On the Efficiency of Classical and Quantum Secure Function Evaluation

We provide bounds on the efficiency of secure one-sided output two-party computation of arbitrary finite functions from trusted distributed randomness in the statistical case. From these results, we derive bounds on the efficiency of protocols that use different variants of oblivious transfer (OT) as a black box. When applied to implementations of OT, these bounds generalize most known results to the statistical case. Our results hold in particular for transformations between a finite number of primitives and for any error. In the second part, we study the efficiency of quantum protocols implementing OT. While most classical lower bounds for perfectly secure reductions of OT to distributed randomness still hold in the quantum setting, we present a statistically secure protocol that violates these bounds by an arbitrarily large factor. We then prove a weaker lower bound that does hold in the statistical quantum setting and implies that even quantum protocols cannot extend OT. Finally, we present two lower bounds for reductions of OT to commitments and a protocol based on string commitments that is optimal with respect to both of these bounds.