Impossibility of Quantum Virtual Black-Box Obfuscation of Classical Circuits

Virtual black-box obfuscation is a strong cryptographic primitive: it encrypts a circuit while maintaining its full input/output functionality. A remarkable result by Barak et al. (Crypto 2001) shows that a general obfuscator that obfuscates classical circuits into classical circuits cannot exist. A promising direction that circumvents this impossibility result is to obfuscate classical circuits into quantum states, which would potentially be better capable of hiding information about the obfuscated circuit. We show that, under the assumption that learning-with-errors (LWE) is hard for quantum computers, this quantum variant of virtual black-box obfuscation of classical circuits is generally impossible. On the way, we show that under the presence of dependent classical auxiliary input, even the small class of classical point functions cannot be quantum virtual black-box obfuscated.Comment: v2: Add the notion of decomposable public keys, which allows our impossibility to hold without assuming circular security for QFHE. We also fix an auxiliary lemma (2.9 in v2) where a square root was missing (this does not influence the main result

Indistinguishability Obfuscation of Null Quantum Circuits and Applications

We study the notion of indistinguishability obfuscation for null quantum circuits (quantum null-iO). We present a construction assuming: - The quantum hardness of learning with errors (LWE). - Post-quantum indistinguishability obfuscation for classical circuits. - A notion of "dual-mode" classical verification of quantum computation (CVQC). We give evidence that our notion of dual-mode CVQC exists by proposing a scheme that is secure assuming LWE in the quantum random oracle model (QROM). Then we show how quantum null-iO enables a series of new cryptographic primitives that, prior to our work, were unknown to exist even making heuristic assumptions. Among others, we obtain the first witness encryption scheme for QMA, the first publicly verifiable non-interactive zero-knowledge (NIZK) scheme for QMA, and the first attribute-based encryption (ABE) scheme for BQP

An Alternative View of the Graph-Induced Multilinear Maps

In this paper, we view multilinear maps through the lens of ``homomorphic obfuscation . In specific, we show how to homomorphically obfuscate the kernel-test and affine subspace-test functionalities of high dimensional matrices. Namely, the evaluator is able to perform additions and multiplications over the obfuscated matrices, and test subspace memberships on the resulting code. The homomorphic operations are constrained by the prescribed data structure, e.g. a tree or a graph, where the matrices are stored. The security properties of all the constructions are based on the hardness of Learning with errors problem (LWE). The technical heart is to ``control the ``chain reactions\u27\u27 over a sequence of LWE instances. Viewing the homomorphic obfuscation scheme from a different angle, it coincides with the graph-induced multilinear maps proposed by Gentry, Gorbunov and Halevi (GGH15). Our proof technique recognizes several ``safe modes of GGH15 that are not known before, including a simple special case: if the graph is acyclic and the matrices are sampled independently from binary or error distributions, then the encodings of the matrices are pseudorandom

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

Implementing conjunction obfuscation under entropic ring LWE

We address the practicality challenges of secure program obfuscation by implementing, optimizing, and experimentally assessing an approach to securely obfuscate conjunction programs proposed in [1]. Conjunction programs evaluate functions $f (x_1, . . . , x_L) = \wedge_{i \in I}$ $y_i$, where $y_i$ is either $x_i$ or $\neg x_i$ and $I \subseteq [L]$, and can be used as classifiers. Our obfuscation approach satisfies distributional Virtual Black Box (VBB) security based on reasonable hardness assumptions, namely an entropic variant of the Ring Learning with Errors (Ring-LWE) assumption. Prior implementations of secure program obfuscation techniques support either trivial programs like point functions, or support the obfuscation of more general but less efficient branching programs to satisfy Indistinguishability Obfuscation (IO), a weaker security model. Further, the more general implemented techniques, rather than relying on standard assumptions, base their security on conjectures that have been shown to be theoretically vulnerable. Our work is the first implementation of non-trivial program obfuscation based on polynomial rings. Our contributions include multiple design and implementation advances resulting in reduced program size, obfuscation runtime, and evaluation runtime by many orders of magnitude. We implement our design in software and experimentally assess performance in a commercially available multi-core computing environment. Our implementation achieves runtimes of 6.7 hours to securely obfuscate a 64-bit conjunction program and 2.5 seconds to evaluate this program over an arbitrary input. We are also able to obfuscate a 32-bit conjunction program with 53 bits of security in 7 minutes and evaluate the obfuscated program in 43 milliseconds on a commodity desktop computer, which implies that 32-bit conjunction obfuscation is already practical. Our graph-induced (directed) encoding implementation runs up to 25 levels, which is higher than previously reported in the literature for this encoding. Our design and implementation advances are applicable to obfuscating more general compute-and-compare programs and can also be used for many cryptographic schemes based on lattice trapdoors

Quantum Lightning Never Strikes the Same State Twice

Public key quantum money can be seen as a version of the quantum no-cloning theorem that holds even when the quantum states can be verified by the adversary. In this work, investigate quantum lightning, a formalization of "collision-free quantum money" defined by Lutomirski et al. [ICS'10], where no-cloning holds even when the adversary herself generates the quantum state to be cloned. We then study quantum money and quantum lightning, showing the following results: - We demonstrate the usefulness of quantum lightning by showing several potential applications, such as generating random strings with a proof of entropy, to completely decentralized cryptocurrency without a block-chain, where transactions is instant and local. - We give win-win results for quantum money/lightning, showing that either signatures/hash functions/commitment schemes meet very strong recently proposed notions of security, or they yield quantum money or lightning. - We construct quantum lightning under the assumed multi-collision resistance of random degree-2 systems of polynomials. - We show that instantiating the quantum money scheme of Aaronson and Christiano [STOC'12] with indistinguishability obfuscation that is secure against quantum computers yields a secure quantum money schem

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

Reusable garbled circuits and succinct functional encryption

Garbled circuits, introduced by Yao in the mid 80s, allow computing a function f on an input x without leaking anything about f or x besides f(x). Garbled circuits found numerous applications, but every known construction suffers from one limitation: it offers no security if used on multiple inputs x. In this paper, we construct for the first time reusable garbled circuits. The key building block is a new succinct single-key functional encryption scheme. Functional encryption is an ambitious primitive: given an encryption Enc(x) of a value x, and a secret key sk_f for a function f, anyone can compute f(x) without learning any other information about x. We construct, for the first time, a succinct functional encryption scheme for {\em any} polynomial-time function f where succinctness means that the ciphertext size does not grow with the size of the circuit for f, but only with its depth. The security of our construction is based on the intractability of the Learning with Errors (LWE) problem and holds as long as an adversary has access to a single key sk_f (or even an a priori bounded number of keys for different functions). Building on our succinct single-key functional encryption scheme, we show several new applications in addition to reusable garbled circuits, such as a paradigm for general function obfuscation which we call token-based obfuscation, homomorphic encryption for a class of Turing machines where the evaluation runs in input-specific time rather than worst-case time, and a scheme for delegating computation which is publicly verifiable and maintains the privacy of the computation.Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant)United States. Defense Advanced Research Projects Agency (DARPA award FA8750-11-2-0225)United States. Defense Advanced Research Projects Agency (DARPA award N66001-10-2-4089)National Science Foundation (U.S.) (NSF award CNS-1053143)National Science Foundation (U.S.) (NSF award IIS-1065219)Google (Firm