Correcting Subverted Random Oracles

The random oracle methodology [7] has proven to be a powerful tool for designing and reasoning about cryptographic schemes. It consists of the following two steps: (i) design a scheme \(\varPi \) in which all parties (including the adversary) have oracle access to a common truly random function, and establish the security of \(\varPi \) in this favorable setting; (ii) instantiate the random oracle in \(\varPi \) with a suitable cryptographic hash function (such as SHA256) to obtain an instantiated scheme \(\varPi '\). The random oracle heuristic states that if the original scheme \(\varPi \) is secure, then the instantiated scheme \(\varPi '\) is also secure. While this heuristic can fail in various settings [19] the basic framework remains a fundamental design and analysis tool. In this work we focus on the problem of correcting faulty—or adversarially corrupted—random oracles so that they can be confidently applied for such cryptographic purposes.

Specifically, given a function \(\tilde{h}\) drawn from a distribution which agrees in most places with a uniform function, we would like to produce a corrected version that has stronger uniformity properties. Our problem shares some features with the classical “self-checking and self-correcting program” paradigm [9,10,11]: we wish to transform a program that is faulty at a small fraction of inputs (modeling an evasive adversary) to a program that is correct at all points. In this light, our model can be viewed as an adaptation of the classical theory that considers the problem of “self-correcting a probability distribution.” Notably, in our setting the functions to be corrected are structureless—specifically, drawn from the uniform distribution—rather than heavily structured. Despite that, the basic procedure for correction and portions of the technical development are analogous.

One particular motivation for correcting random oracles in a cryptographic context arises from recent work studying security in the kleptographic setting. In this setting, the various components of a cryptographic scheme may be subverted by an adversary so long as the tampering cannot be detected via blackbox testing. This is a challenging setting for a number of reasons highlighted by  [6, 49, 50]: one particular difficulty is that the random oracle paradigm is directly undermined. In terms of the discussion above, the random oracle—which is eventually to be replaced with a concrete function—is subject to adversarial subversion which complicates even the first step (i) of the random oracle methodology above. Our goal is to provide a generic approach that can rigorously “protect” random oracles from subversion.

1.1 Our Contributions

We first give two concrete attacking scenarios where hash functions are subverted in the kleptographic setting. We then express the security properties by adapting the successful framework of indifferentiability [23, 41] to our setting with adversarial subversion. This framework provides a satisfactory guarantee of modularity—that is, that the resulting object can be directly employed by other constructions demanding a random oracle. We call this new notion “crooked” indifferentiability to reflect the role of adversary in the modeling; see below. (A formal definition appears in Sect. 2.)

We prove that a simple construction involving only public randomness can boost a “subverted” random oracle into a construction that is indifferentiable from a random function (Sects. 3 and 4). We remark that our technical development establishes a novel “rejection re-sampling” lemma, controlling the distribution emerging from adversarial re-sampling of product distributions. This may be a technique of independent interest. We expand on these contributions below.

Consequences of kleptographic hash subversion. We first illustrate the damages that are caused by using hash functions that are subverted at only a negligible fraction of inputs with two concrete examples:

(1) Chain take-over attack on blockchain. For simplicity, consider a proof-of-work blockchain setting where miners compete to find a solution s to the “puzzle” \(h(\textsf {pre}||\textsf {transactions}||s) \le d\), where pre denotes the hash of previous block, transactions denotes the set of valid transactions in the current block, and d denotes the difficulty parameter. Here h is intended to be a strong hash function. Note that, the mining machines use a program \(\tilde{h}(\cdot )\) (or a dedicated hardware module) which could be designed by a clever adversary. Now if \(\tilde{h}\) has been subverted so that \(\tilde{h}(*||z)=0\) for a randomly chosen z—and \(\tilde{h}(x)=h(x)\) in all other cases—this will be difficult to detect by prior black-box testing; on the other hand, the adversary who created \(\tilde{h}\) has the luxury of solving the proof of work without any effort for any challenge, and thus can completely control the blockchain. (A fancier subversion can tune the “backdoor” z to other parts of the input so that it cannot be reused by other parties; e.g., \(\tilde{h}(w||z) = 0\) if \(z=f(w)\) for a secret pseudorandom function known to the adversary.)

(2) System sneak-in attack on password authentication. In Unix-style system, during system initialization, the root user chooses a master password \(\alpha \) and the system stores the digest \(\rho =h(\alpha )\), where h is a given hash function normally modeled as a random oracle. During login, the operating system receives input x and accepts this password if \(h(x) =\rho \). An attractive feature of this practice is that it is still secure if \(\rho \) is accidentally leaked. In the presence of kleptographic attacks, however, the module that implements the hash function h may be strategically subverted, yielding a new function \(\tilde{h}\) which destroys the security of the scheme above: for example, the adversary may choose a relatively short random string z and define \(\tilde{h}(y) = h(y)\) unless y begins with z, in which case \(\tilde{h}(zx) = x\). As above, h and \(\tilde{h}\) are indistinguishable by black-box testing; on the other hand, the adversary can login as the system administrator using \(\rho \) and its knowledge of the backdoor z (without knowing the actual password \(\alpha \), presenting \(z \rho \) instead).

The model of “crooked” indifferentiability. The problem of cleaning defective randomness has a long history in computer science. Our setting requires that the transformation must be carried out by a local rule and involve an exponentially small amount of public randomness (in the sense that we wish to clean a defective random function \(h: \{0,1\}^n \rightarrow \{0,1\}^n\) with only a polynomial length random string). The basic framework of correcting a subverted random oracle is the following:

First, a function \(h: \{0,1\}^n \rightarrow \{0,1\}^n\) is drawn uniformly at random. Then, an adversary may subvert the function h, yielding a new function \(\tilde{h}\). The subverted function \(\tilde{h}(x)\) is described by an adversarially-chosen (polynomial-time) algorithm \(\tilde{H}^h(x)\), with oracle access to h. We insist that \(\tilde{h}(x) \ne h(x)\) only at a negligible fraction of inputs.^{Footnote 1} Next, the function \(\tilde{h}\) is “publicly corrected” to a function \(\tilde{h}_R\) (defined below) that involves some public randomness R selected after \(\tilde{h}\) is supplied.^{Footnote 2}

We wish to show that the resulting function (construction) is “as good as” a random oracle, in the sense of indifferentiability. We say a construction \( C ^H\) (having oracle access to an ideal primitive H) is indifferentiable from another ideal primitive \(\mathcal {F} \), if there exists a simulator \(\mathcal {S} \) so that \(( C ^H,H)\) and \((\mathcal {F},\mathcal {S})\) are indistinguishable to any distinguisher \(\mathcal {D} \).

To reflect our setting, an H-crooked-distinguisher \({\widehat{\mathcal {D}}}\) is introduced; the H-crooked-distinguisher \({\widehat{\mathcal {D}}}\) first prepares the subverted implementation \(\tilde{H}\) (after querying H first); then a fixed amount of (public) randomness R is drawn and published; the construction \( C \) uses only subverted implementation \(\tilde{H}\) and R. Now following the indifferentiability framework, we will ask for a simulator \(\mathcal {S} \), such that \(( C ^{\tilde{H}^H}(\cdot ,R),H)\) and \((\mathcal {F},\mathcal {S} ^{\tilde{H}}(R))\) are indistinguishable to any H-crooked-distinguisher \(\widehat{\mathcal {D}} \) who even knows R. A similar security preserving theorem [23, 41] also holds in our model. See Sect. 2 for details.

The construction. The construction depends on a parameter \(\ell = \mathrm {poly} (n)\) and public randomness \(R = (r_1, \ldots , r_\ell )\), where each \(r_i\) is an independent and uniform element of \(\{0,1\}^n\). For simplicity, the construction relies on a family of independent random oracles \(h_i(x)\), for \(i \in \{0, \ldots , \ell \}\). (Of course, these can all be extracted from a single random oracle with slightly longer inputs by defining \(\tilde{h}_i(x) = \tilde{h}(i,x)\) and treating the output of \(h_i(x)\) as n bits long.) Then we define

Note that the adversary is permitted to subvert the function(s) \(h_i\) by choosing an algorithm \(H^{h_*}(x)\) so that \(\tilde{h}_i(x) = H^{h_*}(i,x)\). Before diving into the analysis, let us first quick demonstrate how some simpler constructions fail.

Simple constructions and their shortcomings. Although during the stage of manufacturing the hash functions \(\tilde{h}_*=\{\tilde{h}_i\}_{i=0}^\ell \), the randomness \(R:=r_1,\ldots ,r_\ell \) are not known to the adversary, they become public in the second query phase. If the “mixing” operation is not carefully designed, the adversary could choose inputs accordingly, trying to “peel off” R. We discuss a few examples:

1. 1.

