We consider the latter approach — in particular, analyzing the security of the FX and double encryption constructions. Classically, these constructs were considered as key-length extension techniques for DES. FX was proven to be a secure key-length extension technique, while double encryption was shown to be no more secure than single encryption due to a meet-in-the-middle attack. In this work we provide positive results, with concrete and tight bounds, for the security of both of these constructions against quantum attackers in ideal models.
For FX, we consider security in the so-called “Q1 model,” a natural model in which the attacker has quantum access to the ideal primitive, but only classic access to FX. We provide two partial results for FX in this model. The first establishes the security of FX against non-adaptive attackers. The second establishes security against fully adaptive attackers when considering a variant of FX using a random oracle in place of an ideal cipher. This result relies on the techniques of Zhandry (CRYPTO ’19) for lazily sampling a quantum random oracle and are thus hard to extend to the true FX construction because it is currently unknown if a quantum random permutation can be lazily sampled. To the best of our knowledge, this result also is the first to introduce techniques to handle Q1 security in ideal models without analyzing the classical and quantum oracles separately, which may be of broader interest.
For double encryption we apply a technique of Tessaro and Thiruvengadam (TCC ’18) to establish that security reduces to the difficulty of solving the list disjointness problem, which we are able to reduce through a chain of results to the known quantum difficulty of the element distinctness problem.