How can the AND gate be executed efficiently in the GMW protocol for the MPC setting?

JAAAY

11/24/23, 6:20 PM

GMW is a pretty straight forward protocol for the setting of 2PC. It is still straight forward for the case of MPC except for the AND gate. I'm struggling to understand how to obtain a "good" communication complexity. To be more specific, the AND gate is described by the following function.

\begin{aligned} c &=a \wedge b=\left(a_1 \oplus \cdots \oplus a_n\right) \wedge\left(b_1 \oplus \cdots \oplus b_n\right) \\ &=\left(\bigoplus_{i=1}^n a_i \wedge b_i\right) \oplus\left(\bigoplus_{i \neq j} a_i \wedge b_j\right) \end{aligned}

Of course the first term can be computed locally, but the second term from my intuition requires $\mathcal{O}(n^2)$ OTs. Is there a better way to evaluate this term?

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