   \(\tilde{h}_R(x)\) is simply defined as \(\tilde{h}_1(x\oplus r_1)\). A straightforward attack is as follows: the adversary can subvert \(h_1\) in a way that \(\tilde{h}_1(m)=0\) for a random input m; the adversary then queries \(m\oplus r_1\) on \(\tilde{h}_R(\cdot )\) and can trivially distinguish \(\tilde{h}\) from a random function.

2. 2.

   \(\tilde{h}_R(x)\) is defined as \(\tilde{h}_1(x\oplus r_1) \oplus \tilde{h}_2 (x\oplus r_2)\). Now a slightly more complex attack can still succeed: the adversary subverts \(h_1\) so that \(\tilde{h}_1(x)=0\) if \(x=m||*\), that is, when the first half of x equals to a randomly selected string m with length n / 2; likewise, \(h_2\) is subverted so that \(\tilde{h}_2(x)=0\) if \(x=*||m\), that is, the second half of x equals m. Then, the adversary queries \(m_1||m_2\) on \(\tilde{h}_R(\cdot )\), where \(m_1=m\oplus r_{1,0}\), and \(m_2=m\oplus r_{2,1}\), and \(r_{1,0}\) is the first half of \(r_1\), and \(r_{2,1}\) is the second half of \(r_2\). Again, trivially, it can be distinguished from a random function.

   This attack can be generalized in a straightforward fashion to any \(\ell \le n/\lambda \): the input can be divided in into consecutive substrings each with length \(\lambda \), and the “trigger” substrings can be planted in each chunk.

Challenges in the analysis. To analyze security in the “crooked” indifferentiability framework, our simulator needs to ensure consistency between two ways of generating output values: one is directly from the construction \(C^{\tilde{H}^h}(x,R)\); the other calls for an “explanation” of F—a truly random function—via reconstruction from related queries to H (in a way consistent with the subverted implementation \(\tilde{H}\)). To ensure a correct simulation, the simulator must suitably answer related queries (defining one value of \(C^{\tilde{H}^h}(x,R)\)). We develop a theorem establishing an unpredictability property of the internal function \(\tilde{g}_R(x)\) to guarantee the success of simulation. In particular, we prove that for any input x (if not yet “fully decided” by previous queries), the output of \(\tilde{g}_R(x)\) is unpredicatable to the distinguisher even if she knows the public randomness R (even conditioned on adaptive queries generated by \(\widehat{\mathcal {D}} \)).

Section 4 develops the detailed security analysis for the property of the internal function \(\tilde{g}_R(x)\). The proof of correctness for this construction is complicated by the fact that the “defining” algorithm \(\tilde{H}\) is permitted to make adaptive queries to h during the definition of \(\tilde{h}\); in particular, this means that even when a particular “constellation” of points (the \(h_i(x \oplus r_i)\)) contains a point that is left alone by \(\tilde{H}\) (which is to say that it agrees with \(h_i()\)) there is no guarantee that \(\bigoplus _i h_i(x \oplus r_i)\) is uniformly random. This suggests focusing the analysis on demonstrating that the constellation associated with every \(x \in \{0,1\}^n\) will have at least one “good” component, which is (i.) not queried by \(\tilde{H}^h(\cdot )\) when evaluated on the other terms, and (ii.) answered honestly. Unfortunately, actually identifying such a good point with certainty appears to require that we examine all of the points in the constellation for x, and this interferes with the standard “exposure martingale” proof that is so powerful in the random oracle setting (which capitalizes on the fact that “unexamined” values of h can be treated as independent and uniform values).

To sidestep this difficulty, we prove a “resampling” lemma, which lets us examine all points in a particular constellation, identify one “good” one of interest, and then resample this point so as to “forget” about all possible conditioning this value might have. The resampling lemma gives a precise bound on the effects of such conditioning.

Immediate applications: Our correction function can be easily applied to save the faulty hash implementation in several important application scenarios, as explained in the motivational examples.

1. (1)

   For proof-of-work based blockchains, as discussed above, miners may rely on a common library \(\tilde{h}\) for the hash evaluation, perhaps cleverly implemented by an adversary. Here \(\tilde{h}\) is determined before the chain has been deployed. We can then prevent the adversary from capitalizing on this subversion by applying our correction function. In particular, the public randomness R can be embedded in the genesis block; the function \(\tilde{h}_R(\cdot )\) is then used for mining (and verification) rather than \(\tilde{h}\).

2. (2)

   The system sneak-in can also be resolved immediately by applying our correcting random oracle. During system initialization (or even when the operating system is released), the system administrator generates some randomness R and wraps the hash module \(\tilde{h}\) (potentially subverted) to define \(\tilde{h}_R(\cdot )\). The password \(\alpha \) then gives rise to the digest \(\rho =\tilde{h}_R(\alpha )\) together with the randomness R. Upon receiving input x, the system first “recovers” \(\tilde{h}_R(\cdot )\) based on the previously stored R, and then tests if \(\rho =\tilde{h}_R(x)\). The access will be enabled if the test is valid. As the corrected random oracle ensures the output to be uniform for every input point, this remains secure in the face of subversion.^{Footnote 3}

1.2 Related Work

Related work on indifferentiability. The notion of indifferentiability was proposed by Maurer et al. [41], as an extension of the classical concept of indistinguishability when one or more oracles are publicly available (such as a random oracle). It was later adapted by Coron et al. [23] and generalized to several other variants in [29, 33, 34, 47, 53]. Notably, a line of elegant work demonstrated the equivalence of the random oracle model and the ideal cipher model; in particular, the Feistel construction (with a small constant number of rounds) is indifferentiable from an ideal cipher, see [24,25,26]. Our work adapts the indifferentiability framework to the setting where the construction uses only a subverted implementation, which we call “crooked indifferentiability,” where the construction aims to be indifferentiable from another repaired random oracle.

Related work on self-correcting programs. The theory of program self-testing, and self-correcting, was pioneered by the work of Blum et al. [9,10,11]. This theory addresses the basic problem of program correctness by verifying relationships between the outputs of the program on randomly selected inputs; a similar problem is to turn an almost correct program into one that is correct at every point with overwhelming probability. Rubinfeld’s thesis [48] is an authoritative survey of the basic framework and results. Our results can be seen as a distributional analogue of this theory but with two main differences: (i). we insist on using only public randomness drawn once for the entire “correction”; (ii). our target object is a distribution, rather than a particular function.

Related work on random oracles. The random oracle methodology [7] can significantly simplify both cryptographic constructions and proofs, even though there exist schemes which are secure using random oracles, but cannot be instantiated in the standard model, [19]. On the other hand, efforts have been made to identify instantiable assumptions/models in which we may analyze interesting cryptographic tasks [4, 12,13,14, 16, 18, 20, 39]. Also, we note that research efforts have also been made to investigate weakened idealized models [37, 38, 40, 45]. Finally, there are several recent nice works about random oracle in the auxiliary input model (or with pre-processing) [31, 51]. Our model shares some similarities that the adversary may embed some preprocessed information into the subverted implementation, but our subverted implementation can further misbehave. Our results strengthen the random oracle methodology in the sense that using our construction, we can even tolerate a faulty hash implementation.

Related work on kleptographic security. Kleptographic attacks were originally introduced by Young and Yung [54, 55]; In such attacks, the adversary provides subverted implementations of the cryptographic primitive, trying to learn secret without being detected. In recent years, several remarkable allegations of cryptographic tampering [42, 46], including detailed investigations [21, 22], have produced a renewed interest in both kleptographic attacks and in techniques for preventing them [1,2,3, 5, 6, 8, 15, 27, 28, 30, 32, 43, 49, 50, 52]. None of those work considered how to actually correct a subverted random oracle.

Concurrently, Fischlin et al. [36] also considered backdoored (keyed) hash functions, and how to immunize them particularly for the settings of HMAC and HKDF. They focused on preserving some weaker property of weak pseudorandomness for the building block of the compression function. We aim at correcting all the properties of a subverted random oracle, and moreover, our correction function can be applied to immunize backdoored public hash functions, which was left open in [36].

Similar constructions in other context. Our construction follows the simple intuition by mixing and input and output by XORing multiple terms. This share similarities in constructions in several other scenarios, e.g., about hardness amplification, notably the famous Yao XOR lemma, and for weak PRF [44]; and randomizers in the bounded storage model [35]. Our construction has to have an external layer of \(h_0\) to wrap the XOR of terms, and our analysis is very different from them due to that our starting point of a subverted implementation.

2.1 Preliminary: Indifferentiability

The notion of indifferentiability introduced by Maurer et al. [41] has been found very useful for studying the security of hash function and many other primitives, especially model them as idealized objectives. This notion is an extension of the classical notion of indistinguishability, when one or more oracles are publicly available. The indifferentiability notion in [41] is given in the framework of random systems providing interfaces to other systems. Coron et al. [23] demonstrate an equivalent indifferentiability notion for random oracles but in the framework of Interactive Turing Machines (as in [17]). The indifferentiability formulation in this subsection is essentially taken from [23]. In the next subsection, we will introduce our new notion, crooked indifferentiability.

