The dual-execution protocol of Mohassel and Franklin (PKC 2006) is a highly efficient (each party garbling only one circuit) 2PC protocol that achieves malicious security apart from leaking an arbitrary, adversarially-chosen predicate about the honest party's input. We present two practical and orthogonal approaches to improve the security of the dual-execution technique. First, we show how to greatly restrict the predicate that an adversary can learn in the protocol, to a natural notion of ``only computation leaks''-style leakage. Along the way, we identify a natural security property of garbled circuits called property-enforcing that may be of independent interest. Second, we address a complementary direction of reducing the probability that the leakage occurs. We propose a new dual-execution protocol --- with a very light cheating-detection phase and each party garbling s+1 circuits --- in which a cheating party learns a bit with probability only 2^{-s}. Our concrete measurements show approximately 35% reduction in communication for the AES circuit, compared to the best combination of state of the art techniques for achieving the same security notion. Combining the two results, we achieve a rich continuum of practical trade-offs between efficiency and security, connecting the covert, dual-execution and full-malicious guarantees.
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