Universal Computational Extractors and the Superfluous Padding Assumption for Indistinguishability Obfuscation

Universal Computational Extractors (UCEs), introduced by Bellare, Hoang and Keelveedhi (CRYPTO 2013), are a framework of assumptions on hash functions that allow to instantiate random oracles in a large variety of settings. Brzuska, Farshim and Mittelbach (CRYPTO 2014) showed that a large class of UCE assumptions with \emph{computationally} unpredictable sources cannot be achieved, if indistinguishability obfuscation exists. In the process of circumventing obfuscation-based attacks, new UCE notions emerged, most notably UCEs with respect to \emph{statistically} unpredictable sources that suffice for a large class of applications. However, the only standard model constructions of UCEs are for a small subclass considering only $q$-query sources which are \emph{strongly statistically} unpredictable (Brzuska, Mittelbach; Asiacrypt 2014). The contributions of this paper are threefold: 1) We show a surprising equivalence for the notions of strong unpredictability and (plain) unpredictability thereby lifting the construction from Brzuska and Mittelbach to achieve $q$-query UCEs for statistically unpredictable sources. This yields standard model instantiations for various ($q$-query) primitives including, deterministic public-key encryption, message-locked encryption, multi-bit point obfuscation, CCA-secure encryption, and more. For some of these, our construction yields the first standard model candidate. 2) We study the blow-up that occurs in indistinguishability obfuscation proof techniques due to puncturing and state the \emph{Superfluous Padding Assumption} for indistinguishability obfuscation which allows us to lift the $q$-query restriction of our construction. We validate the assumption by showing that it holds for virtual black-box obfuscation. 3) Brzuska and Mittelbach require a strong form of point obfuscation secure in the presence of auxiliary input for their construction of UCEs. We show that this assumption is indeed necessary for the construction of injective UCEs

Indistinguishability Obfuscation and UCEs : The Case of Computationally Unpredictable Sources

Random oracles are powerful cryptographic objects. They facilitate the security proofs of an impressive number of practical cryptosystems ranging from KDM-secure and deterministic encryption to point-function obfuscation and many more. However, due to an uninstantiability result of Canetti, Goldreich, and Halevi (STOC 1998) random oracles have become somewhat controversial. Recently, Bellare, Hoang, and Keelveedhi (BHK; CRYPTO 2013 and ePrint 2013/424, August 2013) introduced a new abstraction called Universal Computational Extractors (UCEs), and showed that they suffice to securely replace random oracles in a number of prominent applications, including all those mentioned above, without suffering from the aforementioned uninstantiability result. This, however, leaves open the question of constructing UCEs in the standard model. We show that the existence of indistinguishability obfuscation (iO) implies (non-black-box) attacks on all the definitions that BHK proposed within their UCE framework in the original version of their paper, in the sense that no concrete hash function can satisfy them. We also show that this limitation can be overcome, to some extent, by restraining the class of admissible adversaries via a statistical notion of unpredictability. Following our attack, BHK (ePrint 2013/424, September 2013), independently adopted this approach in their work. In the updated version of their paper, BHK (ePrint 2013/424, September 2013) also introduce two other novel source classes, called bounded parallel sources and split sources, which aim at recovering the computational applications of UCEs that fall outside the statistical fix. These notions keep to a computational notion of unpredictability, but impose structural restrictions on the adversary so that our original iO attack no longer applies. We extend our attack to show that indistinguishability obfuscation is sufficient to also break the UCE security of any hash function against bounded parallel sources. Towards this goal, we use the randomized encodings paradigm of Applebaum, Ishai, and Kushilevitz (STOC 2004) to parallelize the obfuscated circuit used in our attack, so that it can be computed by a bounded parallel source whose second stage consists of constant-depth circuits. BHK, in the latest version of their paper (ePrint 2013/424, May 2014), have subsequently replace bounded parallel sources with new source classes. We conclude by discussing the composability and feasibility of hash functions secure against split sources