Defining indifferentiability. We consider ideal primitives. An ideal primitive is an algorithmic entity which receives inputs from one of the parties and returns its output immediately to the querying party. We now proceed to the definition of indifferentiability [23, 41]:

Definition 1

(Indifferentiability [23, 41]). A Turing machine \( C \) with oracle access to an ideal primitive \(\mathcal {G} \) is said to be \((t_\mathcal {D},t_\mathcal {S},q,\epsilon )\)-indifferentiable from an ideal primitive \(\mathcal {F} \), if there is a simulator \(\mathcal {S} \), such that for any distinguisher \(\mathcal {D} \), it holds that :

The simulator \(\mathcal {S} \) has oracle access to \(\mathcal {F} \) and runs in time at most \(t_\mathcal {S} \). The distinguisher \(\mathcal {D} \) runs in time at most \(t_\mathcal {D} \) and makes at most q queries. Similarly, \( C ^\mathcal {G} \) is said to be (computationally) indifferentiable from \(\mathcal {F} \) if \(\epsilon \) is a negligible function of the security parameter \(\lambda \) (for polynomially bounded \(t_\mathcal {D} \) and \(t_\mathcal {S} \)). See Fig. 1.

The indifferentiability notion: the distinguisher \(\mathcal {D} \) either interacts with algorithm \( C \) and ideal primitive \(\mathcal {G} \), or with ideal primitive \(\mathcal {F} \) and simulator \(\mathcal {S} \). Algorithm \( C \) has oracle access to \(\mathcal {G} \), while simulator \(\mathcal {S} \) has oracle access to \(\mathcal {F} \).

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As illustrated in Fig. 1, the role of the simulator is to simulate the ideal primitive \(\mathcal {G} \) so that no distinguisher can tell whether it is interacting with \( C \) and \(\mathcal {G} \), or with \(\mathcal {F} \) and \(\mathcal {S} \); in other words, the output of \(\mathcal {S} \) should look “consistent” with what the distinguisher can obtain from \(\mathcal {F} \). Note that normally the simulator does not see the distinguisher’s queries to \(\mathcal {F} \); however, it can call \(\mathcal {F} \) directly when needed for the simulation.

Replacement. It is shown in [41] that if \( C ^\mathcal {G} \) is indifferentiable from \(\mathcal {F} \), then \( C ^\mathcal {G} \) can replace \(\mathcal {F} \) in any cryptosystem, and the resulting cryptosystem is at least as secure in the \(\mathcal {G} \) model as in the \(\mathcal {F} \) model.

We use the definition of [41] to specify what it means for a cryptosystem to be at least as secure in the \(\mathcal {G} \) model as in the \(\mathcal {F} \) model. A cryptosystem is modeled as an Interactive Turing Machine with an interface to an adversary \(\mathcal {A} \) and to a public oracle. The cryptosystem is run by an environment \(\mathcal {E} \) which provides a binary output and also runs the adversary. In the \(\mathcal {G} \) model, cryptosystem \(\mathcal {P} \) has oracle access to \( C \) (which has oracle access to \(\mathcal {G} \)), whereas attacker \(\mathcal {A} \) has oracle access to \(\mathcal {G} \). In the \(\mathcal {F} \) model, both \(\mathcal {P} \) and \(\mathcal {S}_\mathcal {A} \) (the simulator) has direct oracle access to \(\mathcal {F} \). The definition is illustrated in Fig. 2.

The environment \(\mathcal {E} \) interacts with cryptosystem \(\mathcal {P} \) and attacker \(\mathcal {A} \). In the \(\mathcal {G} \) model (left), \(\mathcal {P} \) has oracle access to \( C \) whereas \(\mathcal {A} \) has oracle access to \(\mathcal {G} \). In the \(\mathcal {F} \) model, both \(\mathcal {P} \) and \(\mathcal {S}_\mathcal {A} \) have oracle access to \(\mathcal {F} \).

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Definition 2

A cryptosystem \(\mathcal {P} \) is said to be at least as secure in the \(\mathcal {G} \) model with algorithm \( C \), as in the \(\mathcal {F} \) model, if for any environment \(\mathcal {E} \) and any attacker \(\mathcal {A} \) in the \(\mathcal {G} \) model, there exists an attacker \(\mathcal {S}_\mathcal {A} \) in the \(\mathcal {F} \) model, such that:

where \(\epsilon \) is a negligible function of the security parameter \(\lambda \), and the notation \(\mathcal {E} (\mathcal {P} ^{ C ^{}},\mathcal {A} ^{\mathcal {G}})\) defines the output of \(\mathcal {E} \) after interacting with \(\mathcal {P},\mathcal {A} \) as on the left side of Fig. 2 (similarly we can define the right hand side). Moreover, a cryptosystem is said to be computationally at least as secure, etc., if \(\mathcal {E} \), \(\mathcal {A} \) and \(\mathcal {S}_\mathcal {A} \) are polynomial-time in \(\lambda \).

We have the following security preserving (replacement) theorem, which says that when an ideal primitive is replaced by an indifferentiable one, the security of the “bigger” cryptosystem remains:

Theorem 1

([23, 41]). Let \(\mathcal {P} \) be a cryptosystem with oracle access to an ideal primitive \(\mathcal {F} \). Let \( C \) be an algorithm such that \( C ^{\mathcal {G}}\) is indifferentiable from \(\mathcal {F} \). Then cryptosystem \(\mathcal {P} \) is at least as secure in the \(\mathcal {G} \) model with algorithm \( C \) as in the \(\mathcal {F} \) model.

2.2 Crooked Indifferentiability

The ideal primitives that we focus on in this paper are random oracles. A random oracle [7] is an ideal primitive which provides a random output for each new query, and for the identical input queries the same answer will be given. Next we will formalize a new notion called crooked indifferentiability to characterize subversion. For simplicity, our formalization here is for random oracles, we remark that the formalization can be easily extended for other ideal primitives.

Crooked indifferentiability for random oracles. Let us briefly recall our goal: as mentioned in the Introduction, we are considering to repair a subverted/faulty random oracle, such that the corrected construction can be used as good as a random oracle. It is thus natural to consider the indifferentiability notion. However, we need to adjust the notion to properly model the subversion and to avoid trivial impossibility.

We use H to denote the original random oracle and \(\tilde{H}_z\) to be the subverted implementation (where z could be the potential backdoor hardcoded in the implementation and we often ignore it using \(\tilde{H}\) for simplicity). There will be several modifications to the original indifferentiability notion. (1) The deterministic construction C will have the oracle access to the random oracle via the subverted implementation \(\tilde{H}\), not via the original ideal primitive H; This creates lots of difficulty (and even impossibility) for us to develop a suitable construction. For that reason, the construction is allowed to access to trusted but public randomness r (see Remark 1 below). (2) The simulator will also have the oracle access to the subverted implementation \(\tilde{H}\) and also the public randomness r. Item (2) is necessary as it is clearly impossible to have an indifferentiability definition with a simulator that has no access to \(\tilde{H}\), as the distinguisher can simply make query an input such that \( C \) will use a value that is modified by \(\tilde{H}\) while \(\mathcal {S} \) has no way to reproduce it. More importantly, we will show below that, the security will still be preserved to replace an ideal random oracle with a construction satisfying our definition (with an augmented simulator). We will prove the security preserving (i.e., replacement) theorem from [23, 41] similarly with our adapted notions. (3) To model the whole process of the subversion and correction, we consider a two-stage adversary: subverting and distinguishing. For simplicity, we simply consider them as parts of one distinguisher, and do not use separate the notations and state passing.

Definition 3

(H-crooked indifferentiability). Consider a distinguisher \(\widehat{\mathcal {D}} \) and the following multi-phase real execution. Initially, the distinguisher \(\widehat{\mathcal {D}} \) who has oracle access to ideal primitive H, publishes a subverted implementation of H (denoted as \(\tilde{H}\)). Secondly, a uniformly random string r is sampled and published. Thirdly, a deterministic construction \( C \) is then developed: the construction \( C \) has random string r as input, and has the oracle access to \(\tilde{H}\) (the crooked version of H). Finally, the distinguisher \(\widehat{\mathcal {D}} \), also having random string r as input, and the oracle access to the pair \(( C , H)\), returns a decision bit b. Often, we call \(\widehat{\mathcal {D}} \), the H-crooked-distinguisher.

