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On Generating the Initial Key in the Bounded-Storage Model

Stefan Dziembowski and Ueli Maurer

In the bounded-storage model (BSM) for information-theoretically secure encryption and key-agreement one uses a random string $R$ whose length $t$ is greater than the assumed bound $s$ on the adversary Eve's storage capacity. The legitimate parties Alice and Bob share a short initial secret key $K$ which they use to select and combine certain bits of $R$ to obtain a derived key $X$ which is much longer than $K$. Eve can be proved to obtain essentially no information about $X$ even if she has infinite computing power and even if she learns $K$ after having performed the storage operation and lost access to $R$.

This paper addresses the problem of generating the initial key $K$ and makes two contributions. First, we prove that without such a key, secret key agreement in the BSM is impossible unless Alice and Bob have themselves very high storage capacity, thus proving the optimality of a scheme proposed by Cachin and Maurer. Second, we investigate the hybrid model where $K$ is generated by a computationally secure key agreement protocol. The motivation for the hybrid model is to achieve provable security under the sole assumption that Eve cannot break the key agreement scheme during} the storage phase, even if afterwards she may gain infinite computing power (or at least be able to break the key agreement scheme). In earlier work on the BSM, it was suggested such a hybrid scheme is secure because if Eve has no information about $K$ during the storage phase, then she has missed any opportunity to know anything about $X$, even when later learning $K$. We show that this very intuitive and apparently correct reasoning is false by giving an example of a secure (according to the standard definition) computational key-agreement scheme for which the BSM-scheme is nevertheless completely insecure. One of the surprising consequences of this example is that existing definitions for the computational security of key-agreement and encryption are still too weak and therefore new, stronger definitions are needed.