The Random Oracle Methodology, Revisited

We take a critical look at the relationship between the security of cryptographic schemes in the Random Oracle Model, and the security of the schemes that result from implementing the random oracle by so called "cryptographic hash functions". The main result of this paper is a negative one: There exist signature and encryption schemes that are secure in the Random Oracle Model, but for which any implementation of the random oracle results in insecure schemes. In the process of devising the above schemes, we consider possible definitions for the notion of a "good implementation" of a random oracle, pointing out limitations and challenges.Comment: 31 page

Naor-Reingold Goes Public: The Complexity of Known-key Security

We study the complexity of building secure block ciphers in the setting where the key is known to the attacker. In particular, we consider two security notions with useful implications, namely public-seed pseudorandom permutations (or psPRPs, for short) (Soni and Tessaro, EUROCRYPT \u2717) and correlation-intractable ciphers (Knudsen and Rijmen, ASIACRYPT \u2707; Mandal, Seurin, and Patarin, TCC \u2712). For both these notions, we exhibit constructions which make only two calls to an underlying non-invertible primitive, matching the complexity of building a pseudorandom permutation in the secret-key setting. Our psPRP result instantiates the round functions in the Naor-Reingold (NR) construction with a secure UCE hash function. For correlation intractability, we instead instantiate them from a (public) random function, and replace the pairwise-independent permutations in the NR construction with (almost) $O(k^2)$-wise independent permutations, where $k$ is the arity of the relations for which we want correlation intractability. Our constructions improve upon the current state of the art, requiring five- and six-round Feistel networks, respectively, to achieve psPRP security and correlation intractability. To do so, we rely on techniques borrowed from Impagliazzo-Rudich-style black-box impossibility proofs for our psPRP result, for which we give what we believe to be the first constructive application, and on techniques for studying randomness with limited independence for correlation intractability

Hiding secrets in public random functions

Constructing advanced cryptographic applications often requires the ability of privately embedding messages or functions in the code of a program. As an example, consider the task of building a searchable encryption scheme, which allows the users to search over the encrypted data and learn nothing other than the search result. Such a task is achievable if it is possible to embed the secret key of an encryption scheme into the code of a program that performs the "decrypt-then-search" functionality, and guarantee that the code hides everything except its functionality. This thesis studies two cryptographic primitives that facilitate the capability of hiding secrets in the program of random functions. 1. We first study the notion of a private constrained pseudorandom function (PCPRF). A PCPRF allows the PRF master secret key holder to derive a public constrained key that changes the functionality of the original key without revealing the constraint description. Such a notion closely captures the goal of privately embedding functions in the code of a random function. Our main contribution is in constructing single-key secure PCPRFs for NC^1 circuit constraints based on the learning with errors assumption. Single-key secure PCPRFs were known to support a wide range of cryptographic applications, such as private-key deniable encryption and watermarking. In addition, we build reusable garbled circuits from PCPRFs. 2. We then study how to construct cryptographic hash functions that satisfy strong random oracle-like properties. In particular, we focus on the notion of correlation intractability, which requires that given the description of a function, it should be hard to find an input-output pair that satisfies any sparse relations. Correlation intractability captures the security properties required for, e.g., the soundness of the Fiat-Shamir heuristic, where the Fiat-Shamir transformation is a practical method of building signature schemes from interactive proof protocols. However, correlation intractability was shown to be impossible to achieve for certain length parameters, and was widely considered to be unobtainable. Our contribution is in building correlation intractable functions from various cryptographic assumptions. The security analyses of the constructions use the techniques of secretly embedding constraints in the code of random functions

The random oracle methodology, revisited

We take a critical look at the relationship between the security of cryptographic schemes in the Random Oracle Model, and the security of the schemes that result from implementing the random oracle by so called “cryptographic hash functions”. The main result of this paper is a negative one: There exist signature and encryption schemes that are secure in the Random Oracle Model, but for which any implementation of the random oracle results in insecure schemes. In the process of devising the above schemes, we consider possible definitions for the notion of a “good implementation” of a random oracle, pointing out limitations and challengesAccepted manuscrip

Correlation Intractability and SNARGs from Sub-exponential DDH

We provide the first constructions of SNARGs for Batch-NP and P based solely on the sub-exponential Decisional Diffie Hellman (DDH) assumption. Our schemes achieve poly-logarithmic proof sizes. Central to our results and of independent interest is a new construction of correlation-intractable hash functions for ``small input\u27\u27 product relations verifiable in $\mathsf{TC}^0$, based on sub-exponential DDH

