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Breaking the Circuit Size Barrier for Secure Computation Under Quasi-Polynomial LPN

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Advances in Cryptology – EUROCRYPT 2021 (EUROCRYPT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12697))

Abstract

In this work we introduce a new (circuit-dependent) homomorphic secret sharing (HSS) scheme for all \(\log /\log \log \)-local circuits, with communication proportional only to the width of the circuit, and polynomial computation, assuming the super-polynomial hardness of learning parity with noise (\(\mathsf {LPN}\)). At the heart of our new construction is a pseudorandom correlation generator (PCG), which allows two partie to locally stretch, from short seeds, pseudorandom instances of an arbitrary \(\log /\log \log \)-local additive correlation.

Our main application, and the main motivation behind this work, is a generic two-party secure computation protocol for every layered (boolean or arithmetic) circuit of size s with total communication \(O(s/\log \log s)\) and polynomial computation, assuming the super-polynomial hardness of the standard learning parity with noise assumption (a circuit is layered if its nodes can be partitioned in layers, such that any wire connects adjacent layers). This expands the set of assumptions under which the ‘circuit size barrier’ can be broken, for a large class of circuits. The strength of the underlying assumption is tied to the sublinearity factor: we achieve communication O(s/k(s)) under the \(s^{2^{k(s)}}\)-hardness of LPN, for any \(k(s) \le \log \log s / 4\).

Previously, the set of assumptions known to imply a PCG for correlations of degree \(\omega (1)\) or generic secure computation protocols with sublinear communication was restricted to LWE, DDH, and a circularly secure variant of DCR.

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Notes

  1. 1.

    A depth-d circuit is layered if it can be divided into d layers such that any wire connects adjacent layers.

  2. 2.

    BKW and its variants [BKW00, Lyu05] do not improve over information set decoding attacks in this regime of parameters, due to the very low number of samples.

  3. 3.

    i.e. the \(j^\text {th}\) coordinate of the image by \(f_i\), seen as \(f_i:\mathbb {F}^n \times \mathbb {F}^{s_i} \rightarrow \mathbb {F}^{s_{i+1}+m_i}\).

  4. 4.

    If \(\bigcup _{i=1}^{\mathsf {n}_{\mathsf {s}}}S_i\ne \emptyset \), and with the notations of the rest of the section, the vector \(\vec r\) we generate is equal to \(0_\mathbb {F}\) on \([w]\smallsetminus \bigcup _{i=1}^{\mathsf {n}_{\mathsf {s}}}S_i\), hence not pseudorandom. However, we can simply have the parties generate another mask \(\vec r'=H'\cdot \vec e'\), pseudorandom under \(\mathsf {xLPN}\), to cover \([w]\smallsetminus \bigcup _{i=1}^{\mathsf {n}_{\mathsf {s}}}S_i\). Since the parties do not need shares of \((\vec r')^{\otimes \mathsf {tpp}}\), the communication complexity of generating the \(\lambda \)-sparse \(\vec e'\) using an MPFSS is not an issue.

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Authors and Affiliations

  1. CNRS, IRIF, Université de Paris, Paris, France

    Geoffroy Couteau

  2. École Normale Supérieure de Lyon and IDC Herzliya, Herzliya, Israel

    Pierre Meyer

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  1. Geoffroy Couteau
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Correspondence to Geoffroy Couteau or Pierre Meyer .

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  1. Inria, Paris, France

    Anne Canteaut

  2. UCLouvain, Louvain-la-Neuve, Belgium

    François-Xavier Standaert

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Couteau, G., Meyer, P. (2021). Breaking the Circuit Size Barrier for Secure Computation Under Quasi-Polynomial LPN. In: Canteaut, A., Standaert, FX. (eds) Advances in Cryptology – EUROCRYPT 2021. EUROCRYPT 2021. Lecture Notes in Computer Science(), vol 12697. Springer, Cham. https://doi.org/10.1007/978-3-030-77886-6_29

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  • DOI: https://doi.org/10.1007/978-3-030-77886-6_29

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