## CryptoDB

### Papers from TCC 2023

**Year**

**Venue**

**Title**

2023

TCC

(Verifiable) Delay Functions from Lucas Sequences
Abstract

Lucas sequences are constant-recursive integer sequences with a long history of applications in cryptography, both in the design of cryptographic schemes and cryptanalysis. In this work, we study the sequential hardness of computing Lucas sequences over an RSA modulus.
First, we show that modular Lucas sequences are at least as sequentially hard as the classical delay function given by iterated modular squaring proposed by Rivest, Shamir, and Wagner (MIT Tech. Rep. 1996) in the context of time-lock puzzles. Moreover, there is no obvious reduction in the other direction, which suggests that the assumption of sequential hardness of modular Lucas sequences is strictly weaker than that of iterated modular squaring. In other words, the sequential hardness of modular Lucas sequences might hold even in the case of an algorithmic improvement violating the sequential hardness of iterated modular squaring.
Second, we demonstrate the feasibility of constructing practically-efficient verifiable delay functions based on the sequential hardness of modular Lucas sequences. Our construction builds on the work of Pietrzak (ITCS 2019) by leveraging the intrinsic connection between the problem of computing modular Lucas sequences and exponentiation in an appropriate extension field.

2023

TCC

3-Party Secure Computation for RAMs: Optimal and Concretely Efficient
Abstract

A distributed oblivious RAM (DORAM) is a method for accessing a secret-shared memory while hiding the accessed locations. DORAMs are the key tool for secure multiparty computation (MPC) for RAM programs that avoids expensive RAM-to-circuit transformations.
We present new and improved 3-party DORAM protocols. For a logical memory of size N and for each logical operation, our DORAM requires O(log N) local CPU computation steps. This is known to be asymptotically optimal. Our DORAM satisfies passive security in the honest majority setting. Our technique results with concretely-efficient protocols and does not use expensive cryptography (such as re-randomizable or homomorphic encryption). Specifically, our DORAM is 25X faster than the known most efficient DORAM in the same setting.
Lastly, we extend our technique to handle malicious attackers at the expense of using slightly larger blocks (i.e., ω(log2 N) vs. Ω(log N)). To the best of our knowledge, this is the first concretely-efficient maliciously secure DORAM.
Technically, our construction relies on a novel concretely-efficient 3-party oblivious permutation protocol. We combine it with efficient non-oblivious hashing techniques (i.e., Cuckoo hashing) to get a distributed oblivious hash table. From this, we build a full-fledged DORAM using a distributed variant of the hierarchical approach of Goldreich and Ostrovsky (J. ACM ’96). These ideas, and especially the permutation protocol, are of independent interest.

2023

TCC

Agile Cryptography: A Universally Composable Approach
Abstract

Being capable of updating cryptographic algorithms is an inevitable and essential practice in cryptographic engineering. This cryptographic agility, as it has been called, is a fundamental desideratum for long term cryptographic system security that still poses significant challenges from a modeling perspective. For instance, current formulations of agility fail to express the fundamental security that is expected to stem from timely implementation updates, namely the fact that the system retains some of its security properties provided that the update is performed prior to the deprecated implementation becoming exploited.
In this work we put forth a novel framework for expressing updateability in the context of cryptographic primitives within the universal composition model. Our updatable ideal functionality framework provides a general template for expressing the security we expect from cryptographic agility capturing in a fine grained manner all the properties that can be retained across implementation updates. We exemplify our framework over two basic cryptographic primitives, digital signatures and non-interactive zero-knowledge (NIZK), where we demonstrate how to achieve updateability with consistency and backwards-compatibility across updates in a composable manner. We also illustrate how our notion is a continuation of a much broader scope of the concept of agility introduced by Acar, Belenkiy, Bellare, and Cash in Eurocrypt 2010 in the context of symmetric cryptographic primitives.