Point-Function Obfuscation: A Framework and Generic Constructions

We give a definitional framework for point-function obfuscation in which security is parameterized by a class of algorithms we call target generators. Existing and new notions are captured and explained as corresponding to different choices of this class. This leads to an elegant question: Is it possible to provide a generic construction, meaning one that takes an arbitrary class of target generators and returns a point-function obfuscator secure for it? We answer this in the affirmative with three generic constructions, the first based on indistinguishability obfuscation, the second on deterministic public-key encryption and the third on universal computational extractors. By exploiting known constructions of the primitives assumed, we obtain new point-function obfuscators, including many under standard assumptions. We end with a broader look that relates different known and possible notions of point function obfuscation to each other and to ours

A Unified Approach to Idealized Model Separations via Indistinguishability Obfuscation

It is well known that the random oracle model is not sound in the sense that there exist cryptographic systems that are secure in the random oracle model but when instantiated by any family of hash functions become insecure. However, all known separation results require the attacker to send an appropriately crafted message to the challenger in order to break security. Thus, this leaves open the possibility that some cryptographic schemes, such as bit-encryption, are still sound in the random oracle model. In this work we refute this possibility, assuming the existence of indistinguishability obfuscation. We do so in the following way. First, we present a random oracle separation for bit-encryption; namely, we show that there exists a bit-encryption protocol secure in the random oracle model but \emph{completely insecure} when the random oracle is instantiated by any concrete function. Second, we show how to adapt this separation to work for most natural simulation-based and game-based definitions. Our techniques can easily be adapted to other idealized models, and thus we present a \emph{unified approach} to showing separations for most protocols of interest in most idealized models

Indistinguishability Obfuscation from Well-Founded Assumptions

In this work, we show how to construct indistinguishability obfuscation from subexponential hardness of four well-founded assumptions. We prove: Let $\tau \in (0,\infty), \delta \in (0,1), \epsilon \in (0,1)$ be arbitrary constants. Assume sub-exponential security of the following assumptions, where $\lambda$ is a security parameter, and the parameters $\ell,k,n$ below are large enough polynomials in $\lambda$: - The SXDH assumption on asymmetric bilinear groups of a prime order $p = O(2^\lambda)$, - The LWE assumption over $\mathbb{Z}_{p}$ with subexponential modulus-to-noise ratio $2^{k^\epsilon}$, where $k$ is the dimension of the LWE secret, - The LPN assumption over $\mathbb{Z}_p$ with polynomially many LPN samples and error rate $1/\ell^\delta$, where $\ell$ is the dimension of the LPN secret, - The existence of a Boolean PRG in $\mathsf{NC}^0$ with stretch $n^{1+\tau}$, Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists

Public-Seed Pseudorandom Permutations

A number of cryptographic schemes are built from (keyless) permutations, which are either designed in an ad-hoc fashion or are obtained by fixing the key in a block cipher. Security proofs for these schemes, however, idealize this permutation, i.e., making it random and accessible, as an oracle, to all parties. Finding plausible concrete assumptions on such permutations that guarantee security of the resulting schemes has remained an elusive open question. This paper initiates the study of standard-model assumptions on permutations -- or more precisely, on families of permutations indexed by a {\em public} seed. We introduce the notion of a {\em public-seed pseudorandom permutation} (psPRP), which is inspired by the UCE notion by Bellare, Hoang, and Keelveedhi (CRYPTO \u2713). It considers a two-stage security game, where only the second stage learns the seed, and the first-stage adversary, known as the source, is restricted to prevent trivial attacks -- the security notion is consequently parameterized by the class of allowable sources. To this end, we define in particular unpredictable and reset-secure sources analogous to similar notions for UCEs. We first study the relationship between psPRPs and UCEs. To start with, we provide efficient constructions of UCEs from psPRPs for both reset-secure and unpredictable sources, thus showing that most applications of the UCE framework admit instantiations from psPRPs. We also show a converse of this statement, namely that the five-round Feistel construction yields a psPRP for reset-secure sources when the round function is built from UCEs for reset-secure sources, hence making psPRP and UCE equivalent notions for such sources. In addition to studying such reductions, we suggest generic instantiations of psPRPs from both block ciphers and (keyless) permutations, and analyze them in ideal models. Also, as an application of our notions, we show that a simple modification of a recent highly-efficient garbling scheme by Bellare et al. (S&P \u2713) is secure under our psPRP assumption

