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Three Halves Make a Whole? Beating the Half-Gates Lower Bound for Garbled Circuits

Part of the Lecture Notes in Computer Science book series (LNSC,volume 12825)

Abstract

We describe a garbling scheme for boolean circuits, in which XOR gates are free and AND gates require communication of \(1.5\kappa + 5\) bits. This improves over the state-of-the-art “half-gates” scheme of Zahur, Rosulek, and Evans (Eurocrypt 2015), in which XOR gates are free and AND gates cost \(2\kappa \) bits. The half-gates paper proved a lower bound of \(2\kappa \) bits per AND gate, in a model that captured all known garbling techniques at the time. We bypass this lower bound with a novel technique that we call slicing and dicing, which involves slicing wire labels in half and operating separately on those halves. Ours is the first to bypass the lower bound while being fully compatible with free-XOR, making it a drop-in replacement for half-gates. Our construction is proven secure from a similar assumption to prior free-XOR garbling (circular correlation-robust hash), and uses only slightly more computation than half-gates.

First author partially supported by NSF award #1617197. Second author supported by a DoE CSGF Fellowship.

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Notes

  1. 1.

    These constructions require the input labels to have a certain correlation that they do not guarantee for the gate’s output labels.

  2. 2.

    We assume that all gates take two inputs. NOT gates can be merged into downstream gates—e.g. if x goes into a NOT gate, and then into an AND gate with another input y, this is equivalent to a single \(\overline{x} \wedge y\) gate.

  3. 3.

    Most garbling schemes actually do not have perfect correctness. If an output wire has labels \(W_0, W_1\), then d will contain both \(H(W_0)\) and \(H(W_1)\). Correctness is violated if \(H(W_0) = H(W_1)\).

  4. 4.

    Equivalently, \(\mathcal {U}\) is \(2^{-\kappa }\)-almost-XOR-universal (AXU).

  5. 5.

    For now, assume H is a random oracle. We ignore including the gate ID as an additional argument to H.

  6. 6.

    More generally, multiplying by a left-inverse of the matrix on the left-hand side “just works,” as in the case where the matrix on the left-hand side is invertible.

  7. 7.

    Hence the title: “Three Halves Make a Whole”.

  8. 8.

    Note also that the calls to H have globally distinct tweaks.

  9. 9.

    Circuits were obtained from https://homes.esat.kuleuven.be/~nsmart/MPC/.

  10. 10.

    This can happen, e.g., when for every \(a \wedge b\) gate there is a corresponding \(a \vee b = \overline{ \overline{a} \wedge \overline{b}}\) gate.

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Rosulek, M., Roy, L. (2021). Three Halves Make a Whole? Beating the Half-Gates Lower Bound for Garbled Circuits. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12825. Springer, Cham. https://doi.org/10.1007/978-3-030-84242-0_5

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  • DOI: https://doi.org/10.1007/978-3-030-84242-0_5

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