International Association
for Cryptologic Research

The sum of two $n$-bit pseudorandom permutations is known to behave like a pseudorandom function with $n$ bits of security. A recent line of research has investigated the security of two public $n$-bit permutations and its degree of indifferentiability. Mandal et al. (INDOCRYPT 2010) proved $2n/3$-bit security, Mennink and Preneel (ACNS 2015) pointed out a non-trivial flaw in their analysis and re-proved $2n/3$-bit security. Bhattacharya and Nandi (EUROCRYPT 2018) eventually improved the result to $n$-bit security. Recently, Gunsing at CRYPTO 2022 already observed that a proof technique used in this line of research only holds for sequential indifferentiability. We revisit the line of research in detail, and observe that the strongest bound of $n$-bit security has two other serious issues in the reasoning, the first one is actually the same non-trivial flaw that was present in the work of Mandal et al., while the second one discards biases in the randomness influenced by the distinguisher. More concretely, we introduce two attacks that show limited potential of different approaches. We (i) show that the latter issue that discards biases only holds up to $2^{3n/4}$ queries, and (ii) perform a differentiability attack against their simulator in $2^{5n/6}$ queries. On the upside, we revive the result of Mennink and Preneel and show $2n/3$-bit regular indifferentiability security of the sum of public permutations.

In this paper we characterize all $2n$-bit-to-$n$-bit Pseudorandom Functions (PRFs) constructed with the minimum number of calls to $n$-bit-to-$n$-bit PRFs and arbitrary number of linear functions. First, we show that all two-round constructions are either classically insecure, or vulnerable to quantum period-finding attacks. Second, we categorize three-round constructions depending on their vulnerability to these types of attacks. This allows us to identify classes of constructions that could be proven secure. We then proceed to show the security of the following three candidates against any quantum distinguisher that makes at most $ 2^{n/4} $ (possibly superposition) queries: \begin{align*} TNT(x_1,x_2) &:= f_3(x_2 \oplus f_2(x_2 \oplus f_1(x_1)));\\ LRQ(x_1,x_2) &:= f_2(x_2) \oplus f_3(x_2 \oplus f_1(x_1));\\ LRWQ(x_1,x_2) &:= f_3( f_1(x_1) \oplus f_2(x_2)). \end{align*} Note that the first construction is a classically secure tweakable block-cipher due to Bao et al., and the third construction was shown to be a quantum-secure tweakable block-cipher by Hosoyamada and Iwata with similar query limits. Of note is our proof framework, an adaptation of Chung et al.'s rigorous formulation of Zhandry's compressed oracle technique in the indistinguishability setup, which could be of independent interest. This framework gives very compact and mostly classical-looking proofs as compared to Hosoyamada-Iwata interpretation of Zhandry's compressed oracle.

It was long thought that symmetric cryptography was only mildly affected by quantum attacks, and that doubling the key length was sufficient to restore security. However, recent works have shown that Simon's quantum period finding algorithm breaks a large number of MAC and authenticated encryption algorithms when the adversary can query the MAC/encryption oracle with a quantum superposition of messages. In particular, the OCB authenticated encryption mode is broken in this setting, and no quantum-secure mode is known with the same efficiency (rate-one and parallelizable). In this paper we generalize the previous attacks, show that a large class of OCB-like schemes is unsafe against superposition queries, and discuss the quantum security notions for authenticated encryption modes. We propose a new rate-one parallelizable mode named QCB inspired by TAE and OCB and prove its security against quantum superposition queries.

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’15 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.’s ZHash proposal at CRYPTO’17, 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; we 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.

CAESAR has caused a heated discussion regarding the merits of one-pass encryption and online ciphers. The latter is a keyed, length preserving function which outputs ciphertext blocks as soon as the respective plaintext block is available as input. The immediacy of an online cipher affords a clear performance advantage, but it comes at a price: ciphertext blocks cannot depend on later plaintext blocks, limiting diffusion and hence security. We show how one can attain the best of both worlds by providing provably secure constructions, achieving full cipher security, based on applications of an online cipher around blockwise reordering layers. Explicitly, we show that with just two calls to the online cipher, prp security up to the birthday bound is both attainable and maximal. Moreover, we demonstrate that three calls to the online cipher suffice to obtain beyond birthday bound security. We provide a full proof of this for a prp construction, and, in the ±prp setting, security against adversaries who make queries of any single length. As part of our investigation, we extend an observation by Rogaway and Zhang by further highlighting the close relationship between online ciphers and tweakable blockciphers with variable-length tweaks.

Online ciphers, in spite of being insecure against an sprp adversary, can be desirable at places because of their ease of implementation and speed. Here we propose a single-keyed inverse-free construction that achieves online sprp security with an optimal number of blockcipher calls. We also include a partial block construction, without requiring any extra key.