Tight Security Proofs for the Bounded-Storage Model
Stefan Dziembowski and Ueli Maurer
In the bounded-storage model for information-theoretically secure encryption and key-agreement one can prove the security of a cipher based on the sole assumption that the adversary's storage capacity is bounded, say by $s$ bits, even if her computational power is unlimited. Assume that a random $t$-bit string $R$ is either publicly available (e.g. the signal of a deep space radio source) or broadcast by one of the legitimate parties. If $s<t$, the adversary can store only partial information about $R$. The legitimate sender Alice and receiver Bob, sharing a short secret key $K$ initially, can therefore potentially generate a very long $n$-bit one-time pad $X$ with $n\gg|K|$ about which the adversary has essentially no information, thus at first glance apparently contradicting Shannon's bound on the key size of a perfect cipher.
All previous results in the bounded-storage model were partial or far from optimal, for one of the following reasons: either the secret key $K$ had in fact to be longer than the derived one-time pad, or $t$ had to be extremely large ($t>ns$), or the adversary was assumed to be able to store only actual bits of $R$ rather than arbitrary $s$ bits of information about $R$, or the adversary could obtain a non-negligible amount of information about $X$.
In this paper we give the first fully satisfactory security proof in the bounded-storage model, exploiting the full potential of the model: $K$ is short, $X$ is very long (e.g. gigabytes), $t$ needs to be only moderately larger than $s$, and the security proof is optimally strong. This solves the main open problem of both Maurer's 1992 paper which introduced the bounded-storage model and of the recent paper by Aumann, Ding, and Rabin. In fact, we prove that $s/t$ can be arbitrarily close to $1$ and hence the storage bound is essentially optimal.