Backdoors in Pseudorandom Number Generators:Possibility and Impossibility Results

Inspired by the Dual EC DBRG incident, Dodis et al. (Eurocrypt 2015) initiated the formal study of backdoored PRGs, showing that backdoored PRGs are equivalent to public key encryption schemes, giving constructions for backdoored PRGs (BPRGs), and showing how BPRGs can be ``immunised\u27\u27 by careful post-processing of their outputs. In this paper, we continue the foundational line of work initiated by Dodis et al., providing both positive and negative results. We first revisit the backdoored PRG setting of Dodis et al., showing that PRGs can be more strongly backdoored than was previously envisaged. Specifically, we give efficient constructions of BPRGs for which, given a single generator output, Big Brother can recover the initial state and, therefore, all outputs of the BPRG. Moreover, our constructions are forward-secure in the traditional sense for a PRG, resolving an open question of Dodis et al. in the negative. We then turn to the question of the effectiveness of backdoors in robust PRNGs with input (c.f. Dodis et al., ACM-CCS 2013): generators in which the state can be regularly refreshed using an entropy source, and in which, provided sufficient entropy has been made available since the last refresh, the outputs will appear pseudorandom. The presence of a refresh procedure might suggest that Big Brother could be defeated, since he would not be able to predict the values of the PRNG state backwards or forwards through the high-entropy refreshes. Unfortunately, we show that this intuition is not correct: we are also able to construct robust PRNGs with input that are backdoored in a backwards sense. Namely, given a single output, Big Brother is able to rewind through a number of refresh operations to earlier ``phases\u27\u27, and recover all the generator\u27s outputs in those earlier phases. Finally, and ending on a positive note, we give an impossibility result: we provide a bound on the number of previous phases that Big Brother can compromise as a function of the state-size of the generator: smaller states provide more limited backdooring opportunities for Big Brother

Populating the Zoo of Rugged Pseudorandom Permutations

A Rugged Pseudorandom Permutation (RPRP) is a variable-input-length tweakable cipher satisfying a security notion that is intermediate between tweakable PRP and tweakable SPRP. It was introduced at CRYPTO 2022 by Degabriele and Karadžić, who additionally showed how to generically convert such a primitive into nonce-based and nonce-hiding AEAD schemes satisfying either misuse-resistance or release-of-unverified-plaintext security as well as Nonce-Set AEAD which has applications in protocols like QUIC and DTLS. Their work shows that RPRPs are powerful and versatile cryptographic primitives. However, the RPRP security notion itself can seem rather contrived, and the motivation behind it is not immediately clear. Moreover, they only provided a single RPRP construction, called UIV, which puts into question the generality of their modular approach and whether other instantiations are even possible. In this work, we address this question positively by presenting new RPRP constructions, thereby validating their modular approach and providing further justification in support of the RPRP security definition. Furthermore, we present a more refined view of their results by showing that strictly weaker RPRP variants, which we introduce, suffice for many of their transformations. From a theoretical perspective, our results show that the well-known three-round Feistel structure achieves stronger security as a permutation than a mere pseudorandom permutation---as was established in the seminal result by Luby and Rackoff. We conclude on a more practical note by showing how to extend the left domain of one RPRP construction for applications that require larger values in order to meet the desired level of security

Overloading the Nonce: Rugged PRPs, Nonce-Set AEAD, and Order-Resilient Channels

We introduce a new security notion that lies right in between pseudorandom permutations (PRPs) and strong pseudorandom permutations (SPRPs). We call this new security notion and any (tweakable) cipher that satisfies it a $\textit{rugged pseudorandom permutation}$ (RPRP). Rugged pseudorandom permutations lend themselves to some interesting applications, have practical benefits, and lead to novel cryptographic constructions. Our focus is on variable-length tweakable RPRPs, and analogous to the encode-then-encipher paradigm of Bellare and Rogaway, we can generically transform any such cipher into different AEAD schemes with varying security properties. However, the benefit of RPRPs is that they can be constructed more efficiently as they are weaker primitives than SPRPs (the notion traditionally required by the encode-then-encipher paradigm). We can construct RPRPs using only two layers of processing, whereas SPRPs typically require three layers of processing over the input data. We also identify a new transformation that yields RUP-secure AEAD schemes with more compact ciphertexts than previously known. Further extending this approach, we arrive at a new generalized notion of authenticated encryption and a matching construction, which we refer to as $\textit{nonce-set AEAD}$. Nonce-set AEAD is particularly well-suited in the context of secure channels, like QUIC and DTLS, that operate over unreliable transports and employ a window mechanism at the receiver\u27s end of the channel. We conclude by presenting a generic construction for transforming a nonce-set AEAD scheme into an order-resilient secure channel. Our channel construction sheds new light on order-resilient channels and additionally leads to more compact ciphertexts when instantiated from RPRPs

