Functional Encryption and Property Preserving Encryption: New Definitions and Positive Results

Functional Encryption (FE) is an exciting new paradigm that extends the notion of public key encryption. In this work we explore the security of Inner Product Functional Encryption schemes with the goal of achieving the highest security against practically feasible attacks. In addition, we improve efficiency/ underlying assumptions/ security achieved by existing inner product Functional Encryption and Property Preserving Encryption schemes, in both the private and public key setting. Our results can be summarized as follows: - We study whether known impossibilities for achieving strong SIM based security imply actual real world attacks. For this, we present a new UC-style SIM based definition of security that captures both data and function hiding, both public key and symmetric key settings and represents the dream security of FE. While known impossibilities rule out its achievability in the standard model, we show, surprisingly, that it can be achieved in the generic group model for Inner Product FE (Katz et al., Eurocrypt 2008). This provides evidence that FE implementations may enjoy extremely strong security against a large class of real world attacks, namely generic attacks. - We provide several improvements to known constructions of Inner Product FE. In the private key setting, the construction by Shen et al. (TCC 2009) was based on non-standard assumptions, used composite order groups, and only achieved selective security. We give the first construction of a symmetric key inner product FE which is built using prime order groups, and is fully secure under the standard DLIN assumption. Our scheme is more efficient in the size of key and ciphertext than Shen et al.\u27s, when the latter is converted to prime-order groups. - We give the first construction of a property preserving encryption (PPE) scheme for inner-products. Our scheme is secure under the DLIN assumption and satisfies the strongest definition of security -- Left-or-Right security in the standard model. Note that the only previously known construction for PPE by Pandey et al. (Eurocrypt 2012), which was claimed to be secure in the generic group model, was recently attacked Chatterjee and Das, making our construction the first candidate for PPE

Forward-secure hierarchical predicate encryption

Secrecy of decryption keys is an important pre-requisite for security of any encryption scheme and compromised private keys must be immediately replaced. \emph{Forward Security (FS)}, introduced to Public Key Encryption (PKE) by Canetti, Halevi, and Katz (Eurocrypt 2003), reduces damage from compromised keys by guaranteeing confidentiality of messages that were encrypted prior to the compromise event. The FS property was also shown to be achievable in (Hierarchical) Identity-Based Encryption (HIBE) by Yao, Fazio, Dodis, and Lysyanskaya (ACM CCS 2004). Yet, for emerging encryption techniques, offering flexible access control to encrypted data, by means of functional relationships between ciphertexts and decryption keys, FS protection was not known to exist.\smallskip In this paper we introduce FS to the powerful setting of \emph{Hierarchical Predicate Encryption (HPE)}, proposed by Okamoto and Takashima (Asiacrypt 2009). Anticipated applications of FS-HPE schemes can be found in searchable encryption and in fully private communication. Considering the dependencies amongst the concepts, our FS-HPE scheme implies forward-secure flavors of Predicate Encryption and (Hierarchical) Attribute-Based Encryption.\smallskip Our FS-HPE scheme guarantees forward security for plaintexts and for attributes that are hidden in HPE ciphertexts. It further allows delegation of decrypting abilities at any point in time, independent of FS time evolution. It realizes zero-inner-product predicates and is proven adaptively secure under standard assumptions. As the ``cross-product" approach taken in FS-HIBE is not directly applicable to the HPE setting, our construction resorts to techniques that are specific to existing HPE schemes and extends them with what can be seen as a reminiscent of binary tree encryption from FS-PKE

Bi-Deniable Inner Product Encryption from LWE

Deniable encryption (Canetti et al. CRYPTO \u2797) is an intriguing primitive that provides a security guarantee against not only eavesdropping attacks as required by semantic security, but also stronger coercion attacks performed after the fact. The concept of deniability has later demonstrated useful and powerful in many other contexts, such as leakage resilience, adaptive security of protocols, and security against selective opening attacks. Despite its conceptual usefulness, our understanding of how to construct deniable primitives under standard assumptions is restricted. In particular, from standard assumptions such as Learning with Errors (LWE), we have only multi-distributional or non-negligible advantage deniable encryption schemes, whereas with the much stronger assumption of indistinguishable obfuscation, we can obtain at least fully-secure sender-deniable PKE and computation. How to achieve deniability for other more advanced encryption schemes under standard assumptions remains an interesting open question. In this work, we construct a bi-deniable inner product encryption (IPE) in the multi-distributional model without relying on obfuscation as a black box. Our techniques involve new ways of manipulating Gaussian noise, and lead to a significantly tighter analysis of noise growth in Dual Regev type encryption schemes. We hope these ideas can give insight into achieving deniability and related properties for further, advanced cryptographic constructions under standard assumptions

