## CryptoDB

### Sanjam Garg

#### Publications

**Year**

**Venue**

**Title**

2023

CRYPTO

Correlation Intractability and SNARGs from Sub-exponential DDH
Abstract

We provide the first constructions of SNARGs for Batch-NP and P based solely on the sub-exponential Decisional Diffie Hellman (DDH) assumption. Our schemes achieve poly-logarithmic proof sizes.
We obtain our results by following the correlation-intractability framework for secure instantiation of the Fiat-Shamir paradigm. The centerpiece of our results and of independent interest is a new construction of correlation-intractable hash functions for ``small input'' product relations verifiable in TC0, based on sub-exponential DDH.

2023

PKC

Credibility in Private Set Membership
Abstract

A private set membership (PSM) protocol allows a ``receiver'' to learn whether its input $x$ is contained in a large database $\algo{DB}$ held by a ``sender''. In this work, we define and construct \emph{credible private set membership (C-PSM)} protocols: in addition to the conventional notions of privacy, C-PSM provides a soundness guarantee that it is hard for a sender (that does not know $x$) to convince the receiver that $x \in \algo{DB}$.
Furthermore, the communication complexity must be logarithmic in the size of $\algo{DB}$.
We provide 2-round (i.e., round-optimal) C-PSM constructions based on standard assumptions:
\begin{itemize}[itemsep=0pt]
\item We present a black-box construction in the plain model based on DDH or LWE.
\item Next, we consider protocols that support predicates $f$ beyond string equality, i.e., the receiver can learn if there exists $w \in \algo{DB}$ such that $f(x,w) = 1$. We present two results with transparent setups: (1) A black-box protocol, based on DDH or LWE, for the class of NC$^1$ functions $f$ which are efficiently searchable. (2) An LWE-based construction for all bounded-depth circuits. The only non-black-box use of cryptography in this construction is through the bootstrapping procedure in fully homomorphic encryption.
\end{itemize}
As an application, our protocols can be used to build enhanced leaked password notification services, where unlike existing solutions, a dubious sender {\em cannot} fool a receiver into changing its password.

2023

EUROCRYPT

End to End Secure Messaging with Traceability Only for Illegal Content
Abstract

As end-to-end encrypted messaging services become widely adopted, law enforcement agencies have increasingly expressed concern that such services interfere with their ability to maintain public safety. Indeed, there is a direct tension between preserving user privacy and enabling content moderation on these platforms. Recent research has begun to address this tension, proposing systems that purport to strike a balance between the privacy of ``honest’’ users and traceability of ``malicious’’ users. Unfortunately, these systems suffer from a lack of protection against malicious or coerced service providers.
In this work, we address the privacy vs. content moderation question through the lens of pre-constrained cryptography [Ananth et al., ITCS 2022]. We introduce the notion of {\em set pre-constrained} (SPC) {\em group signatures} that guarantees security against \emph{malicious key generators}.
SPC group signatures offer the ability to trace users in messaging systems who originate pre-defined illegal content (such as child sexual abuse material), while providing security against malicious service providers.
We construct concretely efficient protocols for SPC group signatures, and demonstrate the real-world feasibility of our approach via an implementation. The starting point for our solution is the recently introduced Apple PSI system, which we significantly modify to improve security and expand functionality.

2023

CRYPTO

Cryptography with Weights: MPC, Encryption and Signatures
Abstract

The security of many powerful cryptographic systems such as secure multiparty computation, threshold encryption, and threshold signatures rests on trust assumptions about the parties. The de-facto model treats all parties equally and requires that a certain fraction of the parties are honest. While this paradigm of one-person-one-vote has been very successful over the years, current and emerging practical use cases suggest that it is outdated.
In this work, we consider {\em weighted} cryptosystems where every party is assigned a certain weight and the trust assumption is that a certain fraction of the total weight is honest. This setting can be translated to the standard setting (where each party has a unit weight) via virtualization. However, this method is quite expensive, incurring a multiplicative overhead in the weight.
We present new weighted cryptosystems with significantly better efficiency: our proposed schemes incur only an {\em additive} overhead in weights.
\begin{itemize}
\item We first present a weighted ramp secret-sharing scheme (WRSS) where the size of a secret share is $O(w)$ (where $w$ corresponds to the weight). In comparison, Shamir's secret sharing with virtualization requires secret shares of size $w\cdot\lambda$, where $\lambda=\log |\bbF|$ is the security parameter.
\item Next, we use our WRSS to construct weighted versions of (semi-honest) secure multiparty computation (MPC), threshold encryption, and threshold signatures. All these schemes inherit the efficiency of our WRSS and incur only an additive overhead in weights.
\end{itemize}
Our WRSS is based on the Chinese remainder theorem-based secret-sharing scheme. Interestingly, this secret-sharing scheme is {\em non-linear} and only achieves statistical privacy. These distinct features introduce several technical hurdles in applications to MPC and threshold cryptosystems. We resolve these challenges by developing several new ideas.

2023

JOFC

Candidate iO from Homomorphic Encryption Schemes
Abstract

We propose a new approach to construct general-purpose indistinguishability obfuscation (iO). Our construction is obtained via a new intermediate primitive that we call split fully homomorphic encryption (split FHE), which we show to be sufficient for constructing iO. Specifically, split FHE is FHE where decryption takes the following two-step syntactic form: (i) a secret decryption step that uses the secret key and produces a hint which is (asymptotically) shorter than the length of the encrypted message, and (ii) a public decryption step that only requires the ciphertext and the previously generated hint (and not the entire secret key) and recovers the encrypted message. In terms of security, the hints for a set of ciphertexts should not allow one to violate semantic security for any other ciphertexts. Next, we show a generic candidate construction of split FHE based on three building blocks: (i) A standard FHE scheme with linear decrypt-and-multiply (which can be instantiated with essentially all LWE-based constructions), (ii) a linearly homomorphic encryption scheme with short decryption hints (such as the Damgård-Jurik encryption scheme, based on the DCR problem), and (iii) a cryptographic hash function (which can be based on a variety of standard assumptions). Our approach is heuristic in the sense that our construction is not provably secure and makes implicit assumptions about the interplay between these underlying primitives. We show evidence that this construction is secure by providing an argument in an appropriately defined oracle model. We view our construction as a big departure from the state-of-the-art constructions, and it is in fact quite simple.

