Combiners for Backdoored Random Oracles

International audienceWe formulate and study the security of cryptographic hash functions in the backdoored random-oracle (BRO) model, whereby a big brother designs a "good" hash function, but can also see arbitrary functions of its table via backdoor capabilities. This model captures intentional (and unintentional) weaknesses due to the existence of collision-finding or inversion algorithms, but goes well beyond them by allowing, for example, to search for structured preimages. The latter can easily break constructions that are secure under random inversions. BROs make the task of bootstrapping cryptographic hardness somewhat challenging. Indeed, with only a single arbitrarily backdoored function no hardness can be bootstrapped as any construction can be inverted. However, when two (or more) independent hash functions are available, hardness emerges even with unrestricted and adaptive access to all backdoor oracles. At the core of our results lie new reductions from cryptographic problems to the communication complexities of various two-party tasks. Along the way we establish a communication complexity lower bound for set-intersection for cryptographically relevant ranges of parameters and distributions and where set-disjointness can be easy

Data Structures Meet Cryptography: 3SUM with Preprocessing

This paper shows several connections between data structure problems and cryptography against preprocessing attacks. Our results span data structure upper bounds, cryptographic applications, and data structure lower bounds, as summarized next. First, we apply Fiat--Naor inversion, a technique with cryptographic origins, to obtain a data structure upper bound. In particular, our technique yields a suite of algorithms with space $S$ and (online) time $T$ for a preprocessing version of the $N$-input 3SUM problem where $S^3\cdot T = \widetilde{O}(N^6)$. This disproves a strong conjecture (Goldstein et al., WADS 2017) that there is no data structure that solves this problem for $S=N^{2-\delta}$ and $T = N^{1-\delta}$ for any constant $\delta>0$. Secondly, we show equivalence between lower bounds for a broad class of (static) data structure problems and one-way functions in the random oracle model that resist a very strong form of preprocessing attack. Concretely, given a random function $F: [N] \to [N]$ (accessed as an oracle) we show how to compile it into a function $G^F: [N^2] \to [N^2]$ which resists $S$-bit preprocessing attacks that run in query time $T$ where $ST=O(N^{2-\varepsilon})$ (assuming a corresponding data structure lower bound on 3SUM). In contrast, a classical result of Hellman tells us that $F$ itself can be more easily inverted, say with $N^{2/3}$-bit preprocessing in $N^{2/3}$ time. We also show that much stronger lower bounds follow from the hardness of kSUM. Our results can be equivalently interpreted as security against adversaries that are very non-uniform, or have large auxiliary input, or as security in the face of a powerfully backdoored random oracle. Thirdly, we give non-adaptive lower bounds for 3SUM and a range of geometric problems which match the best known lower bounds for static data structure problems

Immunizing Backdoored PRGs

A backdoored Pseudorandom Generator (PRG) is a PRG which looks pseudorandom to the outside world, but a saboteur can break PRG security by planting a backdoor into a seemingly honest choice of public parameters, $pk$, for the system. Backdoored PRGs became increasingly important due to revelations about NIST’s backdoored Dual EC PRG, and later results about its practical exploitability. Motivated by this, at Eurocrypt\u2715 Dodis et al. [21] initiated the question of immunizing backdoored PRGs. A $k$-immunization scheme repeatedly applies a post-processing function to the output of $k$ backdoored PRGs, to render any (unknown) backdoors provably useless. For $k=1$, [21] showed that no deterministic immunization is possible, but then constructed seeded $1$-immunizer either in the random oracle model, or under strong non-falsifiable assumptions. As our first result, we show that no seeded $1$-immunization scheme can be black-box reduced to any efficiently falsifiable assumption. This motivates studying $k$-immunizers for $k\ge 2$, which have an additional advantage of being deterministic (i.e., seedless ). Indeed, prior work at CCS\u2717 [37] and CRYPTO\u2718 [7] gave supporting evidence that simple $k$-immunizers might exist, albeit in slightly different settings. Unfortunately, we show that simple standard model proposals of [37, 7] (including the XOR function [7]) provably do not work in our setting. On a positive, we confirm the intuition of [37] that a (seedless) random oracle is a provably secure $2$-immunizer. On a negative, no (seedless) $2$-immunization scheme can be black-box reduced to any efficiently falsifiable assumption, at least for a large class of natural $2$-immunizers which includes all cryptographic hash functions. In summary, our results show that $k$-immunizers occupy a peculiar place in the cryptographic world. While they likely exist, and can be made practical and efficient, it is unlikely one can reduce their security to a clean standard-model assumption

