We study key agreement in the bounded-storage model, where the participants and the adversary can use an a priori fixed bounded amount of space, and receive a large stream of data. While key agreement is known to exist unconditionally in this model (Cachin and Maurer, Crypto'97), there are strong lower bounds on the space complexity of the participants, round complexity, and communication complexity that unconditional protocols can achieve. In this work, we explore how a minimal use of cryptographic assumptions can help circumvent these lower bounds. We obtain several contributions:

-Assuming one-way functions, we construct a one-round key agreement in the bounded-storage model, with arbitrary polynomial space gap between the participants and the adversary, and communication slightly larger than the adversarial storage. Additionally, our protocol can achieve everlasting security using a second streaming round. -In the other direction, we show that one-way functions are necessary for key agreement in the bounded-storage model with large space gaps.

We further extend our results to the setting of fully-streaming adversaries, and to the setting of key agreement with multiple streaming rounds. Our results rely on a combination of information-theoretic arguments and technical ingredients such as pseudorandom generators for spacebounded computation, and a tight characterization of the space efficiency of known reductions between standard Minicrypt primitives (from distributional one-way functions to pseudorandom functions), which might be of independent interest.

In the Theorem above, poly(λ) denotes a fixed polynomial independent of a. The BSM-KA uses two rounds of communication with one stream; it can alternatively use a single simultaneous round of streaming (in both directions),