2022

PKC

Reusable Two-Round MPC from LPN
📺
Abstract

We present a new construction of maliciously-secure, two-round multiparty computation (MPC) in the CRS model, where the first message is reusable an unbounded number of times. The security of the protocol relies on the Learning Parity with Noise (LPN) assumption with inverse polynomial noise rate $1/n^{1-\epsilon}$ for small enough constant $\epsilon$, where $n$ is the LPN dimension. Prior works on reusable two-round MPC required assumptions such as DDH or LWE that imply some flavor of homomorphic computation. We obtain our result in two steps:
- In the first step, we construct a two-round MPC protocol in the {\it silent pre-processing model} (Boyle et al., Crypto 2019). Specifically, the parties engage in a computationally inexpensive setup procedure that generates some correlated random strings. Then, the parties commit to their inputs. Finally, each party sends a message depending on the function to be computed, and these messages can be decoded to obtain the output. Crucially, the complexity of the pre-processing phase and the input commitment phase do not grow with the size of the circuit to be computed. We call this {\it multiparty silent NISC} (msNISC), generalizing the notion of two-party silent NISC of Boyle et al. (CCS 2019). We provide a construction of msNISC from LPN in the random oracle model.
- In the second step, we give a transformation that removes the pre-processing phase and use of random oracle from the previous protocol. This transformation additionally adds (unbounded) reusability of the first round message, giving the first construction of reusable two-round MPC from the LPN assumption. This step makes novel use of randomized encoding of circuits (Applebaum et al., FOCS 2004) and a variant of the ``tree of MPC messages" technique of Ananth et al. and Bartusek et al. (TCC 2020).

2022

CRYPTO

A More Complete Analysis of the Signal Double Ratchet Algorithm
📺
Abstract

Seminal works by Cohn-Gordon, Cremers, Dowling, Garratt, and Stebila [Journal of Cryptology 2020] and Alwen, Coretti, and Dodis [EUROCRYPT 2019] provided the first formal frameworks for studying the widely-used Signal Double Ratchet (DR for short) algorithm.
In this work, we develop a new Universally Composable (UC) definition F_DR that we show is provably achieved by the DR protocol. Our definition captures not only the security and correctness guarantees of the DR already identified in the prior state-of-the-art analyses of Cohn-Gordon et al. and Alwen et al., but also more guarantees that are absent from one or both of these works. In particular, we construct six different modified versions of the DR protocol, all of which are insecure according to our definition F_DR, but remain secure according to one (or both) of their definitions. For example, our definition is the first to capture CCA-style attacks possible immediately after a compromise — attacks that, as we show, the DR protocol provably resists, but were not captured by prior definitions.
We additionally show that multiple compromises of a party in a short time interval, which the DR should be able to withstand, as we understand from its whitepaper, nonetheless introduce a new non-trivial (albeit minor) weakness of the DR. Since the definitions in the literature (including our F_DR above) do not capture security against this more nuanced scenario, we define a new stronger definition F_TR that does.
Finally, we provide a minimalistic modification to the DR (that we call the Triple Ratchet, or TR for short) and show that the resulting protocol securely realizes the stronger functionality F_TR. Remarkably, the modification incurs no additional communication cost and virtually no additional computational cost. We also show that these techniques can be used to improve communication costs in other scenarios, e.g. practical Updatable Public Key Encryption schemes and the re-randomized TreeKEM protocol of Alwen et al. [CRYPTO 2020] for Secure Group Messaging.

2022

TCC

IBE with Incompressible Master Secret and Small Identity Secrets
Abstract

Side-stepping the protection provided by cryptography, exfiltration attacks are becoming a considerable real-world threat. With the goal of mitigating the exfiltration of cryptographic keys, big-key cryptosystems have been developed over the past few years. These systems come with very large secret keys which are thus hard to exfiltrate. Typically, in such systems, the setup time must be large as it generates the large secret key. However, subsequently, the encryption and decryption operations, that must be performed repeatedly, are required to be efficient. Specifically, the encryption uses only a small public key and the decryption only accesses small ciphertext-dependent parts of the full secret key. Nonetheless, these schemes require decryption to have access to the entire secret key. Thus, using such big-key cryptosystems necessitate that users carry around large secret keys on their devices, which can be a hassle and in some cases might also render exfiltration easy.
With the goal of removing this problem, in this work, we initiate the study of big-key identity-based encryption (bk-IBE). In such a system, the master secret key is allowed to be large but we require that the identity-based secret keys are short. This allows users to use the identity-based short keys as the ephemeral secret keys that can be more easily carried around and allow for decrypting ciphertexts matching a particular identity, e.g. messages that were encrypted on a particular date. In particular:
-We build a new definitional framework for bk-IBE capturing a range of applications. In the case when the exfiltration is small our definition promises stronger security --- namely, an adversary can break semantic security for only a few identities, proportional to the amount of leakage it gets. In contrast, in the catastrophic case where a large fraction of the master secret key has been ex-filtrated, we can still resort to a guarantee that the ciphertexts generated for a randomly chosen identity (or, an identity with enough entropy) remain protected. We demonstrate how this framework captures the best possible security guarantees.
-We show the first construction of such a bk-IBE offering strong security properties. Our construction is based on standard assumptions on groups with bilinear pairings and brings together techniques from seemingly different contexts such as leakage resilient cryptography, reusable two-round MPC, and laconic oblivious transfer. We expect our techniques to be of independent interest.

2022

TCC

On the Worst-Case Inefficiency of CGKA
Abstract

Continuous Group Key Agreement (CGKA) is the basis of modern Secure Group Messaging (SGM) protocols. At a high level, a CGKA protocol enables a group of users to continuously compute a shared (evolving) secret while members of the group add new members, remove other existing members, and perform state updates. The state updates allow CGKA to offer desirable security features such as forward secrecy and post-compromise security.
CGKA is regarded as a practical primitive in the real-world. Indeed, there is an IETF Messaging Layer Security (MLS) working group devoted to developing a standard for SGM protocols, including the CGKA protocol at their core. Though known CGKA protocols seem to perform relatively well when considering natural sequences of performed group operations, there are no formal guarantees on their efficiency, other than the O(n) bound which can be achieved by trivial protocols, where n is the number of group numbers. In this context, we ask the following questions and provide negative answers.
1. Can we have CGKA protocols that are efficient in the worst case? We start by answering this basic question in the negative. First, we show that a natural primitive that we call Compact Key Exchange (CKE) is at the core of CGKA, and thus tightly captures CGKA’s worst-case communication cost. Intuitively, CKE requires that: first, n users non-interactively generate key pairs and broadcast their public keys, then, some other special user securely communicates to these n users a shared key. Next, we show that CKE with communication cost o(n) by the special user cannot be realized in a black-box manner from public-key encryption and one-way functions, thus implying the same for CGKA, where n is the corresponding number of group members.
2. Can we realize one CGKA protocol that works as well as possible in all cases? Here again, we present negative evidence showing that no such protocol based on black-box use of public-key encryption and one-way functions exists. Specifically, we show two distributions over sequences of group operations such that no CGKA protocol obtains optimal communication costs on both sequences.

