), we have by niceness that \(|q_{ij}| \leqslant \Vert q\Vert _2 \leqslant C n\). Hence \(\mathcal Q\) is O(n)-bounded.
One implication is that \(\mathcal Q\) cannot be a distribution on where the all-zero polynomial appears with probability, say, \(1-1/n\), as otherwise its support would also have to contain polynomials with coefficients \(\gg n\). Our main theorem could not apply to such a distribution, since clearly at least \(\varOmega (n^2)\) independent samples would be needed to get enough information to recover x from \(\{q_i,q_i(x)\}\), while we assume \(m \leqslant n(\log n)^{O(1)} \ll n^2\).
- 4.
For any \(q(x) = \sum _{i,j}q_{i\leqslant j} x_i x_j\), we define \(Q:\mathbb R^{n\times n}\rightarrow \mathbb R\) by \(Q_{i,j} = Q_{j,i}= q_{i,j}/2\). Then,
is a linear map on \(\mathbb R^{n \times n}\).
is a linear map on References
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Acknowledgements
Boaz Barak was supported by NSF awards CCF 1565264 and CNS 1618026 and a Simons Investigator Fellowship. Samuel B. Hopkins was supported by a Miller Postdoctoral Fellowship and NSF award CCF 1408673. Pravesh Kothari was supported in part by Ma fellowship from the Schmidt Foundation and Avi Wigderson’s NSF award CCF-1412958. Amit Sahai and Aayush Jain were supported in part from a DARPA/ARL SAFEWARE award, NSF Frontier Award 1413955, and NSF grant 1619348, BSF grant 2012378, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. Aayush Jain was also supported by Google PhD Fellowship 2018, in the area of Privacy and Security. This material is based upon work supported by the Defense Advanced Research Projects Agency through the ARL under Contract W911NF-15-C- 0205. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, the U.S. Government or Google.
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Barak, B., Hopkins, S.B., Jain, A., Kothari, P., Sahai, A. (2019). Sum-of-Squares Meets Program Obfuscation, Revisited. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11476. Springer, Cham. https://doi.org/10.1007/978-3-030-17653-2_8
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