Augmented Random Oracles

We propose a new paradigm for justifying the security of random oracle-based protocols, which we call the Augmented Random Oracle Model (AROM). We show that the AROM captures a wide range of important random oracle impossibility results. Thus a proof in the AROM implies some resiliency to such impossibilities. We then consider three ROM transforms which are subject to impossibilities: Fiat-Shamir (FS), Fujisaki-Okamoto (FO), and Encrypt-with-Hash (EwH). We show in each case how to obtain security in the AROM by strengthening the building blocks or modifying the transform. Along the way, we give a couple other results. We improve the assumptions needed for the FO and EwH impossibilities from indistinguishability obfuscation to circularly secure LWE; we argue that our AROM still captures this improved impossibility. We also demonstrate that there is no best possible hash function, by giving a pair of security properties, both of which can be instantiated in the standard model separately, which cannot be simultaneously satisfied by a single hash function

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 Magic of ELFs

We introduce the notion of an \emph{Extremely Lossy Function} (ELF). An ELF is a family of functions with an image size that is tunable anywhere from injective to having a polynomial-sized image. Moreover, for any efficient adversary, for a sufficiently large polynomial $r$ (necessarily chosen to be larger than the running time of the adversary), the adversary cannot distinguish the injective case from the case of image size $r$. We develop a handful of techniques for using ELFs, and show that such extreme lossiness is useful for instantiating random oracles in several settings. In particular, we show how to use ELFs to build secure point function obfuscation with auxiliary input, as well as polynomially-many hardcore bits for any one-way function. Such applications were previously known from strong knowledge assumptions --- for example polynomially-many hardcore bits were only know from differing inputs obfuscation, a notion whose plausibility has been seriously challenged. We also use ELFs to build a simple hash function with \emph{output intractability}, a new notion we define that may be useful for generating common reference strings. Next, we give a construction of ELFs relying on the \emph{exponential} hardness of the decisional Diffie-Hellman problem, which is plausible in pairing-based groups. Combining with the applications above, our work gives several practical constructions relying on qualitatively different --- and arguably better --- assumptions than prior works

Instantiability of Classical Random-Oracle-Model Encryption Transforms

Extending work leveraging program obfuscation to instantiate random-oracle-based transforms (e.g., Hohenberger et al., EUROCRYPT 2014, Kalai et al., CRYPTO 2017), we show that, using obfuscation and other assumptions, there exist standard-model hash functions that suffice to instantiate the classical RO-model encryption transforms OAEP (Bellare and Rogaway, EUROCRYPT 1994) and Fujisaki-Okamoto (CRYPTO 1999, J. Cryptology 2013) for specific public-key encryption (PKE) schemes to achieve IND-CCA security. Our result for Fujisaki-Okamoto employs a simple modification to the scheme. Our instantiations do not require much stronger assumptions on the base schemes compared to their corresponding RO-model proofs. For example, to instantiate low-exponent RSA-OAEP, the assumption we need on RSA is sub-exponential partial one-wayness, matching the assumption (partial one-wayness) on RSA needed by Fujisaki et al. (J. Cryptology 2004) in the RO model up to sub-exponentiality. For the part of Fujisaki-Okamoto that upgrades public-key encryption satisfying indistinguishability against plaintext checking attack to IND-CCA, we again do not require much stronger assumptions up to sub-exponentiality. We obtain our hash functions in a unified way, extending a technique of Brzuska and Mittelbach (ASIACRYPT 2014). We incorporate into their technique: (1) extremely lossy functions (ELFs), a notion by Zhandry (CRYPTO 2016), and (2) multi-bit auxiliary-input point function obfuscation (MB-AIPO). While MB-AIPO is impossible in general (Brzuska and Mittelbach, ASIACRYPT 2014), we give plausible constructions for the special cases we need, which may be of independent interest