2021

CRYPTO

Compact Ring Signatures from Learning With Errors
📺
Abstract

Ring signatures allow a user to sign a message on behalf of a ``ring'' of signers, while hiding the true identity of the signer. As the degree of anonymity guaranteed by a ring signature is directly proportional to the size of the ring, an important goal in cryptography is to study constructions that minimize the size of the signature as a function of the number of ring members.
In this work, we present the first compact ring signature scheme (i.e., where the size of the signature grows logarithmically with the size of the ring) from the (plain) learning with errors (LWE) problem. The construction is in the standard model and it does not rely on a trusted setup or on the random oracle heuristic. In contrast with the prior work of Backes
\etal~[EUROCRYPT'2019], our scheme does not rely on bilinear pairings, which allows us to show that the scheme is post-quantum secure assuming the quantum hardness of LWE.
At the heart of our scheme is a new construction of compact and statistically witness-indistinguishable ZAP arguments for NP $\cap$ coNP, that we show to be sound based on the plain LWE assumption. Prior to our work, statistical ZAPs (for all of NP) were known to exist only assuming \emph{sub-exponential} LWE. We believe that this scheme might find further applications in the future.

2021

ASIACRYPT

How to Build a Trapdoor Function from an Encryption Scheme
📺
Abstract

In this work we ask the following question: Can we transform any encryption scheme into a trapdoor function (TDF)? Alternatively stated, can we make any encryption scheme randomness recoverable? We propose a generic compiler that takes as input any encryption scheme with pseudorandom ciphertexts and adds a trapdoor to invert the encryption, recovering also the random coins. This universal TDFier only assumes in addition the existence of a hinting pseudorandom generator (PRG). Despite the simplicity, our transformation is quite general and we establish a series of new feasibility results:
- The first identity-based TDF [Bellare et al. EUROCRYPT 2012] from the CDH assumption in pairing-free groups (or from factoring), thus matching the state of the art for identity-based encryption schemes. Prior works required pairings or LWE.
- The first collusion-resistant attribute-based TDF (AB-TDF) for all ($NC^1$, resp.) circuits from LWE (bilinear maps, resp.). Moreover, the first single-key AB-TDF from CDH. To the best of our knowledge, no AB-TDF was known in the literature (not even for a single key) from any assumption. We obtain the same results for predicate encryption.
As an additional contribution, we define and construct a trapdoor garbling scheme: A simulation secure garbling scheme with a hidden ``trigger'' that allows the evaluator to fully recover the randomness used by the garbling algorithm. We show how to construct trapdoor garbling from the DDH or LWE assumption with an interplay of key-dependent message (KDM) and randomness-dependent message (RDM) techniques.
Trapdoor garbling allows us to obtain alternative constructions of (single-key) AB-TDFs with additional desirable properties, such as adaptive security (in the choice of the attribute) and projective keys. We expect trapdoor garbling to be useful in other contexts, e.g. in case where, upon successful execution, the evaluator needs to immediately verify that the garbled circuit was well-formed.

2021

TCC

Laconic Private Set Intersection and Applications
📺
Abstract

Consider a server with a \emph{large} set $S$ of strings $\{x_1,x_2\ldots,x_N\}$ that would like to publish a \emph{small} hash $h$ of its set $S$ such that any client with a string $y$ can send the server a \emph{short} message allowing it to learn $y$ if $y \in S$ and nothing otherwise. In this work, we study this problem of two-round private set intersection (PSI) with low (asymptotically optimal) communication cost, or what we call \emph{laconic} private set intersection ($\ell$PSI) and its extensions. This problem is inspired by the recent general frameworks for laconic cryptography [Cho et al. CRYPTO 2017, Quach et al. FOCS'18].
We start by showing the first feasibility result for realizing $\ell$PSI~ based on the CDH assumption, or LWE with polynomial noise-to-modulus ratio. However, these feasibility results use expensive non-black-box cryptographic techniques leading to significant inefficiency. Next, with the goal of avoiding these inefficient techniques, we give a construction of $\ell$PSI~schemes making only black-box use of cryptographic functions. Our construction is secure against semi-honest receivers, malicious senders and reusable in the sense that the receiver's message can be reused across any number of executions of the protocol. The scheme is secure under the $\phi$-hiding, decisional composite residuosity and subgroup decision assumptions.
Finally, we show natural applications of $\ell$PSI~to realizing a semantically-secure encryption scheme that supports detection of encrypted messages belonging to a set of ``illegal'' messages (e.g., an illegal video) circulating online.
Over the past few years, significant effort has gone into realizing laconic cryptographic protocols. Nonetheless, our work provides the first black-box constructions of such protocols for a natural application setting.