Black-Box Uselessness: Composing Separations in Cryptography

Black-box separations have been successfully used to identify the limits of a powerful set of tools in cryptography, namely those of black-box reductions. They allow proving that a large set of techniques are not capable of basing one primitive ? on another ?. Such separations, however, do not say anything about the power of the combination of primitives ??,?? for constructing ?, even if ? cannot be based on ?? or ?? alone. By introducing and formalizing the notion of black-box uselessness, we develop a framework that allows us to make such conclusions. At an informal level, we call primitive ? black-box useless (BBU) for ? if ? cannot help constructing ? in a black-box way, even in the presence of another primitive ?. This is formalized by saying that ? is BBU for ? if for any auxiliary primitive ?, whenever there exists a black-box construction of ? from (?,?), then there must already also exist a black-box construction of ? from ? alone. We also formalize various other notions of black-box uselessness, and consider in particular the setting of efficient black-box constructions when the number of queries to ? is below a threshold. Impagliazzo and Rudich (STOC\u2789) initiated the study of black-box separations by separating key agreement from one-way functions. We prove a number of initial results in this direction, which indicate that one-way functions are perhaps also black-box useless for key agreement. In particular, we show that OWFs are black-box useless in any construction of key agreement in either of the following settings: (1) the key agreement has perfect correctness and one of the parties calls the OWF a constant number of times; (2) the key agreement consists of a single round of interaction (as in Merkle-type protocols). We conjecture that OWFs are indeed black-box useless for general key agreement. We also show that certain techniques for proving black-box separations can be lifted to the uselessness regime. In particular, we show that the lower bounds of Canetti, Kalai, and Paneth (TCC\u2715) as well as Garg, Mahmoody, and Mohammed (Crypto\u2717 & TCC\u2717) for assumptions behind indistinguishability obfuscation (IO) can be extended to derive black-box uselessness of a variety of primitives for obtaining (approximately correct) IO. These results follow the so-called "compiling out" technique, which we prove to imply black-box uselessness. Eventually, we study the complementary landscape of black-box uselessness, namely black-box helpfulness. We put forth the conjecture that one-way functions are black-box helpful for building collision-resistant hash functions. We define two natural relaxations of this conjecture, and prove that both of these conjectures are implied by a natural conjecture regarding random permutations equipped with a collision finder oracle, as defined by Simon (Eurocrypt\u2798). This conjecture may also be of interest in other contexts, such as amplification of hardness

Algorithm Substitution Attacks against Receivers

This work describes a class of Algorithm Substitution Attack (ASA) generically targeting the receiver of a communication between two parties. Our work provides a unified framework that applies to any scheme where a secret key is held by the receiver; in particular, message authentication schemes (MACs), authenticated encryption (AEAD) and public key encryption (PKE). Our unified framework brings together prior work targeting MAC schemes and AEAD schemes; we extend prior work by showing that public key encryption may also be targeted. ASAs were initially introduced by Bellare, Paterson and Rogaway in light of revelations concerning mass surveillance, as a novel attack class against the confidentiality of encryption schemes. Such an attack replaces one or more of the regular scheme algorithms with a subverted version that aims to reveal information to an adversary (engaged in mass surveillance), while remaining undetected by users. Previous work looking at ASAs against encryption schemes can be divided into two groups. ASAs against PKE schemes target key generation by creating subverted public keys that allow an adversary to recover the secret key. ASAs against symmetric encryption target the encryption algorithm and leak information through a subliminal channel in the ciphertexts. We present a new class of attack that targets the decryption algorithm of an encryption scheme for symmetric encryption and public key encryption, or the verification algorithm for an authentication scheme. We present a generic framework for subverting a cryptographic scheme between a sender and receiver, and show how a decryption oracle allows a subverter to create a subliminal channel which can be used to leak secret keys. We then show that the generic framework can be applied to authenticated encryption with associated data, message authentication schemes, public key encryption and KEM/DEM constructions. We consider practical considerations and specific conditions that apply for particular schemes, strengthening the generic approach. Furthermore, we show how the hybrid subversion of key generation and decryption algorithms can be used to amplify the effectiveness of our decryption attack. We argue that this attack represents an attractive opportunity for a mass surveillance adversary. Our work serves to refine the ASA model and contributes to a series of papers that raises awareness and understanding about what is possible with ASAs

