David Reichmuth

Oblivious transfer (OT) is a cryptographic primitive which is universal for multiparty computation. Unfortunately, perfect information-theoretically (IT) secure quantum oblivious transfer is impossible (except with restrictions on cheating parties). Imperfect IT secure quantum oblivious transfer remains possible, but the smallest possible cheating probabilities are not known. Informally, in 1-out-of-2 oblivious transfer, a sender Alice has two bits x0, x1. A receiver Bob obtains one of these, xb, where b= 0 or b= 1. Alice should not be able to guess b, and Bob should not be able to guess the bit value he did not obtain. Bounds on cheating probabilities in quantum oblivious transfer have previously been investigated for complete protocols. “Complete” means that if sender Alice and receiver Bob both follow the protocol, the bit value Bob obtains correctly matches Alice’s bit value. Here we instead investigate incomplete protocols, where Bob obtains an incorrect bit value with probability pf. For complete protocols, both “classical” and quantum, it holds that if one party can cheat no better than with a random guess, then the other party can cheat perfectly. For incomplete protocols, in contrast, even with no restrictions on cheating parties, and when one party can cheat no better than with random guess, it is possible that the other party still cannot cheat perfectly; their cheating probability can be lower than in complete protocols. We find the optimal non-interactive protocols where Alice’s bit values are represented by four symmetric pure quantum states, and where Alice cannot cheat better than with a random guess. “Optimal” means that for a given pf, Bob’s cheating probability pr is as low as possible, and vice versa. We also show that quantum protocols can outperform classical non-interactive protocols. Our results also provide a lower bound on Bob’s cheating probability in interactive quantum protocols. An advantage of the non-interactive protocols we investigate is that they require neither entanglement nor quantum memory. The optimal protocols could be readily implemented using standard optical components.