2021

TCC

Amortizing Rate-1 OT and Applications to PIR and PSI
📺
Abstract

Recent new constructions of rate-1 OT [D\"ottling, Garg, Ishai, Malavolta, Mour, and Ostrovsky, CRYPTO 2019] have brought this primitive under the spotlight and the techniques have led to new feasibility results for private-information retrieval, and homomorphic encryption for branching programs. The receiver communication of this construction consists of a quadratic (in the sender's input size) number of group elements for a single instance of rate-1 OT. Recently [Garg, Hajiabadi, Ostrovsky, TCC 2020] improved the receiver communication to a linear number of group elements for a single string-OT. However, most applications of rate-1 OT require executing it multiple times, resulting in large communication costs for the receiver.
In this work, we introduce a new technique for amortizing the cost of multiple rate-1 OTs. Specifically, based on standard pairing assumptions, we obtain a two-message rate-1 OT protocol for which the amortized cost per string-OT is asymptotically reduced to only four group elements. Our results lead to significant communication improvements in PSI and PIR, special cases of SFE for branching programs.
1. PIR: We obtain a rate-1 PIR scheme with client communication cost of $O(\lambda\cdot\log N)$ group elements for security parameter $\lambda$ and database size $N$. Notably, after a one-time setup (or one PIR instance), any following PIR instance only requires communication cost $O(\log N)$ number of group elements.
2. PSI with unbalanced inputs: We apply our techniques to private set intersection with unbalanced set sizes (where the receiver has a smaller set) and achieve receiver communication of $O((m+\lambda) \log N)$ group elements where $m, N$ are the sizes of the receiver and sender sets, respectively. Similarly, after a one-time setup (or one PSI instance), any following PSI instance only requires communication cost $O(m \cdot \log N)$ number of group elements. All previous sublinear-communication non-FHE based PSI protocols for the above unbalanced setting were also based on rate-1 OT, but incurred at least $O(\lambda^2 m \log N)$ group elements.

2020

EUROCRYPT

Two-Round Oblivious Transfer from CDH or LPN
📺
Abstract

We show a new general approach for constructing maliciously-secure two-round oblivious transfer (OT). Specifically, we provide a generic sequence of transformations to upgrade a very basic notion of two-roundOT, which we call elementary OT, to UC-secure OT. We then give simple constructions of elementary OT under the Computational Diffie-Hellman(CDH) assumption or the Learning Parity with Noise (LPN) assumption, yielding the first constructions of malicious (UC-secure) two-round OT under these assumptions. Since two-round OT is complete for two-round 2-party and multi-party computation in the malicious setting, we also achieve the first constructions of the latter under these assumptions.

2020

EUROCRYPT

Formalizing Data Deletion in the Context of the Right to be Forgotten
📺
Abstract

The right of an individual to request the deletion of their personal data by an entity that might be storing it -- referred to as \emph{the right to be forgotten} -- has been explicitly recognized, legislated, and exercised in several jurisdictions across the world, including the European Union, Argentina, and California. However, much of the discussion surrounding this right offers only an intuitive notion of what it means for it to be fulfilled -- of what it means for such personal data to be deleted.
In this work, we provide a formal definitional framework for the right to be forgotten using tools and paradigms from cryptography. In particular, we provide a precise definition of what could be (or should be) expected from an entity that collects individuals' data when a request is made of it to delete some of this data. Our framework captures most, though not all, relevant aspects of typical systems involved in data processing. While it cannot be viewed as expressing the statements of current laws (especially since these are rather vague in this respect), our work offers technically precise definitions that represent possibilities for what the law could reasonably expect, and alternatives for what future versions of the law could explicitly require.
Finally, with the goal of demonstrating the applicability of our framework and definitions, we consider various natural and simple scenarios where the right to be forgotten comes up. For each of these scenarios, we highlight the pitfalls that arise even in genuine attempts at implementing systems offering deletion guarantees, and also describe technological solutions that provably satisfy our definitions. These solutions bring together techniques built by various communities.

2020

EUROCRYPT

Candidate iO From Homomorphic Encryption Schemes
📺
Abstract

We propose a new approach to construct general-purpose indistinguishability obfuscation (iO). Our construction is obtained via a new intermediate primitive that we call split fully-homomorphic encryption (split FHE), which we show to be sufficient for constructing iO. Specifically, split FHE is FHE where decryption takes the following two-step syntactic form: (i) A secret decryption step uses the secret key and produces a hint which is (asymptotically) shorter than the length of the encrypted message, and (ii) a public decryption step that only requires the ciphertext and the previously generated hint (and not the entire secret key), and recovers the encrypted message. In terms of security, the hints for a set of ciphertexts should not allow one to violate semantic security for any other ciphertexts.
Next, we show a generic candidate construction of split FHE based on three building blocks: (i) A standard FHE scheme with linear decrypt-and-multiply (which can be instantiated with essentially all LWE-based constructions), (ii) a linearly homomorphic encryption scheme with short decryption hints (such as the Damgard-Jurik encryption scheme, based on the DCR problem), and (iii) a cryptographic hash function (which can be based on a variety of standard assumptions). Our approach is heuristic in the sense that our construction is not provably secure and makes implicit assumptions about the interplay between these underlying primitives. We show evidence that this construction is secure by providing an argument in an appropriately defined oracle model.
We view our construction as a big departure from the state-of-the-art constructions, and it is in fact quite simple.

2020

PKC

Master-Key KDM-Secure IBE from Pairings
📺
Abstract

Identity-based encryption (IBE) is a generalization of public-key encryption (PKE) by allowing encryptions to be made to user identities. In this work, we seek to obtain IBE schemes that achieve key-dependent-message (KDM) security with respect to messages that depend on the master secret key. Previous KDM-secure schemes only achieved KDM security in simpler settings, in which messages may only depend on user secret keys. An important motivation behind studying master-KDM security is the application of this notion in obtaining generic constructions of KDM-CCA secure PKE, a primitive notoriously difficult to realize. We give the first IBE that achieves master-KDM security from standard assumptions in pairing groups. Our construction is modular and combines techniques from KDM-secure PKE based from hash-proof systems, together with IBE that admits a tight security proof in the multi-challenge setting, which happens to be unexpectedly relevant in the context of KDM security. In fact, to the best of our knowledge, this is the first setting where techniques developed in the context of realizing tightly secure cryptosystems have led to a new feasibility result. As a byproduct, our KDM-secure IBE, and thus the resulting KDM-CCA-secure PKE both enjoy a tight security reduction, independent of the number of challenge ciphertexts, which was not achieved before.