In addition, consider the corresponding multi-phase ideal execution with the same H-crooked-distinguisher \(\widehat{\mathcal {D}} \). In the ideal execution, ideal primitive \(\mathcal {F} \) is provided. The first two phases are the same (as that in the real execution). In the third phase, a simulator \(\mathcal {S} \) will be developed: the simulator has a random string r as input, and has the oracle access to \(\tilde{H}\), as well as the ideal primitive \(\mathcal {F} \). In the last phase, the H-crooked-distinguisher \(\widehat{\mathcal {D}} \), after having random string r as input, and having the oracle access to an alternative pair \((\mathcal {F}, \mathcal {S})\), returns a decision bit b.

We say that construction \( C \), is \((t_{\widehat{\mathcal {D}}},t_\mathcal {S},q,\epsilon )\)-H-crooked-indifferentiable from ideal primitive \(\mathcal {F} \), if there is a simulator \(\mathcal {S} \) so that for any H-crooked-distinguisher \(\widehat{\mathcal {D}} \), (let u be the coins of \(\widehat{\mathcal {D}} \)), it satisfies that the real execution and the ideal execution are indistinguishable. Specifically,

Here \(H: {{\{0,1\}}^{}} ^\lambda \rightarrow {{\{0,1\}}^{}} ^\lambda \) and \(\mathcal {F}: {{\{0,1\}}^{}} ^k \rightarrow {{\{0,1\}}^{}} ^k\) denote random functions. See Fig. 3 for detailed illustration of the last phase in both real and ideal executions (the distinguishing).

The H-crooked indifferentiability notion: the distinguisher \(\widehat{\mathcal {D}} \), in the first phase, manufactures and publishes a subverted implementation denoted as \(\tilde{H}\), for ideal primitive H; then in the second phase, a random string r is published; after that, in the third phase, construction \( C \), or simulator \(\mathcal {S} \) is developed; the H-crooked-distinguisher \(\widehat{\mathcal {D}} \), in the last phase, either interacting with algorithm \( C \) and ideal primitive H, or with ideal primitive \(\mathcal {F} \) and simulator \(\mathcal {S} \), return a decision bit. Here, algorithm \( C \) has oracle access to \(\tilde{H}\), while simulator \(\mathcal {S} \) has oracle access to \(\mathcal {F} \) and \(\tilde{H}\).

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Remark 1

(The necessity of public randomness). It appears difficult to achieve feasibility without randomness. Intuitively: suppose the corrected hash is represented as \(g\big (\tilde{H}(f(x))\big )\), i.e., \(g(\cdot ), f(\cdot )\) are correction functions, f will be applied to each input x before calling \(\tilde{H}\), and g will be further applied to the corresponding output; then the attacker can plant a trigger using f(z) for a random backdoor z, such that \(\tilde{H}\left( f(z)\right) =0\). It is easy to see that the attacker who has full knowledge of (z, g(0)) and can use this pair to distinguish, as \(\mathcal {F} (z)\) would be a random value that would not hit g(0) with a noticeable probability. Similarly, we can see that it is also infeasible if the randomness is generated before the faulty implementation is provided. For this reason, we allow the model to have a public randomness that is generated after \(\tilde{H}\) is supplied, but such randomness would be available to everyone, including the attacker.

Remark 2

(Comparison with preprocessing). There have been several recent nice works [31, 51] about random oracle with preprocessing, in which the adversary can have some auxiliary input compressing the queries. While in the first phase of our model, we also allow the adversary to generate such an auxiliary string as part of the backdoor z (or part of the instruction of \(\tilde{H}\)). We further allow the crooked implementation to deviate from the original random oracle. In this sense, the preprocessing model for random oracle can be considered to defend against a similar attacker than us, but the attacker would provide an honest implementation (only treating the backdoor as the auxiliary input). We note that their construction using simple salting mechanism [51] cannot correct a subverted random oracle as in our model: the distinguisher plants a trigger z into the inputs that \(\tilde{H}(z||*)=0\) for a randomly chosen z. In this way, the salt would be subsumed into the \(*\) part and has no effect on the faulty implementation.

Remark 3

(Extensions). For simplicity, our definition is mainly for random oracle. It is not very difficult to extend our crooked indifferentiability notion to the other setting such as ideal cipher, as long as we represent the interfaces properly, while the multi-phase executions can be similarly defined. Another interesting extension is to consider a global random oracle (while in the current definition, there would be an independent instance in the real and ideal execution). We leave those interesting questions to be explored in future works.

Replacement with crooked indifferentiability. Security preserving (replacement) has been shown in the indifferentiability framework [41]: if \( C ^\mathcal {G} \) is indifferentiable from \(\mathcal {F} \), then \( C ^\mathcal {G} \) can replace \(\mathcal {F} \) in any cryptosystem, and the resulting cryptosystem in the \(\mathcal {G} \) model is at least as secure as that in the \(\mathcal {F} \) model. We next show that the replacement property can also hold in our crooked indifferentiability framework. Recall that, in the “standard” indifferentiability framework [23, 41], a cryptosystem can be modeled as an Interactive Turing Machine with an interface to an adversary \(\mathcal {A} \) and to a public oracle. There the cryptosystem is run by a “standard” environment \(\mathcal {E} \) (see Fig. 2). In our crooked indifferentiability framework, a cryptosystem also has the interface to an adversary \(\mathcal {A} \) and to a public oracle. However, now the cryptosystem is run by a environment \(\widehat{\mathcal {E}} \) that can “crook” the oracle.

Consider an ideal primitive \(\mathcal {G} \). Similar to the \(\mathcal {G} \)-crooked-distinguisher, we can define the \(\mathcal {G} \)-crooked-environment \(\widehat{\mathcal {E}} \) as follows: Initially, the \(\mathcal {G} \)-crooked-environment \(\widehat{\mathcal {E}} \) manufactures and then publishes a subverted implementation of the ideal primitive \(\mathcal {G} \), denoted as \(\tilde{\mathcal {G}}\). Then \(\widehat{\mathcal {E}} \) runs the attacker \(\mathcal {A} \), and the cryptosystem \(\mathcal {P} \) is developed. In the \(\mathcal {G} \) model, cryptosystem \(\mathcal {P} \) has oracle access to \( C \) whereas attacker \(\mathcal {A} \) has oracle access to \(\mathcal {G} \); note that, \( C \) has oracle access to \(\tilde{\mathcal {G}}\), not directly to \(\mathcal {G} \). In the \(\mathcal {F} \) model, both \(\mathcal {P} \) and \(\mathcal {S}_\mathcal {A} \) (the simulator) have oracle access to \(\mathcal {F} \). Finally, the \(\mathcal {G} \)-crooked-environment \(\widehat{\mathcal {E}} \) returns a binary decision output. The definition is illustrated in Fig. 4.

The environment \(\widehat{\mathcal {E}} \) interacts with cryptosystem \(\mathcal {P} \) and attacker \(\mathcal {A} \). In the \(\mathcal {G} \) model (left), \(\mathcal {P} \) has oracle access to \( C \) (who has oracle access to \(\mathcal {G} \)) whereas \(\mathcal {A} \) has oracle access to \(\mathcal {G} \); In the \(\mathcal {F} \) model, both \(\mathcal {P} \) and \(\mathcal {S}_\mathcal {A} \) have oracle access to \(\mathcal {F} \).

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Definition 4

Consider ideal primitives \(\mathcal {G} \) and \(\mathcal {F} \). A cryptosystem \(\mathcal {P} \) is said to be at least as secure in the \(\mathcal {G} \)-crooked model with algorithm \( C \) as in the \(\mathcal {F} \) model, if for any \(\mathcal {G} \)-crooked-environment \(\widehat{\mathcal {E}} \) and any attacker \(\mathcal {A} \) in the \(\mathcal {G} \)-crooked model, there exists an attacker \(\mathcal {S}_\mathcal {A} \) in the \(\mathcal {F} \) model, such that:

where \(\epsilon \) is a negligible function of the security parameter \(\lambda \), and \(\widehat{\mathcal {E}} (\mathcal {P} ^{ C ^{\tilde{\mathcal {G}}}},\mathcal {A} ^{\mathcal {G}})\) describes the output of \(\widehat{\mathcal {E}} \) running the experiment in the \(\mathcal {G} \)-world (the left side of Fig. 4), and similarly for \(\widehat{\mathcal {E}} (\mathcal {P} ^\mathcal {F},\mathcal {S}_\mathcal {A} ^\mathcal {F})\).

We now demonstrate the following theorem which shows that security is preserved when replacing an ideal primitive by a crooked-indifferentiable one:

Theorem 2

Let \(\mathcal {P} \) be a cryptosystem with oracle access to an ideal primitive \(\mathcal {F} \). Let \( C \) be an algorithm such that \( C ^{\mathcal {G}}\) is crooked-indifferentiable from \(\mathcal {F} \). Then cryptosystem \(\mathcal {P} \) is at least as secure in the \(\mathcal {G} \)-crooked model with algorithm \( C \) as in the \(\mathcal {F} \) model.

