Paper 2019/007

Tight Security Bounds for Generic Stream Cipher Constructions

Matthias Hamann and Matthias Krause

Abstract

The design of modern stream ciphers is strongly influenced by the fact that Time-Memory-Data tradeoff attacks (TMD-TO attacks) reduce their effective key length to $\mathit{SL}/2$, where $\mathit{SL}$ denotes the inner state length. The classical solution, employed, e.g., by eSTREAM portfolio members Trivium and Grain v1, is to design the cipher in accordance with the Large-State-Small-Key construction, which implies that $\mathit{SL}$ is at least twice as large as the session key length $\mathit{KL}$. In the last years, a new line of research looking for alternative stream cipher constructions guaranteeing a higher TMD-TO resistance with smaller inner state lengths has emerged. So far, this has led to three generic constructions: the LIZARD construction, having a provable TMD-TO resistance of $2\cdot \mathit{SL}/3$; the Continuous-Key-Use construction, underlying the stream cipher proposals Sprout, Plantlet, and Fruit; and the Continuous-IV-Use construction, very recently proposed by Hamann, Krause, and Meier. Meanwhile, it could be shown that the Continuous-Key-Use construction is vulnerable against certain nontrivial distinguishing attacks. In this paper, we present a formal framework for proving security lower bounds on the resistance of generic stream cipher constructions against TMD-TO attacks and analyze two of the constructions mentioned above. First, we derive a tight security lower bound of approximately $\min\{\mathit{KL},\mathit{SL}/2\}$ on the resistance of the Large-State-Small-Key construction. This shows that the feature $\mathit{KL}\le \mathit{SL}/2$ does not open the door for new nontrivial TMD-TO attacks against Trivium and Grain v1 which are more dangerous than the known ones. Second, we prove a maximal security bound on the TMD-TO resistance of the Continuous-IV-Use construction, which shows that designing concrete instantiations of ultra-lightweight Continuous-IV-Use stream ciphers is a hopeful direction of future research.