2023

TCC

Algebraic Group Model with Oblivious Sampling
Abstract

In the algebraic group model (AGM), an adversary has to return with each group element a linear representation with respect to input group elements. In many groups, it is easy to sample group elements obliviously without knowing such linear representations. Since the AGM does not model this, it can be used to prove the security of spurious knowledge assumptions. We propose AGM with oblivious sampling (AGMOS), a variant of the AGM where the adversary has additional access to an oracle that allows sampling group elements obliviously from some distribution. We separate AGM and AGMOS by classifying the family of ``total knowledge-of-exponent'' assumptions, showing that while they are all secure in the AGM (even insecure ones), most are not secure in the AGMOS if the DL holds. We show that many known AGM reductions go through also in the AGMOS, assuming a novel falsifiable assumption $\TOFR$. We prove that $\TOFR$ is secure in a version of GGM with oblivious sampling.

2023

TCC

Anonymous Permutation Routing
Abstract

The Non-Interactive Anonymous Router (NIAR) model was introduced by Shi and Wu \cite{SW21} as an alternative to conventional solutions to the anonymous routing problem, in which a set of senders wish to send messages to a set of receivers. In contrast to most known approaches to support anonymous routing (e.g. mix-nets, DC-nets, etc.), which rely on a network of routers communicating with users via interactive protocols, the NIAR model assumes a *single* router and is inherently *non-interactive* (after an initial setup phase). In addition to being non-interactive, the NIAR model is compelling due to the security it provides: instead of relying on the honesty of some subset of the routers, the NIAR model requires anonymity even if the router (as well as an arbitrary subset of senders/receivers) is corrupted by an honest-but-curious adversary.
In this paper, we present a protocol for the NIAR model that improves upon the results from \cite{SW21} in two ways:
- Improved computational efficiency (quadratic to near linear): Our protocol matches the communication complexity of \cite{SW21} for each sender/receiver, while reducing the computational overhead for the router to polylog overhead instead of linear overhead.
- Relaxation of assumptions: Security of the protocol in \cite{SW21} relies on the Decisional Linear assumption in bilinear groups; while security for our protocol follows from the existence of any rate-1 oblivious transfer (OT) protocol (instantiations of which are known to exist under the DDH, QR and LWE assumptions \cite{DGI19,GHO20}).

2023

TCC

Beyond MPC-in-the-Head: Black-Box Constructions of Short Zero-Knowledge Proofs
Abstract

In their seminal work, Ishai, Kushilevitz, Ostrovsky, and Sahai (STOC`07) presented the MPC-in-the-Head paradigm, which shows how to design Zero-Knowledge Proofs (ZKPs) from secure Multi-Party Computation (MPC) protocols. This paradigm has since then revolutionized and modularized the design of efficient ZKP systems, with far-reaching applications beyond ZKPs.
However, to the best of our knowledge, all previous instantiations relied on {\em fully-secure} MPC protocols and have not been able to leverage the fact that the paradigm only imposes relatively weak privacy and correctness requirements on the underlying MPC.
In this work, we extend the MPC-in-the-Head paradigm to {\em game-based} cryptographic primitives supporting homomorphic computations (e.g., fully-homomorphic encryption, functional encryption, randomized encodings, homomorphic secret sharing, and more). Specifically, we present a simple yet generic compiler from these primitives to ZKPs which use the underlying primitive as a black box. We also generalize our paradigm to capture commit-and-prove protocols, and use it to devise tight black-box compilers from Interactive (Oracle) Proofs to ZKPs, assuming One-Way Functions (OWFs).
We use our paradigm to obtain several new ZKP constructions:
1. The first constant-round ZKPs for $\NP$ relations computable in (non-uniform) $\NC^1$, with communication approaching witness length (specifically, $n\cdot\poly\left(\secp\right)$, where $n$ is the witness length, and $\secp$ is a security parameter) assuming DCR.
2. Constant-round ZKPs for \NP\ relations computable in bounded polynomial space, with $O\left(n\right)+o\left(m\right)\cdot\poly\left(\secp\right)$ communication assuming OWFs, where $m$ is the instance length.
This gives a black-box alternative to a recent non-black-box construction of Nassar and Ron (CRYPTO`22).
3. ZKPs for \NP\ relations computable by a logspace-uniform family of depth-$d\left(m\right)$ circuits, with $n\cdot\poly\left(\secp,d\left(m\right)\right)$ communication assuming OWFs. This gives a black-box alternative to a result of Goldwasser, Kalai and Rothblum (JACM).