A More Cautious Approach to Security Against Mass Surveillance

At CRYPTO 2014 Bellare, Paterson, and Rogaway (BPR) presented a formal treatment of symmetric encryption in the light of algorithm substitution attacks (ASAs), which may be employed by `big brother\u27 entities for the scope of mass surveillance. Roughly speaking, in ASAs big brother may bias ciphertexts to establish a covert channel to leak vital cryptographic information. In this work, we identify a seemingly benign assumption implicit in BPR\u27s treatment and argue that it artificially (and severely) limits big brother\u27s capabilities. We then demonstrate the critical role that this assumption plays by showing that even a slight weakening of it renders the security notion completely unsatisfiable by any, possibly deterministic and/or stateful, symmetric encryption scheme. We propose a refined security model to address this shortcoming, and use it to restore the positive result of BPR, but caution that this defense does not stop most other forms of covert-channel attacks

The Security of ChaCha20-Poly1305 in the Multi-user Setting

The ChaCha20-Poly1305 AEAD scheme is being increasingly widely deployed in practice. Practitioners need proven security bounds in order to set data limits and rekeying intervals for the scheme. But the formal security analysis of ChaCha20-Poly1305 currently lags behind that of AES-GCM. The only extant analysis (Procter, 2014) contains a flaw and is only for the single-user setting. We rectify this situation. We prove a multi-user security bound on the AEAD security of ChaCha20-Poly1305 and establish the tightness of each term in our bound through matching attacks. We show how our bound differs both qualitatively and quantitatively from the known bounds for AES-GCM, highlighting how subtle design choices lead to distinctive security properties. We translate our bound to the nonce-randomized setting employed in TLS 1.3 and elsewhere, and we additionally improve the corresponding security bounds for GCM. Finally, we provide a simple yet stronger variant of ChaCha20-Poly1305 that addresses the deficiencies highlighted by our analysis

Lightweight Authenticated Encryption Mode Suitable for Threshold Implementation

This paper proposes tweakable block cipher (TBC) based modes $\mathsf{PFB\_Plus}$ and $\mathsf{PFB}\omega$ that are efficient in threshold implementations (TI). Let $t$ be an algebraic degree of a target function, e.g.~$t=1$ (resp.~$t>1$) for linear (resp.~non-linear) function. The $d$-th order TI encodes the internal state into $d t + 1$ shares. Hence, the area size increases proportionally to the number of shares. This implies that TBC based modes can be smaller than block cipher (BC) based modes in TI because TBC requires $s$-bit block to ensure $s$-bit security, e.g. \textsf{PFB} and \textsf{Romulus}, while BC requires $2s$-bit block. However, even with those TBC based modes, the minimum we can reach is 3 shares of $s$-bit state with $t=2$ and the first-order TI ($d=1$). Our first design $\mathsf{PFB\_Plus}$ aims to break the barrier of the $3s$-bit state in TI. The block size of an underlying TBC is $s/2$ bits and the output of TBC is linearly expanded to $s$ bits. This expanded state requires only 2 shares in the first-order TI, which makes the total state size $2.5s$ bits. We also provide rigorous security proof of $\mathsf{PFB\_Plus}$. Our second design $\mathsf{PFB}\omega$ further increases a parameter $\omega$: a ratio of the security level $s$ to the block size of an underlying TBC. We prove security of $\mathsf{PFB}\omega$ for any $\omega$ under some assumptions for an underlying TBC and for parameters used to update a state. Next, we show a concrete instantiation of $\mathsf{PFB\_Plus}$ for 128-bit security. It requires a TBC with 64-bit block, 128-bit key and 128-bit tweak, while no existing TBC can support it. We design a new TBC by extending \textsf{SKINNY} and provide basic security evaluation. Finally, we give hardware benchmarks of $\mathsf{PFB\_Plus}$ in the first-order TI to show that TI of $\mathsf{PFB\_Plus}$ is smaller than that of \textsf{PFB} by more than one thousand gates and is the smallest within the schemes having 128-bit security