Subvert KEM to Break DEM: Practical Algorithm-Substitution Attacks on Public-Key Encryption

Motivated by the currently widespread concern about mass surveillance of encrypted communications, Bellare \emph{et al.} introduced at CRYPTO 2014 the notion of Algorithm-Substitution Attack (ASA) where the legitimate encryption algorithm is replaced by a subverted one that aims to undetectably exfiltrate the secret key via ciphertexts. Practically implementable ASAs on various cryptographic primitives (Bellare \emph{et al.}, CRYPTO\u2714 \& ACM CCS\u2715; Ateniese \emph{et al.}, ACM CCS\u2715; Berndt and Liśkiewicz, ACM CCS\u2717) have been constructed and analyzed, leaking the secret key successfully. Nevertheless, in spite of much progress, the practical impact of ASAs (formulated originally for symmetric key cryptography) on public-key (PKE) encryption operations remains unclear, primarily since the encryption operation of PKE does not involve the secret key, and also previously known ASAs become relatively inefficient for leaking the plaintext due to the logarithmic upper bound of exfiltration rate (Berndt and Liśkiewicz, ACM CCS\u2717). In this work, we formulate a practical ASA on PKE encryption algorithm which, perhaps surprisingly, turns out to be much more efficient and robust than existing ones, showing that ASAs on PKE schemes are far more effective and dangerous than previously believed. We mainly target PKE of hybrid encryption which is the most prevalent way to employ PKE in the literature and in practice. The main strategy of our ASA is to subvert the underlying key encapsulation mechanism (KEM) so that the session key encapsulated could be efficiently extracted, which, in turn, breaks the data encapsulation mechanism (DEM) enabling us to learn the plaintext itself. Concretely, our non-black-box yet quite general attack enables recovering the plaintext from only two successive ciphertexts and minimally depends on a short state of previous internal randomness. A widely used class of KEMs is shown to be subvertible by our powerful attack. Our attack relies on a novel identification and formalization of certain properties that yield practical ASAs on KEMs. More broadly, it points at and may shed some light on exploring structural weaknesses of other ``composed cryptographic primitives,\u27\u27 which may make them susceptible to more dangerous ASAs with effectiveness that surpasses the known logarithmic upper bound (i.e., reviewing composition as an attack enabler)

Towards Applying Cryptographic Security Models to Real-World Systems

The cryptographic methodology of formal security analysis usually works in three steps: choosing a security model, describing a system and its intended security properties, and creating a formal proof of security. For basic cryptographic primitives and simple protocols this is a well understood process and is performed regularly. For more complex systems, as they are in use in real-world settings it is rarely applied, however. In practice, this often leads to missing or incomplete descriptions of the security properties and requirements of such systems, which in turn can lead to insecure implementations and consequent security breaches. One of the main reasons for the lack of application of formal models in practice is that they are particularly difficult to use and to adapt to new use cases. With this work, we therefore aim to investigate how cryptographic security models can be used to argue about the security of real-world systems. To this end, we perform case studies of three important types of real-world systems: data outsourcing, computer networks and electronic payment. First, we give a unified framework to express and analyze the security of data outsourcing schemes. Within this framework, we define three privacy objectives: \emph{data privacy}, \emph{query privacy}, and \emph{result privacy}. We show that data privacy and query privacy are independent concepts, while result privacy is consequential to them. We then extend our framework to allow the modeling of \emph{integrity} for the specific use case of file systems. To validate our model, we show that existing security notions can be expressed within our framework and we prove the security of CryFS---a cryptographic cloud file system. Second, we introduce a model, based on the Universal Composability (UC) framework, in which computer networks and their security properties can be described We extend it to incorporate time, which cannot be expressed in the basic UC framework, and give formal tools to facilitate its application. For validation, we use this model to argue about the security of architectures of multiple firewalls in the presence of an active adversary. We show that a parallel composition of firewalls exhibits strictly better security properties than other variants. Finally, we introduce a formal model for the security of electronic payment protocols within the UC framework. Using this model, we prove a set of necessary requirements for secure electronic payment. Based on these findings, we discuss the security of current payment protocols and find that most are insecure. We then give a simple payment protocol inspired by chipTAN and photoTAN and prove its security within our model. We conclude that cryptographic security models can indeed be used to describe the security of real-world systems. They are, however, difficult to apply and always need to be adapted to the specific use case