Hierarchical Functional Encryption

Functional encryption provides fine-grained access control for encrypted data, allowing each user to learn only specific functions of the encrypted data. We study the notion of hierarchical functional encryption, which augments functional encryption with delegation capabilities, offering significantly more expressive access control. We present a generic transformation that converts any general-purpose public-key functional encryption scheme into a hierarchical one without relying on any additional assumptions. This significantly refines our understanding of the power of functional encryption, showing that the existence of functional encryption is equivalent to that of its hierarchical generalization. Instantiating our transformation with the existing functional encryption schemes yields a variety of hierarchical schemes offering various trade-offs between their delegation capabilities (i.e., the depth and width of their hierarchical structures) and underlying assumptions. When starting with a scheme secure against an unbounded number of collusions, we can support arbitrary hierarchical structures. In addition, even when starting with schemes that are secure against a bounded number of collusions (which are known to exist under rather minimal assumptions such as the existence of public-key encryption and shallow pseudorandom generators), we can support hierarchical structures of bounded depth and width

Hierarchical Functional Encryption

Functional encryption provides fine-grained access control for encrypted data, allowing each user to learn only specific functions of the encrypted data. We study the notion of \emph{hierarchical} functional encryption, which augments functional encryption with \emph{delegation} capabilities, offering significantly more expressive access control. We present a {\em generic transformation} that converts any general-purpose public-key functional encryption scheme into a hierarchical one without relying on any additional assumptions. This significantly refines our understanding of the power of functional encryption, showing (somewhat surprisingly) that the existence of functional encryption is equivalent to that of its hierarchical generalization. Instantiating our transformation with the existing functional encryption schemes yields a variety of hierarchical schemes offering various trade-offs between their delegation capabilities (i.e., the depth and width of their hierarchical structures) and underlying assumptions. When starting with a scheme secure against an unbounded number of collusions, we can support \emph{arbitrary} hierarchical structures. In addition, even when starting with schemes that are secure against a bounded number of collusions (which are known to exist under rather minimal assumptions such as the existence of public-key encryption and shallow pseudorandom generators), we can support hierarchical structures of bounded depth and width

Watermarkable public key encryption with efficient extraction under standard assumptions

The current state of the art in watermarked public-key encryption schemes under standard cryptographic assumptions suggests that extracting the embedded message requires either linear time in the number of marked keys or the a-priori knowledge of the marked key employed in the decoder. We present the first scheme that obviates these restrictions in the secret-key marking model, i.e., the setting where extraction is performed using a private extraction key. Our construction offers constant time extraction complexity with constant size keys and ciphertexts and is secure under standard assumptions, namely the Decisional Composite Residuosity Assumption [Eurocrypt’99] and the Decisional Diffie Hellman in prime order subgroups of square higher order residues

Watermarkable Public key Encryption With Efficient Extraction Under Standard Assumptions

The current state of the art in watermarked public-key encryption schemes under standard cryptographic assumptions suggests that extracting the embedded message requires either linear time in the number of marked keys or the a-priori knowledge of the marked key employed in the decoder. We present the first scheme that obviates these restrictions in the secret-key marking model, i.e., the setting where extraction is performed using a private extraction key. Our construction offers constant time extraction complexity with constant size keys and ciphertexts and is secure under standard assumptions, namely the Decisional Composite Residuosity Assumption [Eurocrypt \u2799] and the Decisional Diffie Hellman in prime order subgroups of  square higher order residues

Watermarking Public-Key Cryptographic Primitives

A software watermarking scheme enables users to embed a message or mark within a program while preserving its functionality. Moreover, it is difficult for an adversary to remove a watermark from a marked program without corrupting its behavior. Existing constructions of software watermarking from standard assumptions have focused exclusively on watermarking pseudorandom functions (PRFs). In this work, we study watermarking public-key primitives such as the signing key of a digital signature scheme or the decryption key of a public-key (predicate) encryption scheme. While watermarking public-key primitives might seem more challenging than watermarking PRFs, we show how to construct watermarkable variants of these notions while only relying on standard, and oftentimes, minimal, assumptions. Our watermarkable signature scheme relies only on the minimal assumption that one-way functions exist and satisfies ideal properties such as public marking, public mark-extraction, and full collusion resistance. Our watermarkable public-key encryption schemes are built using techniques developed for the closely-related problem of traitor tracing. Notably, we obtain fully collusion resistant watermarkable attribute-based encryption in the private-key setting from the standard learning with errors assumption and a bounded collusion resistant watermarkable predicate encryption scheme with public mark-extraction and public marking from the minimal assumption that public-key encryption exists

