ZCZ - Achieving n-bit SPRP Security with a Minimal Number of Tweakable-block-cipher Calls

Strong Pseudo-random Permutations (SPRPs) are important for various applications. In general, it is desirable to base an SPRP on a single-keyed primitive for minimizing the implementation costs. For constructions built on classical block ciphers, Nandi showed at ASIACRYPT\u2715 that at least two calls to the primitive per processed message block are required for SPRP security, assuming that all further operations are linear. The ongoing trend of using tweakable block ciphers as primitive has already led to MACs or encryption modes with high security and efficiency properties. Thus, three interesting research questions are hovering in the domain of SPRPs: (1) if and to which extent the bound of two calls per block can be reduced with a tweakable block cipher, (2) how concrete constructions could be realized, and (3) whether full $n$-bit security is achievable from primitives with $n$-bit state size. The present work addresses all three questions. Inspired by Iwata et al.\u27s ZHash proposal at CRYPTO\u2717, we propose the ZCZ (ZHash-Counter-ZHash) construction, a single-key variable-input-length SPRP based on a single tweakable block cipher whose tweak length is at least its state size. ZCZ possesses close to optimal properties with regards to both performance and security: not only does it require only asymptotically $3\ell/2$ calls to the primitive for $\ell$-block messages, but we also show that this figure is close to the minimum by an PRP distinguishing attack on any construction with tweak size of $\tau = n$ bits and fewer than $(3\ell-1)/2$ calls to the same primitive. Moreover, it provides optimal $n$-bit security for a primitive with $n$-bit state and tweak size

Tight Security of TNT and Beyond: Attacks, Proofs and Possibilities for the Cascaded LRW Paradigm

Liskov, Rivest and Wagner laid the theoretical foundations for tweakable block ciphers (TBC). In a seminal paper, they proposed two (up to) birthday-bound secure design strategies --- LRW1 and LRW2 --- to convert any block cipher into a TBC. Several of the follow-up works consider cascading of LRW-type TBCs to construct beyond-the-birthday bound (BBB) secure TBCs. Landecker et al. demonstrated that just two-round cascading of LRW2 can already give a BBB security. Bao et al. undertook a similar exercise in context of LRW1 with TNT --- a three-round cascading of LRW1 --- that has been shown to achieve BBB security as well. In this paper, we present a CCA distinguisher on TNT that achieves a non-negligible advantage with $O(2^{n/2})$ queries, directly contradicting the security claims made by the designers. We provide a rigorous and complete advantage calculation coupled with experimental verifications that further support our claim. Next, we provide new and simple proofs of birthday-bound CCA security for both TNT and its single-key variant, which confirm the tightness of our attack. Furthering on to a more positive note, we show that adding just one more block cipher call, referred as 4-LRW1, does not just reestablish the BBB security, but also amplifies it up to $2^{3n/4}$ queries. As a side-effect of this endeavour, we propose a new abstraction of the cascaded LRW-design philosophy, referred to as the LRW+ paradigm, comprising two block cipher calls sandwiched between a pair of tweakable universal hashes. This helps us to provide a modular proof approach covering all cascaded LRW constructions with at least $2$ rounds, including 4-LRW1, and its more established relative, the well-known CLRW2, or more aptly, 2-LRW2