Succinct Predicate and Online-Offline Multi-Input Inner Product Encryptions under Standard Static Assumptions

This paper presents expressive predicate encryption (PE) systems, namely non-zero inner-product-predicate encryption (NIPPE) and attribute-based encryption (ABE) supporting monotone span programs achieving best known parameters among existing similar schemes under well-studied static complexity assumptions. Both the constructions are built in composite order bilinear group setting and involve only 2 group elements in the ciphertexts. More interestingly, our NIPPE scheme, which additionally features only 1 group element in the decryption keys, is the first to attain succinct ciphertexts and decryption keys simultaneously. For proving selective security of these constructions under the Subgroup Decision assumptions, which are the most standard static assumptions in composite order bilinear group setting, we apply the extended version of the elegant D´ej`a Q framework, which was originally proposed as a general technique for reducing the q-type complexity assumptions to their static counter parts. Our work thus demonstrates the power of this framework in overcoming the need of q-type assumptions, which are vulnerable to serious practical attacks, for deriving security of highly expressive PE systems with compact parameters. We further introduce the concept of online-offline multi-input functional encryption (OO-MIFE), which is a crucial advancement towards realizing this highly promising but computationally intensive cryptographic primitive in resource bounded and power constrained devices. We also instantiate our notion of OO-MIFE by constructing such a scheme for the multi-input analog of the inner product functionality, which has a wide range of application in practice. Our OO-MIFE scheme for multiinput inner products is built in asymmetric bilinear groups of prime order and is proven selectively secure under the well-studied k-Linear (k-LIN) assumption

Collusion-Resistant Broadcast Encryption with Tight Reductions and Beyond

The issue of tight security for identity-based encryption schemes ($\mathsf{IBE}$) in bilinear groups has been widely investigated and a lot of optimal properties have been achieved. Recently, a tightly secure IBE scheme in bilinear groups under the multi-challenge setting has been achieved by Chen et al. (to appear in PKC 2017), and their scheme even achieves constant-size public parameters and is adaptively secure. However, we note that the issue of tight security for broadcast encryption schemes ($\mathsf{BE}$) in bilinear groups has received less attention so far. Actually current broadcast encryption systems of bilinear groups are either not tightly secure or based on non-static assumptions. In this work we mainly focus on the issue of tight security for standard broadcast encryption schemes \footnote{We utilize the syntax of broadcast encryption schemes under the key-encapsulation setting in this work and it is easy to be transformed into one under the standard setting.}. We construct the \textit{first} tightly secure broadcast encryption scheme from static assumptions (i.e., decisional subgroup assumptions) in the selective security model by utilizing improved techniques derived from the Déjà Q framework (Eurocrypt 2014, TCC-A 2016). The proof of our construction will lead to only $O(\log n)$ or $O(\log \lambda)$ security loss, where $n$ is the number of users in the system and $\lambda$ is the security parameter. Following this result, we present a tightly secure non-zero inner product encryption scheme ($\mathsf{NIPE}$) from decisional subgroup assumptions in the selective security model. This NIPE scheme has the same parameter sizes as our BE scheme and there is only $O(\log n)$ or $O(\log \lambda)$ security loss as well, where $n$ is the dimension of the inner product space and $\lambda$ is the security parameter. Finally, we further present a tightly secure functional commitment scheme ($\mathsf{FC}$) for linear functions, which was introduced by Libert et al. (ICALP 16). In contrast with their scheme, which also suffers $O(n)$ security loss during the reduction, there is only $O(\log n)$ or $O(\log \lambda)$ security loss in our FC scheme

Embed-Augment-Recover: Function Private Predicate Encryption from Minimal Assumptions in the Public-Key Setting

We present a new class of public-key predicate encryption schemes that are provably function private in the standard model under well-known cryptographic assumptions, and assume predicate distributions satisfying realistic min-entropy requirements. More concretely, we present public-key constructions for identity-based encryption (IBE) and inner-product encryption (IPE) that are computationally function private in the standard model under a family of weaker variants of the DLIN assumption. Existing function private constructions in the public-key setting impose highly stringent requirements on the min-entropy of predicate distributions, thereby limiting their applicability in the context of real-world predicates. For example, the statistically function private constructions of Boneh, Raghunathan and Segev (CRYPTO\u2713 and ASIACRYPT\u2713) are inherently restricted to predicate distributions with min-entropy roughly proportional to $\lambda$, where $\lambda$ is the security parameter. Our constructions allow relaxing this min-entropy requirement to $\omega(\log\lambda)$, while achieving a computational notion of function privacy against probabilistic polynomial-time adversaries, which suffices for most real-world applications. Our constructions also avoid the need for strong assumptions such as indistinguishability obfuscation