Proof

The proof is very similar to that in [23, 41]. Let \(\mathcal {P} \) be any cryptosystem, modeled as an Interactive Turing Machine. Let \(\widehat{\mathcal {E}} \) be any crooked-environment, and \(\mathcal {A} \) be any attacker in the \(\mathcal {G} \)-crooked model. In the \(\mathcal {G} \)-crooked model, \(\mathcal {P} \) has oracle access to \( C \) (who has oracle access to \(\tilde{\mathcal {G}}\), not directly to \(\mathcal {G} \).), whereas \(\mathcal {A} \) has oracle access to ideal primitive \(\mathcal {G} \); moreover, the crooked-environment \(\widehat{\mathcal {E}} \) interacts with both \(\mathcal {P} \) and \(\mathcal {A} \). This is illustrated in Fig. 5 (left part).

Since \( C \) is crooked-indifferentiable from \(\mathcal {F} \) (see Fig. 3), one can replace \(( C ^{\tilde{\mathcal {G}}}, \mathcal {G})\) by \((\mathcal {F}, \mathcal {S})\) with only a negligible modification of the \(\mathcal {G} \)-crooked-environment \(\widehat{\mathcal {E}} \)’s output distribution. As illustrated in Fig. 5, by merging attacker \(\mathcal {A} \) and simulator \(\mathcal {S} \), one obtains an attacker \(\mathcal {S}_\mathcal {A} \) in the \(\mathcal {F} \) model, and the difference in \(\widehat{\mathcal {E}} \)’s output distribution is negligible.    \(\square \)

Construction of attacker \(\mathcal {S}_\mathcal {A} \) from attacker \(\mathcal {A} \) and simulator \(\mathcal {S} \).

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Now we proceed to give the construction. Given subverted implementations of the hash functions \(\{\tilde{h}_i\}_{i=0,\ldots ,\ell }\), (the original version of each is \(h_i(\cdot )\) could be considered as \(h(i,\cdot )\)), the corrected function is defined as:

where \(R=(r_1,\ldots ,r_\ell )\) are sampled uniformly after \(\{\tilde{h}_i(\cdot )\}\) is provided, and then revealed to the public, and the internal function \(\tilde{g}_R(\cdot )\) is defined below:

We wish to show that such a construction will be indifferentiable to an actual random oracle (with the proper input/output length). This implies that the distribution of values taken by \(\tilde{h}_R(\cdot )\) at inputs that have not been queried have negligible distance from the uniform distribution.

Theorem 3

Suppose \(h:\{0,\ldots ,\ell \}\times \{0,1\}^n\rightarrow \{0,1\}^n\) defines a family of random oracles \(h_i:\{0,1\}^n\rightarrow \{0,1\}^n\) as \(h(i,\cdot )\), for \(i=0,\ldots ,\ell \), and \(\ell \ge 3n+1\). Consider a (subversion) algorithm \(\tilde{H}\) and \(\tilde{H}^h(x)\) defines a subverted random oracle \(\tilde{h}\). Assume that for every h (and every i),

The construction \(\tilde{h}_R(\cdot )\) is \((t_{\widehat{\mathcal {D}}},t_S,q,\epsilon )\)-indifferentiable from a random oracle \(F:\{0,1\}^n\rightarrow \{0,1\}^n\), for any \(t_{\widehat{\mathcal {D}}}\), with \(t_S=\mathrm {poly} (q)\), \(\epsilon =\mathsf {negl} (n)\) and q is the number of queries made by the distinguisher \(\widehat{\mathcal {D}} \) as in Definition 3.

Roadmap for the proof. We first describe the simulator algorithm. The main challenge for the simulator is to ensure the consistency of two ways of generating the output values of \(\tilde{h}_R(\cdot )\), (it could also be reconstructed by querying the original random oracle directly together with the subverted implementation \(\tilde{h}\) to replace the potentially corrupted terms). The idea for simulation is fairly simple: for an input x, F(x) would be used to program \(h_0\) on input \(\tilde{g}_R(x)\).

There are two obstacles that hinder the simulation: (1) for some x, \(h_0\) has been queried on \(\tilde{g}_R(x)\) before the actual programing step, thus the simulator has to abort; (2) the distinguisher queries on some input x such that \(\tilde{g}_R(x)\) falls into the incorrect portion of inputs to \(\tilde{h}_0\).

To bound the probability of these two events, we first establish the property of the internal function \(\tilde{g}_R(\cdot )\) that no adversary can find an input value that falls into a small domain (or for any input x, the output is unpredicatable to the adversary if he has not made any related queries.). See Theorem 4 below. Note that the bound is conditioned on adaptive queries of the distinguisher.

Theorem 4

(Informal). Suppose the subverted implementation disagrees with the original oracle at only a negligible fraction of inputs, then with an overwhelming probability in R, conditioned on the \(h(q_1), \ldots , h(q_s)\) (made by any \(\widehat{\mathcal {D}} \)), for all x outside the “queried” set \(\{t \mid h_i(t \oplus r_i)\, \text {was queried}\}\), and every event \(E \subset \{0,1\}^n\),

In particular, if |E| is exponentially small in \(\{0,1\}^n\), the probability \(\tilde{g}_R(x)\) falls into E would be negligible for any x.

Next, our major analysis will focus on proving this theorem for \(\tilde{g}_R(\cdot )\).

We first set down and prove a “rejection resampling” lemma. This is instrumental in our approach to Theorem 5 (the formal version of Theorem 4), showing that this produces unpredictable values, even to an adaptive adversary with access to the (public) randomness R;

Surveying the proof in more detail, recall that a value \(\tilde{g}_R(x)\) is determined as the XOR of a “constellation” of values \(\bigoplus \tilde{h}_i(x \oplus r_i)\); intuitively, if we could be sure that (a) at least one of these terms, say \(x \oplus r_i\), was not queried by \(\tilde{H}^h(\cdot )\) when evaluated on the other terms and, (b) this isolated term \(x \oplus r_i\) was answered “honestly” (that is, \(\tilde{h}_i(x \oplus r_i) = h_{i}(x \oplus r_i)\)), then it seems reasonable to conclude that the resulting value, the XOR of the results, is close to uniform.

However, applying this intuition to rigorously prove Theorem 5 faces a few challenges. Perhaps the principal difficulty is that it is not obvious how to “partially expose” the random oracle h to take advantage of this intuition: specifically, a traditional approach to proving such strong results is to expose the values taken by the h(x) “as needed,” maintaining the invariant that the unexposed values are uniform and conditionally independent on the exposed values.

In our setting, we would ideally like to expose all but one of the values of a particular constellation \(\{h_i(x \oplus r_i)\}\) so as to guarantee that the last (unexposed) value has the properties (a) and (b) above. While randomly guessing an ordering could guarantee this with fairly high probability \(\approx 1 - 1/\ell \) we must have such a favorable event take place for all x, and so must somehow find a way to guarantee exponentially small failure probabilities. The rejection resampling lemma, discussed above, permits us to examine all the points in a particular constellation, identify a good point (satisfying (a) and (b)) and then “pretend” that we never actually evaluated the point in question. In this sense, the resampling lemma quantifies the penalty necessary for “unexposing” a point of interest.

A less challenging difficulty is that, even conditioned on \(\tilde{h}_i(x \oplus r_i) = h_i(x \oplus r_i)\), this value may not be uniform, as the adversary may choose to be “honest” based on some criteria depending on x or, even, other adaptively-queried points. Finally, of course, the subversion algorithm \(\tilde{H}^h(\cdot )\) is fully-adaptive, and only needs to disrupt \(\tilde{g}_R(x)\) at a single value of x.

We begin with an abstract formulation of the properties of our construction and the analysis, and then transition to the detailed description of the simulator algorithm and its effectiveness.

4.1 The Simulator Algorithm

The main task of the simulator is to ensure the answers to \(\{h_i\}\)-queries to be consistent with the value of \(F(\cdot )\), since for each input x, \(\tilde{h}_R(x)\) is determined by a sequence of related queries to \(\{h_i\}\) and \(\tilde{H}\), (or simply the backdoor z) and the value of R. The basic idea is to program the external layer \(h_0\) using values of F(x), such that the value F(x) is set for \(h_0(\tilde{g}_R(x))\). The value \(\tilde{g}_R(x)\) is obtained by \(\mathcal {S} \) executing the subverted implementations \(\{\tilde{h}_i\}\).

Let us define the simulator \(\mathcal {S} \) (answering queries in two stages) as below:

In the first stage, \(\mathcal {A} \) makes random oracle queries when manufacturing the subverted implementations \(\{\tilde{h}_i\}_{i=0,\ldots ,\ell }\).