2020

TCC

Efficient Range-Trapdoor Functions and Applications: Rate-1 OT and More
Abstract

Substantial work on trapdoor functions (TDFs) has led to many powerful notions
and applications. However, despite tremendous work and progress, all known
constructions have prohibitively large public keys.
In this work, we introduce new techniques for realizing so-called range-trapdoor hash functions with short public keys. This notion, introduced by Döttling et al. [Crypto 2019], allows for encoding a range of indices into a public key in a way that the public key leaks no information about the range, yet an associated trapdoor enables recovery of the corresponding input part.
We give constructions of range-trapdoor hash functions, where for a given range $I$ the public key consists of $O(n)$ group elements, improving upon $O(n |I|)$ achieved by Döttling et al. Moreover, by designing our evaluation algorithm in a special way involving Toeplitz matrix multiplication and by showing how to perform fast-Fourier transforms in the exponent, we arrive at $O(n \log n)$ group operations for evaluation, improving upon $O(n^2)$, required of previous constructions. Our constructions rely on power-DDH assumptions in pairing-free groups.
As applications of our results we obtain
--- The first construction of (rate-1) lossy TDFs with public keys consisting of a linear number of group elements (without pairings).
--- Rate-1 string OT with receiver communication complexity of $O(n)$ group elements, where $n$ is the sender's message size, improving upon $O(n^2)$ [Crypto 2019]. This leads to a similar result in the context of private-information retrieval (PIR).
--- Semi-compact homomorphic encryption for branching programs: A construction of homomorphic encryption for branching programs, with ciphertexts consisting of $O(\lambda n d)$ group elements, improving upon $O(\lambda^2 n d)$. Here $\lambda $ denotes the security parameter, $n$ the input size and $d$ the depth of the program.

2020

TCC

Reusable Two-Round MPC from DDH
📺
Abstract

We present a reusable two-round multi-party computation (MPC) protocol from the Decisional Diffie Hellman assumption (DDH). In particular, we show how to upgrade any secure two-round MPC protocol to allow reusability of its first message across multiple computations, using Homomorphic Secret Sharing (HSS) and pseudorandom functions in NC1 — each of which can be instantiated from DDH.
In our construction, if the underlying two-round MPC protocol is secure against semi-honest adversaries (in the plain model) then so is our reusable two-round MPC protocol. Similarly, if the underlying two-round MPC protocol is secure against malicious adversaries (in the common random/reference string model) then so is our reusable two-round MPC protocol. Previously, such reusable two-round MPC protocols were only known under assumptions on lattices.
At a technical level, we show how to upgrade any two-round MPC protocol to a first message succinct two-round MPC protocol, where the first message of the protocol is generated independently of the computed circuit (though it is not reusable). This step uses homomorphic secret sharing (HSS) and low-depth pseudorandom functions. Next, we show a generic transformation that upgrades any first message succinct two-round MPC to allow for reusability of its first message.

2020

TCC

Constant Ciphertext-Rate Non-Committing Encryption from Standard Assumptions
📺
Abstract

Non-committing encryption (NCE) is a type of public key encryption which comes with the ability to equivocate ciphertexts to encryptions of arbitrary messages, i.e., it allows one to find coins for key generation and encryption which ``explain'' a given ciphertext as an encryption of any message. NCE is the cornerstone to construct adaptively secure multiparty computation [Canetti et al. STOC'96] and can be seen as the quintessential notion of security for public key encryption to realize ideal communication channels.
A large body of literature investigates what is the best message-to-ciphertext ratio (i.e., the rate) that one can hope to achieve for NCE. In this work we propose a near complete resolution to this question and we show how to construct NCE with constant rate in the plain model from a variety of assumptions, such as the hardness of the learning with errors (LWE), the decisional Diffie-Hellman (DDH), or the quadratic residuosity (QR) problem. Prior to our work, constructing NCE with constant rate required a trusted setup and indistinguishability obfuscation [Canetti et al. ASIACRYPT'17].

2020

TCC

FHE-Based Bootstrapping of Designated-Prover NIZK
📺
Abstract

We present a novel tree-based technique that can convert any designated-prover NIZK proof system (DP-NIZK) which maintains zero-knowledge only for single statement, into one that allows to prove an unlimited number of statements in ZK, while maintaining all parameters succinct. Our transformation requires leveled fully-homomorphic encryption. We note that single-statement DP-NIZK can be constructed from any one-way function.
We also observe a two-way derivation between DP-NIZK and attribute-based signatures (ABS), and as a result derive now constructions of ABS and homomorphic signatures (HS).
Our construction improves upon the prior construction of lattice-based DP-NIZK by Kim and Wu (Crypto 2018) since we only require leveled FHE as opposed to HS (which also translates to improved LWE parameters when instantiated). Alternatively, the recent construction of NIZK without preprocessing from either circular-secure FHE (Canetti et al., STOC 2019) or polynomial Learning with Errors (Peikert and Shiehian, Crypto 2019) could be used to obtain a similar final statement. Nevertheless, we note that our statement is formally incomparable to these works (since leveled FHE is not known to imply circular secure FHE or the hardness of LWE). We view this as evidence for the potential in our technique, which we hope can find additional applications in future works.

2019

PKC

Registration-Based Encryption from Standard Assumptions
Abstract

The notion of Registration-Based Encryption (RBE) was recently introduced by Garg, Hajiabadi, Mahmoody, and Rahimi [TCC’18] with the goal of removing the private-key generator (PKG) from IBE. Specifically, RBE allows encrypting to identities using a (compact) master public key, like how IBE is used, with the benefit that the PKG is substituted with a weaker entity called “key curator” who has no knowledge of any secret keys. Here individuals generate their secret keys on their own and then publicly register their identities and their corresponding public keys to the key curator. Finally, individuals obtain “rare” decryption-key updates from the key curator as the population grows. In their work, they gave a construction of RBE schemes based on the combination of indistinguishability obfuscation and somewhere statistically binding hash functions. However, they left open the problem of constructing RBE schemes based on standard assumptions.In this work, we resolve the above problem and construct RBE schemes based on standard assumptions (e.g., CDH or LWE). Furthermore, we show a new application of RBE in a novel context. In particular, we show that anonymous variants of RBE (which we also construct under standard assumptions) can be used for realizing abstracts forms of anonymous messaging tasks in simple scenarios in which the parties communicate by writing messages on a shared board in a synchronized way.