New Lower Bounds on Predicate Entropy for Function Private Public-Key Predicate Encryption

We present function private public-key predicate encryption schemes from standard cryptographic assumptions, that achieve new lower bounds on the min-entropy of underlying predicate distributions. Existing function private predicate encryption constructions in the public-key setting can be divided into two broad categories. The first category of constructions are based on standard assumptions, but impose highly stringent requirements on the min-entropy of predicate distributions, thereby limiting their applicability in the context of real-world predicates. For example, the statistically function private constructions of Boneh, Raghunathan and Segev (CRYPTO\u2713 and ASIACRYPT\u2713) are inherently restricted to predicate distributions with min-entropy roughly proportional to the security parameter $\lambda$. The second category of constructions mandate more relaxed min-entropy requirements, but are either based on non-standard assumptions (such as indistinguishability obfuscation) or are secure in the generic group model. In this paper, we affirmatively bridge the gap between these categories by presenting new public-key constructions for identity-based encryption, hidden-vector encryption, and subspace-membership encryption~(a generalization of inner-product encryption) that are both data and function private under variants of the well-known DBDH, DLIN and matrix DDH assumptions, while relaxing the min-entropy requirement on the predicate distributions to $\omega(\log\lambda)$. In summary, we establish that the minimum predicate entropy necessary for any meaningful notion of function privacy in the public-key setting, is in fact, sufficient, for a fairly rich class of predicates

Leakage-resilient Identity-based Encryption in Bounded Retrieval Model with Nearly Optimal Leakage-Ratio

We propose new constructions of leakage-resilient public-key encryption (PKE) and identity-based encryption (IBE) schemes in the bounded retrieval model (BRM). In the BRM, adversaries are allowed to obtain at most $\ell$-bit leakage from a secret key and we can increase $\ell$ only by increasing the size of secret keys without losing efficiency in any other performance measure. We call $\ell/|\textsf{sk}|$ leakage-ratio where $|\textsf{sk}|$ denotes a bit-length of a secret key. Several PKE/IBE schemes in the BRM are known. However, none of these constructions achieve a constant leakage-ratio under a standard assumption in the standard model. Our PKE/IBE schemes are the first schemes in the BRM that achieve leakage-ratio $1-\epsilon$ for any constant $\epsilon>0$ under standard assumptions in the standard model. As previous works, we use identity-based hash proof systems (IB-HPS) to construct IBE schemes in the BRM. It is known that a parameter for IB-HPS called the universality-ratio is translated into the leakage-ratio of the resulting IBE scheme in the BRM. We construct an IB-HPS with universality-ratio $1-\epsilon$ for any constant $\epsilon>0$ based on any inner-product predicate encryption (IPE) scheme with compact secret keys. Such IPE schemes exist under the $d$-linear, subgroup decision, learning with errors, or computational bilinear Diffie-Hellman assumptions. As a result, we obtain IBE schemes in the BRM with leakage-ratio $1-\epsilon$ under any of these assumptions. Our PKE schemes are immediately obtained from our IBE schemes

A New Paradigm for Public-Key Functional Encryption for Degree-2 Polynomials

We give the first public-key functional encryption that supports the generation of functional decryption keys for degree-2 polynomials, with succinct ciphertexts, whose semi-adaptive simulation-based security is proven under standard assumptions. At the heart of our new paradigm lies a so-called partially function-hiding functional encryption scheme for inner products, which admits public-key instances, and that is sufficient to build functional encryption for degree-2 polynomials. Doing so, we improve upon prior works, such as the constructions from Lin (CRYPTO 17) or Ananth Sahai (EUROCRYPT 17), both of which rely on function-hiding inner product FE, that can only exist in the private-key setting. The simplicity of our construction yields the most efficient FE for quadratic functions from standard assumptions (even those satisfying a weaker security notion). The interest of our methodology is that the FE for quadratic functions that builds upon any partially function-hiding FE for inner products inherits the security properties of the latter. In particular, we build a partially function-hiding FE for inner products that enjoys simulation security, in the semi-adaptive setting, where the challenge sent from the adversary can be chosen adaptively after seeing the public key (but before corrupting functional decryption keys). This is in contrast from prior public-key FE for quadratic functions from Baltico et al. (CRYPTO 17), which only achieved an indistinguishability-based, selective security. As a bonus, we show that we can obtain security against Chosen-Ciphertext Attacks straightforwardly. Even though this is the de facto security notion for encryption, this was not achieved by prior functional encryption schemes for quadratic functions, where the generic Fujisaki Okamoto transformation (CRYPTO 99) does not apply

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