Re-encryption, functional re-encryption, and multi-hop re-encryption: A framework for achieving obfuscation-based security and instantiations from lattices

In this work we define multiple relaxations to the definition of correctness in secure obfuscation. While still remaining meaningful, these relaxations provide ways to obfuscate many primitives in a more direct and efficient way. In particular, we first show how to construct a secure obfuscator for the re-encryption primitive from the Decisional Learning with Errors (DLWE) assumption, without going through fully homomorphic encryption. This can be viewed as a meaningful way to trade correctness for efficiency. Next, we show how our tools can be used to construct secure obfuscators for the functional re-encryption and multi-hop unidirectional re-encryption primitives. In the former case, we improve upon the efficiency of the only previously known construction that satisfies the stronger notion of collusion-resistant obfuscation (due to Chandran et al. - TCC 2012) and obtain a construction with input ciphertexts of constant length. In the latter case, we provide the first known obfuscation-based definition and construction; additionally, our scheme is the first scheme where the size of the ciphertexts does not grow with every hop

Upgrading to Functional Encryption

The notion of Functional Encryption (FE) has recently emerged as a strong primitive with several exciting applications. In this work, we initiate the study of the following question: Can existing public key encryption schemes be ``upgraded\u27\u27 to Functional Encryption schemes without changing their public keys or the encryption algorithm? We call a public-key encryption with this property to be FE-compatible. Indeed, assuming ideal obfuscation, it is easy to see that every CCA-secure public-key encryption scheme is FE-compatible. Despite the recent success in using indistinguishability obfuscation to replace ideal obfuscation for many applications, we show that this phenomenon most likely will not apply here. We show that assuming fully homomorphic encryption and the learning with errors (LWE) assumption, there exists a CCA-secure encryption scheme that is provably not FE-compatible. We also show that a large class of natural CCA-secure encryption schemes proven secure in the random oracle model are not FE-compatible in the random oracle model. Nevertheless, we identify a key structure that, if present, is sufficient to provide FE-compatibility. Specifically, we show that assuming sub-exponentially secure iO and sub-exponentially secure one way functions, there exists a class of public key encryption schemes which we call Special-CCA secure encryption schemes that are in fact, FE-compatible. In particular, each of the following popular CCA secure encryption schemes (some of which existed even before the notion of FE was introduced) fall into the class of Special-CCA secure encryption schemes and are thus FE-compatible: 1) The scheme of Canetti, Halevi and Katz (Eurocrypt 2004) when instantiated with the IBE scheme of Boneh-Boyen (Eurocrypt 2004). 2) The scheme of Canetti, Halevi and Katz (Eurocrypt 2004) when instantiated with any Hierarchical IBE scheme. 3) The scheme of Peikert and Waters (STOC 2008) when instantiated with any Lossy Trapdoor Function

Ad Hoc Multi-Input Functional Encryption

Consider sources that supply sensitive data to an aggregator. Standard encryption only hides the data from eavesdroppers, but using specialized encryption one can hope to hide the data (to the extent possible) from the aggregator itself. For flexibility and security, we envision schemes that allow sources to supply encrypted data, such that at any point a dynamically-chosen subset of sources can allow an agreed-upon joint function of their data to be computed by the aggregator. A primitive called multi-input functional encryption (MIFE), due to Goldwasser et al. (EUROCRYPT 2014), comes close, but has two main limitations: - it requires trust in a third party, who is able to decrypt all the data, and - it requires function arity to be fixed at setup time and to be equal to the number of parties. To drop these limitations, we introduce a new notion of ad hoc MIFE. In our setting, each source generates its own public key and issues individual, function-specific secret keys to an aggregator. For successful decryption, an aggregator must obtain a separate key from each source whose ciphertext is being computed upon. The aggregator could obtain multiple such secret-keys from a user corresponding to functions of varying arity. For this primitive, we obtain the following results: - We show that standard MIFE for general functions can be bootstrapped to ad hoc MIFE for free, i.e. without making any additional assumption. - We provide a direct construction of ad hoc MIFE for the inner product functionality based on the Learning with Errors (LWE) assumption. This yields the first construction of this natural primitive based on a standard assumption. At a technical level, our results are obtained by combining standard MIFE schemes and two-round secure multiparty computation (MPC) protocols in novel ways highlighting an interesting interplay between MIFE and two-round MPC

Order-Revealing Encryption and the Hardness of Private Learning

An order-revealing encryption scheme gives a public procedure by which two ciphertexts can be compared to reveal the ordering of their underlying plaintexts. We show how to use order-revealing encryption to separate computationally efficient PAC learning from efficient $(\epsilon, \delta)$-differentially private PAC learning. That is, we construct a concept class that is efficiently PAC learnable, but for which every efficient learner fails to be differentially private. This answers a question of Kasiviswanathan et al. (FOCS '08, SIAM J. Comput. '11). To prove our result, we give a generic transformation from an order-revealing encryption scheme into one with strongly correct comparison, which enables the consistent comparison of ciphertexts that are not obtained as the valid encryption of any message. We believe this construction may be of independent interest.Comment: 28 page

On Extractability (a.k.a. Differing-Inputs) Obfuscation

We initiate the study of {\em extractability obfuscation}, a notion first suggested by Barak et al. (JACM 2012): An extractability obfuscator eO for a class of algorithms M guarantees that if an efficient attacker A can distinguish between obfuscations eO(M_1), eO(M_2) of two algorithms M_1,M_2 \in M, then A can efficiently recover (given M_1 and M_2) an input on which M_1 and M_2 provide different outputs. - We rely on the recent candidate virtual black-box obfuscation constructions to provide candidate constructions of extractability obfuscators for NC^1; next, following the blueprint of Garg et~al. (FOCS 2013), we show how to bootstrap the obfuscator for NC^1 to an obfuscator for all non-uniform polynomial-time Turing machines. In contrast to the construction of Garg et al., which relies on indistinguishability obfuscation for NC^1, our construction enables succinctly obfuscating non-uniform {\em Turing machines} (as opposed to circuits), without turning running-time into description size. - We introduce a new notion of {\em functional witness encryption}, which enables encrypting a message m with respect to an instance x, language L, and function f, such that anyone (and only those) who holds a witness w for x\in L can compute f(m,w) on the message and particular known witness. We show that functional witness encryption is, in fact, equivalent to extractability obfuscation. - We demonstrate other applications of extractability extraction, including the first construction of fully (adaptive-message) indistinguishability-secure functional encryption for an unbounded number of key queries and unbounded message spaces. - We finally relate indistinguishability obfuscation and extractability obfuscation and show special cases when indistinguishability obfuscation can be turned into extractability obfuscation