Better concrete security for half-gates garbling (in the multi-instance setting)

We study the concrete security of high-performance implementations of half-gates garbling, which all rely on (hardware-accelerated) AES. We find that current instantiations using k-bit wire labels can be completely broken—in the sense that the circuit evaluator learns all the inputs of the circuit garbler—in time O(2^k/C), where C is the total number of (non-free) gates that are garbled, possibly across multiple independent executions. The attack can be applied to existing circuit-garbling libraries using k=80 when C ≈ 10⁹, and would require 267 machine-months and cost about $3500 to implement on the Google Cloud Platform. Since the attack can be fully parallelized, it could be carried out in about a month using ≈250 machines. With this as our motivation, we seek a way to instantiate the hash function in the half-gates scheme so as to achieve better concrete security. We present a construction based on AES that achieves optimal security in the single-instance setting (when only a single circuit is garbled). We also show how to modify the half-gates scheme so that its concrete security does not degrade in the multi-instance setting. Our modified scheme is as efficient as prior work in networks with up to 2 Gbps bandwidth.