Security evaluation of quantum distance-bounding protocols via semidefinite programming

Quantum distance-bounding (QDB) protocols let a verifier check that a prover is both genuine and physically nearby. During a timed fast phase of quantum communication, the verifier measures round-trip times to obtain an upper bound on the prover's distance. For a uniform comparison, we isolate the fast phase and study one-round distance-fraud (DF) and mafia-fraud (MF) games. For discrete-variable QDB, we show that these games reduce to convex optimization problems and can therefore be solved exactly with semidefinite programming; each MF value comes with an explicit attack achieving it and a matching certificate that no attack does better. This contrasts with quantum position verification, where an attack is split between two separated parties, so its optimization is nonconvex and analyses rely on relaxations. In our MF game, the cooperating pair collapses to a single sequential strategy, which keeps the game convex and its exact value computable. Across the discrete-variable protocols we examine, the best one-round DF attack succeeds with the same probability (1/2) for every protocol, whereas MF clearly separates the protocols. For continuous-variable QDB, we report estimated attack success probabilities from a calibrated Gaussian attack model. The benchmark covers protocols whose fast phase itself authenticates the prover; designs that follow Brands and Chaum and instead bind the fast phase with a final authenticated message, like the earliest QDB proposal, fall outside it and are treated separately. Of the four protocols studied, two had no previously known one-round attack values, and we report the first ones; for the other two, we find MF attacks with higher success probability than previously reported. Overall, one-round MF resistance depends on whether an attacker can use information revealed early by the prover to answer a fresh challenge from the verifier.