Quantum Leakage from Shamir Sharing with Correlated Resources

Abstract

We study leakage from Shamir secret sharing when each device uses its share to produce one qubit from an initial quantum resource independent of the secret and sharing randomness. We prove exponentially small trace distance between the leakage states of any two secrets, when reconstruction requires a sufficiently large fraction of the shares, for fixed stationary chains satisfying an upper operator mixing condition and for resources generated by fixed finite quantum memories with primitive transfer channels. These guarantees permit arbitrary extensions of the device marginal as adversary side information and hold uniformly over prime fields larger than the number of shares. For independent device--reference pairs, which provide the quantitative input to the argument, we determine the optimal exponential base when reconstruction requires all shares: $2\sqrt2/π$ for a qubit and $2/π$ for a classical bit, in both cases allowing arbitrary finite references. A comparison of selected marginals with product states converts the bounds for independent device--reference pairs into security guarantees for the correlated resources.