On input queries \(x_{1},\ldots ,x_{q_1}\) (at \(\mathcal {A} \)’s choice on which random oracle to query) that \(\mathcal {A} \) makes before outputting the implementations (and the backdoor), \(\mathcal {S} \) answers all those using uniform strings respectively. \(\mathcal {S} \) maintains a table. See Table 1. (w.l.o.g, we simply assume the adversary asks all the hash queries for each value \(x_i\), if not, the simulator asks himself to prepare the table.) \(\mathcal {S} \) and \(\mathcal {D} \) also both receive a random value for R.

In the second stage, the distinguisher \(\mathcal {D} \) now having input R, will ask both queries to the construction and the random oracles. The simulator \(\mathcal {S} \) now also has these extra information of R and oracle access to the implementation \(\tilde{h}\) and will answer the random oracle queries to ensure consistency. In particular:

On input query \(m_j\) to the \(k_j\)-th random oracle \(h_{k_j}\), \(\mathcal {S} \) defines the adjusted query \(m_j':=m_j\oplus r_{k_j}\), and prepares answers for all related queries, i.e., for each i, the input \(m'_j\oplus r_i\) to \(h_i\); and the input \(\tilde{g}_R(m_j')\) to \(h_0\).

• If \(k_j > 0\):

  \(\mathcal {S} \) runs the implementation \(\tilde{h}_i\) on \(m'_j\oplus r_i = m_j\oplus r_{k_j}\oplus r_i\), for all \(i\in \{1,\ldots ,\ell \}\), to derive the value \(\tilde{g}_R({m'_j})=\bigoplus _{i=1}^\ell \tilde{h}_i({m}'_{j}\oplus r_i)\). During the execution of \(\tilde{h}_i\) on those inputs, \(\mathcal {S} \) also answers the random oracle queries (or read from Table 1) on those values if the implementation makes any. In more detail,

  1. 1.

     \(\mathcal {S} \) first checks in both tables whether \(m'_j\oplus r_i\) has been queried for \(h_i\) (Table 2 first), if queried in either of them, \(\mathcal {S} \) returns the corresponding answer; if not queried, \(\mathcal {S} \) simply returns a random value \(u_{j,i}\) as answer and records it in Table 2;

  2. 2.

     \(\mathcal {S} \) checks whether \(\tilde{g}_R(m'_j)\) has been queried for \(h_0\). If not, \(\mathcal {S} \) queries F on \(m_j'\) and gets a response \(F(m_j')\). \(\mathcal {S} \) then checks whether \(m_j'\) has been queried in stage-I, (i.e., check Table 1). If yes and the corresponding value \(v_{j,0}\) does not equal to \(F(m_j')\), \(\mathcal {S} \) aborts; otherwise, \(\mathcal {S} \) sets \(F(m_j')=u_{j,0}\) as the answer for \(h_0(\tilde{g}_R(m'_j))\).

• If \(k_j = 0\):

  \(\mathcal {S} \) checks whether \(m_j\) has been queried for \(h_0\) in stage-II, i.e., there exists an \(m'_t\) in Table 2 such that \(\tilde{g}_R(m'_t)=m_j\). If yes, \(\mathcal {S} \) simply uses the corresponding value \(u_{t,0}\) as answer; If not, \(\mathcal {S} \) checks whether it has been queried in stage-I, \(\mathcal {S} \) returns the value of \(v_{i,0}\) if \(m_j\) has been queried. Otherwise, \(\mathcal {S} \) chooses a random value \(v_{j,0}\) as the response and records it in Table 2.

Probability analysis. Let us define the event that \(\mathcal {S} \) aborts as \(\mathsf {Abort}\). According to the description, \(\mathcal {S} \) aborts only when the distinguisher \(\mathcal {D} \) finds an input m such that the \(\tilde{g}_R(m)=x\) has either been queried for \(h_0\) in stage-I, or queried in stage-II before any of \(\{m\oplus r_i\}_{i=1,\ldots ,\ell }\) has been queried for \(h_i\). We can define \(T_0=\{x| x \text{ is } \text{ queried } \text{ for } h_0 \}\), and following Theorem 5 (to be proven below), \(\Pr [\mathsf {Abort}]\le \frac{|T_0|}{2^n}\le \frac{q+q_1}{2^n}\le \mathsf {negl} (n)\) for any polynomially large \(q,q_1\).

We also define the event \(\mathsf {Bad}\) as that the distinguisher finds an input m, such that \(\tilde{h}_0(\tilde{g}_R(m))\ne h_0(\tilde{g}_R(m))\), also we define \(T_1=\{m| \tilde{h}_0(\tilde{g}_R(m))\ne h_0(\tilde{g}_R(m))\}\). Following Theorem 5, \(\Pr [\mathsf {Bad}]\le \frac{|T_1|}{2^n}+\mathsf {negl} (n)\le \mathsf {negl} (n)\). The latter inequality comes from the condition \(\tilde{h}\) disagrees with h only at negligible fraction of inputs.

Furthermore, it is easy to see, conditioned on \(\mathcal {S} \) does not abort, and \(\mathsf {Bad}\) does not happen, the simulation is perfect.

In the rest of the paper, we will focus on proving our main theorem about the property of the internal function \(\tilde{g}_R(\cdot )\).

4.2 A Rejection Resampling Lemma

We first prove a general rejection re-sampling lemma, and use it as a machinery to prove our main theorem for \(\tilde{g}_R(\cdot )\). Let \(\varOmega _1, \ldots , \varOmega _k\) be a family of sets and let \(\varOmega = \varOmega _1 \times \cdots \times \varOmega _k\). We treat \(\varOmega \) as a probability space under the uniform probability law: for an event \(E \subset \varOmega \), we let \(\mu (E) = |E|/|\varOmega |\) denote the probability of E. For an element \(x = (x_1, \ldots , x_k) \in \varOmega \) and an index i, we define the random variable \(R_ix = (x_1, \ldots , x_{i-1}, y, x_{i+1}, \ldots , x_k)\) where y is drawn uniformly at random from \(\varOmega _i\). We say that such a random variable arises by “resampling” x at the index i.

We consider the effect that arbitrary “adversarial” resampling can have on the uniform distribution. Specifically, for a function \(A: \varOmega \rightarrow \{ 1, \ldots , k\}\), we consider the random variable \(R_{A(X)}X\), where X is a uniformly distributed random variable and the index chosen for resampling is determined by A (as a function of X). By this device, the function A implicitly defines a probability law \(\mu _A\) on \(\varOmega \), where the probability of an event E is given by

Lemma 1

(Rejection re-sampling). Let X be a random variable uniform on \(\varOmega = \varOmega _1 \times \cdots \times \varOmega _k\). Let \(A: \varOmega \rightarrow \{1, \ldots , , k\}\) and define \(Z = R_{A(X)}X\) and \(\mu _A\) as above. Then, for any event E,

Remark 4

Jumping ahead, such a resampling lemma will be used to define a good event E such that one term of \(\tilde{h}_i(x\oplus r_i)\) will be uniformly chosen (not correlated with any other term), thus yields a uniform distribution for the summation. The actual adversarial distribution \(\mu _A(E)\) is thus bounded not too far from \(\mu (E)\). Let us first prove this useful lemma.

Proof

Consider an event \(E \subset X\). To simplify our discussion of the adversarial resampling process discussed above, we remark that the random variables \(R_i X\) and \(R_{A(X)}X\) can be directly defined over the probability space \(\varOmega \times \varOmega \): Consider two independent random variables, X and Y, each drawn uniformly on \(\varOmega \); then, for any i the random variable \(R_iX\) can be described \((X_1, \ldots , X_{i-1}, Y_i, X_{i+1}, \ldots , X_k)\) and \(R_{A(X)}X = (Z_1, \ldots , Z_k)\) where

Note that for any fixed i, the probability law of \(R_{i}X\) is the uniform law on \(\varOmega \).

Upper bound. It follows that for an event E

which establishes the claimed upper bound on \(\mu _A(E)\).

Lower bound. As for the lower bound, define

As the \(B_i, E_i\) partition \(\varOmega ,E\) respectively, and \(\sum _i \mu _A(E_i) = \mu _A(E)\). Observe that

(2)

To complete the proof, we will prove that for any i and for any event F

Putting aside the proof of (3) for a moment, observe that applying (3) to the events \(E_i\) in the expansion (2) above yields the following by Cauchy-Schwarz.