2019

EUROCRYPT

New Techniques for Efficient Trapdoor Functions and Applications
📺
Abstract

We develop techniques for constructing trapdoor functions (TDFs) with short image size and advanced security properties. Our approach builds on the recent framework of Garg and Hajiabadi [CRYPTO 2018]. As applications of our techniques, we obtainThe first construction of deterministic-encryption schemes for block-source inputs (both for the CPA and CCA cases) based on the Computational Diffie-Hellman (CDH) assumption. Moreover, by applying our efficiency-enhancing techniques, we obtain CDH-based schemes with ciphertext size linear in plaintext size.The first construction of lossy TDFs based on the Decisional Diffie-Hellman (DDH) assumption with image size linear in input size, while retaining the lossiness rate of [Peikert-Waters STOC 2008].
Prior to our work, all constructions of deterministic encryption based even on the stronger DDH assumption incurred a quadratic gap between the ciphertext and plaintext sizes. Moreover, all DDH-based constructions of lossy TDFs had image size quadratic in the input size.At a high level, we break the previous quadratic barriers by introducing a novel technique for encoding input bits via hardcore output bits with the use of erasure-resilient codes. All previous schemes used group elements for encoding input bits, resulting in quadratic expansions.

2019

CRYPTO

Trapdoor Hash Functions and Their Applications
📺
Abstract

We introduce a new primitive, called trapdoor hash functions (TDH), which are hash functions $$\mathsf {H}: \{0,1\}^n \rightarrow \{0,1\}^\lambda $$ with additional trapdoor function-like properties. Specifically, given an index $$i\in [n]$$, TDHs allow for sampling an encoding key $$\mathsf {ek}$$ (that hides i) along with a corresponding trapdoor. Furthermore, given $$\mathsf {H}(x)$$, a hint value $$\mathsf {E}(\mathsf {ek},x)$$, and the trapdoor corresponding to $$\mathsf {ek}$$, the $$i^{th}$$ bit of x can be efficiently recovered. In this setting, one of our main questions is: How small can the hint value $$\mathsf {E}(\mathsf {ek},x)$$ be? We obtain constructions where the hint is only one bit long based on DDH, QR, DCR, or LWE.This primitive opens a floodgate of applications for low-communication secure computation. We mainly focus on two-message protocols between a receiver and a sender, with private inputs x and y, resp., where the receiver should learn f(x, y). We wish to optimize the (download) rate of such protocols, namely the asymptotic ratio between the size of the output and the sender’s message. Using TDHs, we obtain:1.The first protocols for (two-message) rate-1 string OT based on DDH, QR, or LWE. This has several useful consequences, such as:(a)The first constructions of PIR with communication cost poly-logarithmic in the database size based on DDH or QR. These protocols are in fact rate-1 when considering block PIR.(b)The first constructions of a semi-compact homomorphic encryption scheme for branching programs, where the encrypted output grows only with the program length, based on DDH or QR.(c)The first constructions of lossy trapdoor functions with input to output ratio approaching 1 based on DDH, QR or LWE.(d)The first constant-rate LWE-based construction of a 2-message “statistically sender-private” OT protocol in the plain model.2.The first rate-1 protocols (under any assumption) for n parallel OTs and matrix-vector products from DDH, QR or LWE.
We further consider the setting where f evaluates a RAM program y with running time $$T\ll |x|$$ on x. We obtain the first protocols with communication sublinear in the size of x, namely $$T\cdot \sqrt{|x|}$$ or $$T\cdot \root 3 \of {|x|}$$, based on DDH or, resp., pairings (and correlated-input secure hash functions).

2019

TCC

Leveraging Linear Decryption: Rate-1 Fully-Homomorphic Encryption and Time-Lock Puzzles
Abstract

We show how to combine a fully-homomorphic encryption scheme with linear decryption and a linearly-homomorphic encryption schemes to obtain constructions with new properties. Specifically, we present the following new results.
(1)Rate-1 Fully-Homomorphic Encryption: We construct the first scheme with message-to-ciphertext length ratio (i.e., rate) $$1-\sigma $$ for $$\sigma = o(1)$$. Our scheme is based on the hardness of the Learning with Errors (LWE) problem and $$\sigma $$ is proportional to the noise-to-modulus ratio of the assumption. Our building block is a construction of a new high-rate linearly-homomorphic encryption.One application of this result is the first general-purpose secure function evaluation protocol in the preprocessing model where the communication complexity is within additive factor of the optimal insecure protocol.(2)Fully-Homomorphic Time-Lock Puzzles: We construct the first time-lock puzzle where one can evaluate any function over a set of puzzles without solving them, from standard assumptions. Prior work required the existence of sub-exponentially hard indistinguishability obfuscation.

2019

ASIACRYPT

The Broadcast Message Complexity of Secure Multiparty Computation
Abstract

We study the broadcast message complexity of secure multiparty computation (MPC), namely, the total number of messages that are required for securely computing any functionality in the broadcast model of communication.MPC protocols are traditionally designed in the simultaneous broadcast model, where each round consists of every party broadcasting a message to the other parties. We show that this method of communication is sub-optimal; specifically, by eliminating simultaneity, it is, in fact, possible to reduce the broadcast message complexity of MPC.More specifically, we establish tight lower and upper bounds on the broadcast message complexity of n-party MPC for every $$t<n$$ corruption threshold, both in the plain model as well as common setup models. For example, our results show that the optimal broadcast message complexity of semi-honest MPC can be much lower than 2n, but necessarily requires at least three rounds of communication. We also extend our results to the malicious setting in setup models.

2019

ASIACRYPT

Rate-1 Trapdoor Functions from the Diffie-Hellman Problem
Abstract

Trapdoor functions (TDFs) are one of the fundamental building blocks in cryptography. Studying the underlying assumptions and the efficiency of the resulting instantiations is therefore of both theoretical and practical interest. In this work we improve the input-to-image rate of TDFs based on the Diffie-Hellman problem. Specifically, we present: (a)A rate-1 TDF from the computational Diffie-Hellman (CDH) assumption, improving the result of Garg, Gay, and Hajiabadi [EUROCRYPT 2019], which achieved linear-size outputs but with large constants. Our techniques combine non-binary alphabets and high-rate error-correcting codes over large fields.(b)A rate-1 deterministic public-key encryption satisfying block-source security from the decisional Diffie-Hellman (DDH) assumption. While this question was recently settled by Döttling et al. [CRYPTO 2019], our scheme is conceptually simpler and concretely more efficient. We demonstrate this fact by implementing our construction.

