Quantum Key-length Extension

TL;DR

This paper analyzes quantum security of key-length extension techniques like FX and double encryption, providing tight bounds and new proof techniques for their resilience against quantum attacks in ideal models.

Contribution

It offers the first concrete quantum security bounds for FX and double encryption, introduces novel proof techniques for partially-quantum models, and links security to quantum complexity problems.

Findings

FX is secure against non-adaptive quantum attackers in a partially-quantum model.

Double encryption security reduces to the quantum difficulty of element distinctness.

New techniques for partially-quantum proofs without separate oracle analysis.

Abstract

Should quantum computers become available, they will reduce the effective key length of basic secret-key primitives, such as blockciphers. To address this we will either need to use blockciphers which inherently have longer keys or use key-length extension techniques which employ a blockcipher to construct a more secure blockcipher that uses longer keys. We consider the latter approach and revisit the FX and double encryption constructions. Classically, FX is known to be secure, while double encryption is no more secure than single encryption due to a meet-in-the-middle attack. We provide positive results, with concrete and tight bounds, for both of these constructions against quantum attackers in ideal models. For FX, we consider a partially-quantum model, where the attacker has quantum access to the ideal primitive, but only classic access to FX. We provide two results for FX in this model. The first establishes the security of FX against non-adaptive attackers. The second establishes security against general adaptive attacks for 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. An extension to perfectly lazily sampling a quantum random permutation, which would help resolve the adaptive security of standard FX, is an important but challenging open question. We introduce techniques for partially-quantum proofs without relying on analyzing the classical and quantum oracles separately, which is common in existing work. This 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.