Finally, we return to establish (3). Observe that for an event F,

\({\Pr }_{X,Y}[X \in F \,\text {and}\, R_iX \in F]\)

(The leading \(1/|\varOmega _i|\) term cancels the sum over \(x_i\), which not referenced in the argument of the sum.) Under such strong conditioning, however, the two events \(X \in F\) and \(R_iX \in F\) are independent and, moreover, have the same probability. (Conditioned on the other coordinates, the event depends only on coordinate i of the result that is uniform and independent for the two random variables.) As

we may rewrite the sum above as

\({\Pr }_{X,Y}[X \in F \,\text {and}\, R_iX \in F]\)

where the inequality is Cauchy-Schwarz.    \(\square \)

We remark that these bounds are fairly tight. For the lower bound—the case of interest in our applications—let \(E_i \subset \varOmega _i\) be a family of events with small probability \(\epsilon \) and \(E = \{ (\omega _1, \ldots , \omega _k, \omega _{k+1}) \mid \exists \; \text {unique}\, i \le k, \omega _i \in E_i\} \subset \prod _i^{k+1} \varOmega _i\). When \(\epsilon \ll 1/k\), \(\Pr [E] \approx k \epsilon \) while \(\Pr [R_{A(X)} X \in E] \approx k\epsilon ^2 = (k\epsilon )^2/k\) for the strategy which, in case the event occurred, redraws the offending index and, in case the event did not occur, redraws the \(k+1\)st “dummy” index. For the upper bound, consider an event E consisting of a single point x in the hypercube \(\{0,1\}^k\); then \(\Pr [E] = 2^{-k}\) and \(\Pr [R_{A(X)} X \in E] \ge 2^{-k}(k+1)/2\) for the strategy which re-randomizes any coordinate on which the sample and x disagree (the strategy can be defined arbitrarily on the point x itself).

4.3 Establishing Pointwise Unpredictability

In this section, we focus our attention on the “internal” function (for \(\ell >3n\))

In particular, we will prove that for each x, the probability that the adversary can force the output of \(\tilde{g}_R(x)\) to fall into some range E is polynomial in the density of the range (that is, the probability that a uniform element lies in E). Thus, the output will be unpredictable to the adversary if she has not queried the corresponding random oracles.

Intuition about the analysis. As discussed above, we want to show that for every x, there exist at least one term \(h_i(x\oplus r_i)\) satisfying: (i) \(h_i(x\oplus r_i)\) is answered honestly (that is, \(h_i(x\oplus r_i) = \tilde{h}_i(x \oplus r_i)\); (ii) \(h_i(x\oplus r_i)\) is not correlated with other terms. In order to ensure condition (ii), we proceed in two steps. We first turn to analyze the probability that \(h_i(x\oplus r_i)\) has not been queried by \(H^{h_*}(x\oplus r_j)\) for all other index j. This is still not enough to demonstrate perfect independence, as the good term is subject to the condition of (i), but it suffices for our purposes. As discussed above, for analytical purposes we consider an exotic distribution that calls for this “good” term to be independently re-sampled and apply rejection re-sampling lemma to ensure the original (adversarial) distribution is not too far from the exotic one. We first recall the theorem for the internal function:

Theorem 5

Suppose \(h:\{0,\ldots ,\ell \}\times \{0,1\}^n\rightarrow \{0,1\}^n\) defines a family of random oracles \(h_i:\{0,1\}^n\rightarrow \{0,1\}^n\) as \(h(i,\cdot )\), for \(i=0,\ldots ,\ell \), and \(\ell \ge 3n+1\). Consider a (subversion) algorithm H and \(H^h(x)\) defines a subverted random oracle \(\tilde{h}\). Assume that for every h (and every i),

Then, with overwhelming probability in R, h, and conditioned on the \(h(q_1), \ldots ,\) \(h(q_s)\) (made by any \(\widehat{\mathcal {D}} \)), for all x outside the “queried” set \(\{t \mid h_i(t \oplus r_i)\) \(\text {was queried}\}\) and every event \(E \subset \{0,1\}^n\),

Proof

Throughout the estimates, we will assume that \(\ell > 3n\). Here, we overload the notation \(h_*\) to denote the collection of functions \(h_1, \ldots , h_\ell \).

We begin by considering the simpler case where no queries are made, and just focus on controlling the resulting values vis-a-vis a particular event E. At the end of the proof, we explain how to handle an adaptive family of queries.

Guaranteeing honest answers. First, we ensure that with high probability in R and \(h_*\), for every x, there is a contributing term \(\tilde{h}_i(x \oplus r_i)\) that is likely (if the random variable \(h_i(x \oplus r_i)\) is redrawn according to the uniform distribution) to be “honest” in the sense that \(\tilde{h}_i(x \oplus r_i) = h_i(x \oplus r_i)\). The reason that this simple property does not follow straightforwardly is due to the fact that \(\tilde{h}_i\) may adaptively define the “dishonest” points which are not fixed during the manufacturing of \(\tilde{h}_i\).

To begin, let us consider the following random variables defined by random selection of \(h_*\) (denoting the \(\{h_i\}\)) and R, (later used to bound the number of dishonest terms):

(Throughout \({{\mathrm{\mathbb {E}}}}[\cdot ]\) denotes expectation. For a given \(h_*\) and an element \(\alpha \), the value \(D_i(\alpha )\) is defined by redrawing the value of \(h_i(\alpha )\) uniformly at random; equivalently, \(D_i(\alpha )\) is the conditional expectation of \(d_i(\alpha )\) obtained by setting all other values of \(h_*()\) except \(h_i(\alpha )\).) Note that by assumption, for each i,

where \(\alpha \) is chosen uniformly and \(\epsilon \) is the (negligible) disagreement probability of (1) above.

We introduce several events that play a basic role in the proof.

Flat functions. We say that \(h_*\) is flat if, for each i, \({{\mathrm{\mathbb {E}}}}_{\alpha }[D_i(\alpha )] \le {\epsilon }^{1/3}\), where \(\alpha \) is drawn uniformly.

Note that \(\Pr [h_* \,\text {not flat}] = \Pr [\exists i\in [\ell ], {{\mathrm{\mathbb {E}}}}_\alpha [D_i(\alpha )]> \epsilon ^{1/3}]\), thus

by Markov’s inequality and the union bound. Further, observe that if \(h_*\) is flat, then for any \(x \in \{0,1\}^n\), any \(0 < k \le \ell \), and random choices of \(R=\{r_1,\ldots ,r_\ell \}\),

Next, we will use this property to show that with a sufficiently large \(\ell \), e.g., \(\ell =3n\), then for each x, we can find an index i such that \(D_i(x\oplus r_i)\) is small.

Honesty under resampling. For a tuple \(R = (r_1, \ldots , r_\ell )\), functions \(h_*\), and an element \(x \in \{0,1\}^n\), we say that the triple \((R, h_*, x)\) is honest if

If R and \(h_*\) are “universally” honest, which is to say that \((R,h_*,x)\) is honest for all \(x \in \{0,1\}^n\), we simply say \((R,h_*)\) is honest. Then \(\mathop {\Pr }\limits _R[(R, h_*)\, \text {is not honest}] = \mathop {\Pr }\limits _R[\exists x, (R,h_*,x)\, \text {is not honest}]\,.\) When \(h_*\) is flat, we have the following:

by Markov’s inequality (on the random variable \(\sum _{\begin{array}{c} I \subset [\ell ],\\ |I| = 3n \end{array}}\prod _i D_i(x \oplus r_i)\)) and the union bound. Observe that if \((R,h_*)\) is honest, then for every x,

It follows that, for every set I of size 3n, there exists an element \(i \in I\) so that

That said, conditioned on \(h_*\) being flat, with an overwhelming probability (that is, \(1 - \mathsf {negl} (n)\)) in R, the pair \((R, h_*)\) is honest and so gives rise to at least one small \(D_i(x \oplus r_i)\) for each x (recall that the smaller \(D_i(\alpha )\) is, the fewer points that \(h_i\) disagrees \(\tilde{h}_i\)).

Unfortunately, merely ensuring that some term of each “constellation” \(\{ \tilde{h}_i(x \oplus r_i)\}\) is honest with high probability is not enough—it is possible that a clever adversary can adapt other terms to an honest term to interfere with the final value of \(\tilde{g}_R(x)\). The next part focuses on controlling these dependencies.

Controlling dependence among the terms. We now transition to controlling dependence between various values of \(\tilde{h}_i(x)\). In particular, for every x we want to ensure that there exists some i so that \(h_i(x\oplus r_i)\) was never queried by \(\tilde{H}^{h_*}(\cdot )\) when evaluated on all other \(x\oplus r_j\), i.e., for all \(j\in [\ell ]\) and \(j\ne i\).