Compactness vs Collusion Resistance in Functional Encryption

We present two general constructions that can be used to combine any two functional encryption (FE) schemes (supporting a bounded number of key queries) into a new functional encryption scheme supporting a larger number of key queries. By using these constructions iteratively, we transform any primitive FE scheme supporting a single functional key query (from a sufficiently general class of functions) and has certain weak compactness properties to a collusion-resistant FE scheme with the same or slightly weaker compactness properties. Together with previously known reductions, this shows that the compact, weakly compact, collusion-resistant, and weakly collusion-resistant versions of FE are all equivalent under polynomial time reductions. These are all FE variants known to imply the existence of indistinguishability obfuscation, and were previously thought to offer slightly different avenues toward the realization of obfuscation from general assumptions. Our results show that they are indeed all equivalent, improving our understanding of the minimal assumptions on functional encryption required to instantiate indistinguishability obfuscation

Homomorphic Indistinguishability Obfuscation and its Applications

In this work, we propose the notion of homomorphic indistinguishability obfuscation ($\mathsf{HiO}$) and present a construction based on subexponentially-secure $\mathsf{iO}$ and one-way functions. An $\mathsf{HiO}$ scheme allows us to convert an obfuscation of circuit $C$ to an obfuscation of $C\u27\circ C$, and this can be performed obliviously (that is, without knowing the circuit $C$). A naive solution would be to obfuscate $C\u27 \circ \mathsf{iO}(C)$. However, if we do this for $k$ hops, then the size of the final obfuscation is exponential in $k$. $\mathsf{HiO}$ ensures that the size of the final obfuscation remains polynomial after repeated compositions. As an application, we show how to build function-hiding hierarchical multi-input functional encryption and homomorphic witness encryption using $\mathsf{HiO}$

Simpler Constructions of Asymmetric Primitives from Obfuscation

We revisit constructions of asymmetric primitives from obfuscation and give simpler alternatives. We consider public-key encryption, (hierarchical) identity-based encryption ((H)IBE), and predicate encryption. Obfuscation has already been shown to imply PKE by Sahai and Waters (STOC\u2714) and full-fledged functional encryption by Garg et al. (FOCS\u2713). We simplify all these constructions and reduce the necessary assumptions on the class of circuits that the obfuscator needs to support. Our PKE scheme relies on just a PRG and does not need any puncturing. Our IBE and bounded HIBE schemes convert natural key-delegation mechanisms from (recursive) applications of puncturable PRFs to IBE and HIBE schemes. Our most technical contribution is an unbounded HIBE, which uses (public-coin) differing-inputs obfuscation for circuits and whose proof relies on a recent pebbling-based hybrid argument by Fuchsbauer et al. (ASIACRYPT\u2714). All our constructions are anonymous, support arbitrary inputs, and have compact keys and ciphertexts

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

Hierarchical functional encryption for linear transformations

In contrast to the conventional all-or-nothing encryption, functional encryption (FE) allows partial revelation of encrypted information based on the keys associated with different functionalities. Extending FE with key delegation ability, hierarchical functional encryption (HFE) enables a secret key holder to delegate a portion of its decryption ability to others and the delegation can be done hierarchically. All HFE schemes in the literature are for general functionalities and not very practical. In this paper, we focus on the functionality of linear transformations (i.e. matrix product evaluation). We refine the definition of HFE and further extend the delegation to accept multiple keys. We also propose a generic HFE construction for linear transformations with IND-CPA security in the standard model from hash proof systems. In addition, we give two instantiations from the DDH and DCR assumptions which to the best of our knowledge are the first practical concrete HFE constructions