2018

CRYPTO

Two-Round Multiparty Secure Computation Minimizing Public Key Operations
📺
Abstract

We show new constructions of semi-honest and malicious two-round multiparty secure computation protocols using only (a fixed)
$$\mathsf {poly}(n,\lambda )$$
poly(n,λ) invocations of a two-round oblivious transfer protocol (which use expensive public-key operations) and
$$\mathsf {poly}(\lambda , |C|)$$
poly(λ,|C|) cheaper one-way function calls, where
$$\lambda $$
λ is the security parameter, n is the number of parties, and C is the circuit being computed. All previously known two-round multiparty secure computation protocols required
$$\mathsf {poly}(\lambda ,|C|)$$
poly(λ,|C|) expensive public-key operations.

2018

CRYPTO

Limits on the Power of Garbling Techniques for Public-Key Encryption
📺
Abstract

Understanding whether public-key encryption can be based on one-way functions is a fundamental open problem in cryptography. The seminal work of Impagliazzo and Rudich [STOC’89] shows that black-box constructions of public-key encryption from one-way functions are impossible. However, this impossibility result leaves open the possibility of using non-black-box techniques for achieving this goal.One of the most powerful classes of non-black-box techniques, which can be based on one-way functions (OWFs) alone, is Yao’s garbled circuit technique [FOCS’86]. As for the non-black-box power of this technique, the recent work of Döttling and Garg [CRYPTO’17] shows that the use of garbling allows us to circumvent known black-box barriers in the context of identity-based encryption.We prove that garbling of circuits that have OWF (or even random oracle) gates in them are insufficient for obtaining public-key encryption. Additionally, we show that this model also captures (non-interactive) zero-knowledge proofs for relations with OWF gates. This indicates that currently known OWF-based non-black-box techniques are perhaps insufficient for realizing public-key encryption.

2018

CRYPTO

Adaptive Garbled RAM from Laconic Oblivious Transfer
Abstract

We give a construction of an adaptive garbled RAM scheme. In the adaptive setting, a client first garbles a “large” persistent database which is stored on a server. Next, the client can provide garbling of multiple adaptively and adversarially chosen RAM programs that execute and modify the stored database arbitrarily. The garbled database and the garbled program should reveal nothing more than the running time and the output of the computation. Furthermore, the sizes of the garbled database and the garbled program grow only linearly in the size of the database and the running time of the executed program respectively (up to poly logarithmic factors). The security of our construction is based on the assumption that laconic oblivious transfer (Cho et al., CRYPTO 2017) exists. Previously, such adaptive garbled RAM constructions were only known using indistinguishability obfuscation or in random oracle model. As an additional application, we note that this work yields the first constant round secure computation protocol for persistent RAM programs in the malicious setting from standard assumptions. Prior works did not support persistence in the malicious setting.

2018

CRYPTO

On the Round Complexity of OT Extension
📺
Abstract

We show that any OT extension protocol based on one-way functions (or more generally any symmetric-key primitive) either requires an additional round compared to the base OTs or must make a non-black-box use of one-way functions. This result also holds in the semi-honest setting or in the case of certain setup models such as the common random string model. This implies that OT extension in any secure computation protocol must come at the price of an additional round of communication or the non-black-box use of symmetric key primitives. Moreover, we observe that our result is tight in the sense that positive results can indeed be obtained using non-black-box techniques or at the cost of one additional round of communication.

2018

CRYPTO

Trapdoor Functions from the Computational Diffie-Hellman Assumption
📺
Abstract

Trapdoor functions (TDFs) are a fundamental primitive in cryptography. Yet, the current set of assumptions known to imply TDFs is surprisingly limited, when compared to public-key encryption. We present a new general approach for constructing TDFs. Specifically, we give a generic construction of TDFs from any Chameleon Encryption (Döttling and Garg [CRYPTO’17]) satisfying a novel property which we call recyclability. By showing how to adapt current Computational Diffie-Hellman (CDH) based constructions of chameleon encryption to yield recyclability, we obtain the first construction of TDFs with security proved under the CDH assumption. While TDFs from the Decisional Diffie-Hellman (DDH) assumption were previously known, the possibility of basing them on CDH had remained open for more than 30 years.

2018

PKC

New Constructions of Identity-Based and Key-Dependent Message Secure Encryption Schemes
Abstract

Recently, Döttling and Garg (CRYPTO 2017) showed how to build identity-based encryption (IBE) from a novel primitive termed Chameleon Encryption, which can in turn be realized from simple number theoretic hardness assumptions such as the computational Diffie-Hellman assumption (in groups without pairings) or the factoring assumption. In a follow-up work (TCC 2017), the same authors showed that IBE can also be constructed from a slightly weaker primitive called One-Time Signatures with Encryption (OTSE).In this work, we show that OTSE can be instantiated from hard learning problems such as the Learning With Errors (LWE) and the Learning Parity with Noise (LPN) problems. This immediately yields the first IBE construction from the LPN problem and a construction based on a weaker LWE assumption compared to previous works.Finally, we show that the notion of one-time signatures with encryption is also useful for the construction of key-dependent-message (KDM) secure public-key encryption. In particular, our results imply that a KDM-secure public key encryption can be constructed from any KDM-secure secret-key encryption scheme and any public-key encryption scheme.

2018

TCC

Registration-Based Encryption: Removing Private-Key Generator from IBE
Abstract

In this work, we introduce the notion of registration-based encryption (RBE for short) with the goal of removing the trust parties need to place in the private-key generator in an IBE scheme. In an RBE scheme, users sample their own public and secret keys. There will also be a “key curator” whose job is only to aggregate the public keys of all the registered users and update the “short” public parameter whenever a new user joins the system. Encryption can still be performed to a particular recipient using the recipient’s identity and any public parameters released subsequent to the recipient’s registration. Decryption requires some auxiliary information connecting users’ public (and secret) keys to the public parameters. Because of this, as the public parameters get updated, a decryptor may need to obtain “a few” additional auxiliary information for decryption. More formally, if n is the total number of identities and $$\mathrm {\kappa }$$κ is the security parameter, we require the following.Efficiency requirements: (1) A decryptor only needs to obtain updated auxiliary information for decryption at most $$O(\log n)$$O(logn) times in its lifetime, (2) each of these updates are computed by the key curator in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn), and (3) the key curator updates the public parameter upon the registration of a new party in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). Properties (2) and (3) require the key curator to have random access to its data.Compactness requirements: (1) Public parameters are always at most $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) bit, and (2) the total size of updates a user ever needs for decryption is also at most $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) bits.We present feasibility results for constructions of RBE based on indistinguishably obfuscation. We further provide constructions of weakly efficient RBE, in which the registration step is done in $${\text {poly}}(\mathrm {\kappa },n)$$poly(κ,n), based on CDH, Factoring or LWE assumptions. Note that registration is done only once per identity, and the more frequent operation of generating updates for a user, which can happen more times, still runs in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). We leave open the problem of obtaining standard RBE (with $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) registration time) from standard assumptions.

