  Secure two-party cryptography is possible if the adversary's quantum storage
device suffers imperfections. For example, security can be achieved if the
adversary can store strictly less then half of the qubits transmitted during
the protocol. This special case is known as the bounded-storage model, and it
has long been an open question whether security can still be achieved if the
adversary's storage were any larger. Here, we answer this question positively
and demonstrate a two-party protocol which is secure as long as the adversary
cannot store even a small fraction of the transmitted pulses. We also show that
security can be extended to a larger class of noisy quantum memories.