Note that the set of queries made by \(\tilde{H}^{h_*}(u)\) is determined entirely by \(h_*\) and u: thus, conditioned on a particular \(h_*\), the event (over R) that \(\tilde{H}^{h_*}(u)\) queries \(h_s(x \oplus r_s)\) and the event that \(\tilde{H}^{h_*}(u')\) queries \(h_t(x \oplus r_t)\) are independent (for any \(u, u'\) and \(s \ne t\)). We introduce the following notation for these events: for a pair of indices i, j (\(i\ne j\)), we define

In light of the discussion above, for an element x and a fixed value of \(h_*\), consider a subset \(T \subset [\ell ]\) and a function \(s: T \rightarrow [\ell ]\); we treat such a function as a representative for the event that each “target” \(t \in T\) was queried by a “source” s(t). (Note that there could be multiple such functions \(s(\cdot )\) for T). We define

We also introduce a couple of new notions for the ease of presentation:

Independent fingerprints. We say that a representative \(s: T \rightarrow [\ell ]\) is independent if \(s(T) \cap T = \emptyset \), which is to say that the range of the function lies in \([\ell ] \setminus T\). For any such independent fingerprint \(s(\cdot )\), note that (for any \(x, h_*\)):

where \(\tau (n)\) denotes the running time (and, hence, an upper bound on the number of queries) of \(\tilde{H}^h(x)\) on inputs of length n.

We will next use such notion to bound the number of bad terms that were queried by some other term.

Dangerous set. For a fixed x, \(h_*\), and R, we say that a set \(T \subset [\ell ]\) is dangerous if every element t in T is queried by some \(\tilde{H}^{h_*}(x \oplus r_i)\) for \(i \ne t\).

We claim that if T is a dangerous set then we can always identify an independent fingerprint \(s: T' \rightarrow [\ell ]\) for a subset \(T' \subset T\) with \(|T'| \ge |T|/2\).

To see this, we build \(T'\) as follows: write \(T = \{ t_1, \ldots , t_m\}\) and consider the elements in order \(t_1, \ldots , t_m\); for each element \(t_i\), we add it to \(T'\), if \(t_i\) is queried by some elements in T^{Footnote 4}, pick one of them, denoted as \(t_j\) (for \(j>i\)), define \(s(t_i)=t_j\), and remove \(t_j\) from T. Observe now that (i) each element in \(T'\) maps to a value (or was queried by a term) outside of \(T'\); (ii) each element \(t_i\) added to \(T'\) removes at most two elements of T (\(t_i\) and \(s(t_i)\)), and hence \(|T'| \ge |T|/2\).

If follows that for a set T the number of such possible independent fingerprints (whose image is at least half the size of T) is bounded by:

We conclude from (4) that for any fixed set T (and any fixed x and \(h_*\))

By taking the union bound over all sets T of size k, it follows immediately that

(5)

The above bound guarantees that, for any fixed x, with overwhelming probability, there are \(\ell -k\) terms that were never queried by any other terms.

k-sparsity: Finally, we say that the pair \((R, h_*)\) is k-sparse if for all x, the set of queries made by \(H^{h_*}(x \oplus r_i)\) includes no more than k of the \(h_i(x \oplus r_i)\).

Applying the union bound over all \(2^n\) strings x to (5), we conclude that, for even constant k (say \(k=5\)), we have

With the preparatory work behind us, we turn to guaranteeing that each x possesses a good term (one that is both well behaved under resampling and not queried by other terms).

Establishing existence of a good term. Next, we wish to show that for any event E with \(\Pr [E] = \mu (E)\), and for any x,

In particular, if E has negligible density, the probability that \(\tilde{g}_R(x) \in E\) is likewise negligible.

We say that R is flat-honest if \({\Pr }_{h_*}[(R,h_*)\, \text {not honest} \mid \, h_*\, \text {is flat}] \le 2^{-n}\,.\) Observe that by Markov’s inequality a uniformly selected R is flat-honest with probability \(1 - 2^{-n}\).

We also say R is uniformly-k-sparse if \({\Pr }_{h_*}[(R,h_*)\, \text {is not} k-\text {sparse}] \le 2^{-n}\,.\) Assuming k is a sufficiently large constant (e.g., 5), note that by Markov’s inequality a random R is uniformly-k-sparse with probability \(1 - 2^{-n}\).

Now we know that selection of a uniformly random \(R = (r_1, \ldots , r_\ell )\), with probability \(1 - 2^{n-1}\), is both uniformly-k-sparse (for the constant k discussed above) and flat-honest. We condition, for the moment, on such a choice of R. In this case, a random function \(h_*\) is likely to be both k-sparse and honest: it follows that for every x there is some term \(h_i(x \oplus r_i)\) that is not queried by H to determine the value of the other terms and, moreover, it is equal to \(\tilde{h}_i(x \oplus r_i)\) with an overwhelming probability. We say that such a pair \((R,h_*)\) is unpredictable; otherwise, we say that \((R,h_*)\) is predictable.

(6)

For each x, we can be sure there is at least one term \(\tilde{h}_i(x\oplus r_i)\) which is typically answered according to \(h_i(x\oplus r_i)\) (i.e., answered honestly) and never queried by the other terms. Unfortunately, to identify this term, we needed to evaluate \(\tilde{h}\) on the whole constellation of points; the rejection resampling lemma lets us correct for this with a bounded penalty.

To complete the analysis, we consider the following experiment: conditioned on R, consider the probability that \(\tilde{g}_R(x) \in E\) when \(h_*\) is drawn as follows:

• if \((R,h_*)\) is unpredictable, there is a (lexicographically first) index i for which \(h_i(x \oplus r_i)\) is queried by no other \(\tilde{h}_j(x \oplus r_j)\) and is honest. Now, redraw the value of \(h_i(x \oplus r_i)\) uniformly at random.

These rules define a distribution on \(h_*\) that is no longer uniform. Note, however, that redrawing \(h_i(x \oplus r_i)\) does not affect the values of \(\tilde{h}_i(x \oplus r_j)\) (for distinct j); as \(\tilde{g}_R(x) = \bigoplus _i \tilde{h}_i(x \oplus r_i)\), under this exotic distribution (for any x),

where the \(2\ell \epsilon ^{1/3}\) term arises because we have only the guarantee that \(D_i(x) \le 2\ell \epsilon ^{1/3}\) from the condition on honesty.

However, based on the rejection resampling lemma above, we conclude

(7)

and, hence, that

where the \(2^{-n}\) term comes from the cases that a randomly chosen R is not flat-honest or universal-k-sparse.

Conditioning on adaptive queries. Finally, we return to the problem of handling adaptive queries. With R and z fixed, the queries generated by \(Q^{h_*}(R,z)\) depend only on \(h_*\) and we may ramify the probability space of h according to the queries and responses of Q; we say \(\alpha = ((q_1, a_1), \ldots , (q_t,a_t))\) is a transcript for Q if Q queries \(h_*\) at \(q_1, \ldots , q_t\) and receives the responses \(a_1, \ldots , a_t\). We remark that if E is an event for which \(\mathop {\Pr }\limits _h[E \mid R,z] \le \epsilon \), then by considering the natural martingale given by iterative exposure of values of \(h_*\) at the points queried by \(Q^h(R,z)\), we have that \(\epsilon \ge \Pr [E \mid R,z] = \sum _\alpha \Pr [E \mid \alpha , R, z] \cdot \Pr [\alpha \mid R, z]\). In particular, events with negligible probability likewise occur with negligible probability for all but a negligible fraction of transcripts \(\alpha \). Thus, the global properties of h discussed in the previous proof are retained even conditioned on a typical transcript \(\alpha \).

We require one amplification of the high-probability structural statements developed above. Note that, with overwhelming probability in R and h, every constellation \(\{ x \oplus r_i\}\) (an argument to \(h_i\)) contains only a constant number of points that are queried by more than a \(2^{-n}\) fraction of other points in the domain of \(h_*\). (Indeed, the fraction of points in the domain of \(h_*\) that are queried by \(H^h(z,x)\) for at least w(n) values of x can be no more than \(\mathrm {poly} (n)/w(n)\), where the polynomial is determined by the running time of H.) We say that a pair R, z is diffuse if a randomly selected h has this property with probability \(1 - 2^{-n/2}\); note that a random pair (R, z) is diffuse with probability \(1 - 2^{-n/2}\).

Consider then conditioning on the event that (R, z) is flat-honest, uniformly-4-sparse, and diffuse; note that in this case, with high probability in h every x has an member of its constellation which is not queried by other members of the constellation, only queried by H() at a vanishing fraction of other points in the domain, and has \(D_i(x \oplus r_i) \le 2\ell \root 3 \of {\epsilon }\). We emphasize that these properties are global properties, holding for all x in the domain of h. In particular, we can apply the argument above to any x for which none of the \(q_i\) touch its constellation \(\{ x \oplus r_i\}\). This concludes the proof.    \(\square \)

We initiate the study of correcting subverted random oracles, where each subverted version disagrees with the original random oracle at a negligible fraction of inputs. We demonstrate that such an attack is devastating in several real-world scenarios. We give a simple construction that can be proven indifferentiable from a random oracle. Our analysis involves developing a new machinery of rejection resampling lemma which may be with independent interests. Our work provides a general tool to transform a buggy implementation of random oracle into a well-behaved one which can be directly applied to the kleptographic setting.

There are many interesting problems worth further exploring, such as better constructions, correcting other ideal objectives under subversion and more.