2018

TCC

Two-Round MPC: Information-Theoretic and Black-Box
Abstract

We continue the study of protocols for secure multiparty computation (MPC) that require only two rounds of interaction. The recent works of Garg and Srinivasan (Eurocrypt 2018) and Benhamouda and Lin (Eurocrypt 2018) essentially settle the question by showing that such protocols are implied by the minimal assumption that a two-round oblivious transfer (OT) protocol exists. However, these protocols inherently make a non-black-box use of the underlying OT protocol, which results in poor concrete efficiency. Moreover, no analogous result was known in the information-theoretic setting, or alternatively based on one-way functions, given an OT correlations setup or an honest majority.Motivated by these limitations, we study the possibility of obtaining information-theoretic and “black-box” implementations of two-round MPC protocols. We obtain the following results:Two-round MPC from OT correlations. Given an OT correlations setup, we get protocols that make a black-box use of a pseudorandom generator (PRG) and are secure against a malicious adversary corrupting an arbitrary number of parties. For a semi-honest adversary, we get similar information-theoretic protocols for branching programs.New NIOT constructions. Towards realizing OT correlations, we extend the DDH-based non-interactive OT (NIOT) protocol of Bellare and Micali (Crypto’89) to the malicious security model, and present new NIOT constructions from the Quadratic Residuosity Assumption (QRA) and the Learning With Errors (LWE) assumption.Two-round black-box MPC with strong PKI setup. Combining the two previous results, we get two-round MPC protocols that make a black-box use of any DDH-hard or QRA-hard group. The protocols can offer security against a malicious adversary, and require a PKI setup that depends on the number of parties and the size of computation, but not on the inputs or the identities of the participating parties.Two-round honest-majority MPC from secure channels. Given secure point-to-point channels, we get protocols that make a black-box use of a pseudorandom generator (PRG), as well as information-theoretic protocols for branching programs. These protocols can tolerate a semi-honest adversary corrupting a strict minority of the parties, where in the information-theoretic case the complexity is exponential in the number of parties.

2018

TCC

A Simple Construction of iO for Turing Machines
Abstract

We give a simple construction of indistinguishability obfuscation for Turing machines where the time to obfuscate grows only with the description size of the machine and otherwise, independent of the running time and the space used. While this result is already known [Koppula, Lewko, and Waters, STOC 2015] from
$$i\mathcal {O}$$
for circuits and injective pseudorandom generators, our construction and its analysis are conceptually much simpler. In particular, the main technical component in the proof of our construction is a simple combinatorial pebbling argument [Garg and Srinivasan, EUROCRYPT 2018]. Our construction makes use of indistinguishability obfuscation for circuits and
$$\mathrm {somewhere\, statistically\, binding\, hash\, functions}$$
.

2017

ASIACRYPT

2016

CRYPTO

2014

CRYPTO

#### Program Committees

- Asiacrypt 2022
- TCC 2021
- Crypto 2020
- Eurocrypt 2019
- PKC 2017
- PKC 2016
- Crypto 2016
- Asiacrypt 2016
- TCC 2015
- Crypto 2015
- PKC 2014
- Asiacrypt 2014

#### Coauthors

- Navid Alamati (1)
- Saikrishna Badrinarayanan (1)
- Boaz Barak (1)
- James Bartusek (3)
- Raghav Bhaskar (1)
- Alexander Bienstock (2)
- Nir Bitansky (1)
- Elette Boyle (1)
- Zvika Brakerski (5)
- Pedro Branco (2)
- Melissa Chase (1)
- Rahul Chatterjee (1)
- Chongwon Cho (2)
- Arka Rai Choudhuri (1)
- Dana Dachman-Soled (1)
- Yevgeniy Dodis (1)
- Nico Döttling (13)
- Jaiden Fairoze (1)
- Romain Gay (2)
- Craig Gentry (5)
- Aarushi Goel (1)
- Shafi Goldwasser (1)
- Vipul Goyal (2)
- Garrison Grogan (1)
- Divya Gupta (3)
- Mohammad Hajiabadi (16)
- Shai Halevi (5)
- Yuval Ishai (4)
- Abhishek Jain (10)
- Zhengzhong Jin (2)
- Yael Tauman Kalai (3)
- Dakshita Khurana (1)
- Susumu Kiyoshima (2)
- Abishek Kumarasubramanian (1)
- Eyal Kushilevitz (1)
- Jialin Li (1)
- Xiaohui Liang (1)
- Kevin Liu (1)
- Satyanarayana V. Lokam (1)
- Adriana López-Alt (1)
- Mohammad Mahmoody (6)
- Giulio Malavolta (8)
- Daniel Masny (4)
- Izaak Meckler (1)
- Peihan Miao (4)
- Eric Miles (1)
- Ameer Mohammed (3)
- Payman Mohassel (1)
- Tamer Mour (1)
- Pratyay Mukherjee (5)
- Rafail Ostrovsky (8)
- Omkant Pandey (9)
- Omer Paneth (1)
- Charalampos Papamanthou (1)
- Guru Vamsi Policharla (1)
- Antigoni Polychroniadou (3)
- Sihang Pu (1)
- Srinivasan Raghuraman (1)
- Ahmadreza Rahimi (2)
- Vanishree Rao (1)
- Mariana Raykova (1)
- Paul Rösler (1)
- Amit Sahai (11)
- Dominique Schröder (1)
- Sruthi Sekar (2)
- Sina Shiehian (2)
- Rohit Sinha (1)
- Akshayaram Srinivasan (11)
- Rotem Tsabary (1)
- Dominique Unruh (1)
- Prashant Nalini Vasudevan (1)
- Ivan Visconti (2)
- Akshay Wadia (2)
- Mingyuan Wang (2)
- Brent Waters (1)
- Daniel Wichs (3)
- Mark Zhandry (3)
- Jiaheng Zhang (1)
- Yinuo Zhang (2)