On The Black-Box Complexity of Correlation Intractability

Correlation intractability is an emerging cryptographic paradigm that enabled several recent breakthroughs in establishing soundness of the Fiat-Shamir transform and, consequently, basing non-interactive zero-knowledge proofs and succinct arguments on standard cryptographic assumptions. In a nutshell, a hash family is said to be \emph{correlation intractable} for a class of relations $\mathcal{R}$ if, for any relation $R\in\mathcal{R}$, it is hard given a random hash function $h\gets H$ to find an input $z$ s.t. $(z,h(z))\in R$, namely a correlation. Despite substantial progress in constructing correlation intractable hash functions, all constructions known to date are based on highly-structured hardness assumptions and, further, are of complexity scaling with the circuit complexity of the target relation class. In this work, we initiate the study of the barriers for building correlation intractability. Our main result is a lower bound on the complexity of any black-box construction of CIH from collision resistant hash (CRH), or one-way permutations (OWP), for any sufficiently expressive relation class. In particular, any such construction for a class of relations with circuit complexity $t$ must make at least $\Omega(t)$ invocations of the underlying building block. We see this as a first step in developing a methodology towards broader lower bounds

Correlation-Intractable Hash Functions via Shift-Hiding

A hash function family $\mathcal{H}$ is correlation intractable for a $t$-input relation $\mathcal{R}$ if, given a random function $h$ chosen from $\mathcal{H}$, it is hard to find $x_1,\ldots,x_t$ such that $\mathcal{R}(x_1,\ldots,x_t,h(x_1),\ldots,h(x_t))$ is true. Among other applications, such hash functions are a crucial tool for instantiating the Fiat-Shamir heuristic in the plain model, including the only known NIZK for NP based on the learning with errors (LWE) problem (Peikert and Shiehian, CRYPTO 2019). We give a conceptually simple and generic construction of single-input CI hash functions from shift-hiding shiftable functions (Peikert and Shiehian, PKC 2018) satisfying an additional one-wayness property. This results in a clean abstract framework for instantiating CI, and also shows that a previously existing function family (PKC 2018) was already CI under the LWE assumption. In addition, our framework transparently generalizes to other settings, yielding new results: - We show how to instantiate certain forms of multi-input CI under the LWE assumption. Prior constructions either relied on a very strong ``brute-force-is-best\u27\u27 type of hardness assumption (Holmgren and Lombardi, FOCS 2018) or were restricted to ``output-only\u27\u27 relations (Zhandry, CRYPTO 2016). - We construct single-input CI hash functions from indistinguishability obfuscation (iO) and one-way permutations. Prior constructions relied essentially on variants of fully homomorphic encryption that are impossible to construct from such primitives. This result also generalizes to more expressive variants of multi-input CI under iO and additional standard assumptions

Lossy Correlation Intractability and PPAD Hardness from Sub-exponential LWE

We introduce a new cryptographic primitive, a lossy correlation-intractable hash function, and use it to soundly instantiate the Fiat-Shamir transform for the general interactive sumcheck protocol, assuming sub-exponential hardness of the Learning with Errors (LWE) problem. By combining this with the result of Choudhuri et al. (STOC 2019), we show that $\#\mathsf{SAT}$ reduces to end-of-line, which is a $\mathsf{PPAD}$-complete problem, assuming the sub-exponential hardness of LWE

Fiat-Shamir and Correlation Intractability from Strong KDM-Secure Encryption

A hash function family is called correlation intractable if for all sparse relations, it is hard to find, given a random function from the family, an input-output pair that satisfies the relation (Canetti et al., STOC 98). Correlation intractability (CI) captures a strong Random-Oracle-like property of hash functions. In particular, when security holds for all sparse relations, CI suffices for guaranteeing the soundness of the Fiat-Shamir transformation from any constant round, statistically sound interactive proof to a non-interactive argument. However, to date, the only CI hash function for all sparse relations (Kalai et al., Crypto 17) is based on general program obfuscation with exponential hardness properties. We construct a simple CI hash function for arbitrary sparse relations, from any symmetric encryption scheme that satisfies some natural structural properties, and in addition guarantees that key recovery attacks mounted by polynomial-time adversaries have only exponentially small success probability - even in the context of key-dependent messages (KDM). We then provide parameter settings where ElGamal encryption and Regev encryption plausibly satisfy the needed properties. Our techniques are based on those of Kalai et al., with the main contribution being substituting a statistical argument for the use of obfuscation, therefore greatly simplifying the construction and basing security on better-understood intractability assumptions. In addition, we extend the definition of correlation intractability to handle moderately sparse relations so as to capture the properties required in proof-of-work applications (e.g. Bitcoin). We also discuss the applicability of our constructions and analyses in that regime