2023

TCC

Broadcast-Optimal Four-Round MPC in the Plain Model
Abstract

The prior works of Cohen, Garay and Zikas (Eurocrypt 2020), Damgård, Magri, Ravi, Siniscalchi and Yakoubov (Crypto 2021) and Damgård, Ravi, Siniscalchi and Yakoubov (Eurocrypt 2023) study 2-round Multi-Party Computation (where some form of set-up is required). Motivated by the fact that broadcast is an expensive resource, they focus on so-called broadcast optimal MPC, i.e., they give tight characterizations of which security guarantees are achievable, if broadcast is available in the first round, the second round, both rounds, or not at all.
This work considers the natural question of characterizing broadcast optimal MPC in the plain model where no set-up is assumed. We focus on 4-round protocols, since 4 is known to be the minimal number of rounds required to securely realize any functionality with black-box simulation. We give a complete characterization of which security guarantees, (namely selective abort, selective identifiable abort, unanimous abort and identifiable abort) are feasible or not, depending on the exact selection of rounds in which broadcast is available.

2023

TCC

CASE: A New Frontier in Public-Key Authenticated Encryption
Abstract

We introduce a new cryptographic primitive, called Completely Anonymous Signed Encryption(CASE). CASE is a public-key authenticated encryption primitive, that offers anonymity for senders as well as receivers. A “case-packet” should appear, without a (decryption) key for opening it, to be a blackbox that reveals no information at all about its contents. To decase a case-packet fully – so that the message is retrieved and authenticated – a verification key is also required.
Defining security for this primitive is subtle. We present a relatively simple Chosen Objects Attack (COA) security definition. Validating this definition, we show that it implies a comprehensive indistinguishability-preservation definition in the real-ideal paradigm. To obtain the latter definition, we extend the Cryptographic Agents framework to allow maliciously created objects.
We also provide a novel and practical construction for COA-secure CASE under standard assumptions in public-key cryptography, and in the standard model.
We believe CASE can be a staple in future cryptographic libraries, thanks to its robust security guarantees and efficient instantiations based on standard assumptions.

2023

TCC

Chainable Functional Commitments for Unbounded-Depth Circuits
Abstract

A functional commitment (FC) scheme allows one to commit to a vector $\vec{x}$ and later produce a short opening proof of $(f, f(\vec{x}))$ for any admissible function $f$. Since their inception, FC schemes supporting ever more expressive classes of functions have been proposed.
In this work, we introduce a novel primitive that we call chainable functional commitment (CFC), which extends the functionality of FCs by allowing one to 1) open to functions of multiple inputs $f(\vec x_1, \ldots, \vec x_m)$ that are committed independently, 2) while preserving the output also in committed form. We show that CFCs for quadratic polynomial maps generically imply FCs for circuits. Then, we efficiently realize CFCs for quadratic polynomials over pairing groups and lattices, resulting in the first FC schemes for circuits of unbounded depth based on either pairing-based or lattice-based falsifiable assumptions. Our FCs require fixing a-priori only the maximal width of the circuit to be evaluated, and have opening proofs whose size only depends on the depth of the circuit. Additionally, our FCs feature other nice properties such as being additively homomorphic and supporting sublinear-time verification after offline preprocessing.
Using a recent transformation that constructs homomorphic signatures (HS) from FCs, we obtain the first pairing- and lattice-based realisations of HS for bounded-width, but unbounded-depth, circuits. Prior to this work, the only HS for general circuits is lattice-based and requires bounding the circuit depth at setup time.

2023

TCC

Combinatorially Homomorphic Encryption
Abstract

Homomorphic encryption enables public computation over encrypted data. In the past few decades, homomorphic encryption has become a staple of both the theory and practice of cryptography. Nevertheless, while there is a general loose understanding of what it means for a scheme to be homomorphic, to date there is no single unifying minimal definition that captures all schemes.
In this work, we propose a new definition, which we refer to as \emph{combinatorially homomorphic encryption}, which attempts to give a broad base that captures the intuitive meaning of homomorphic encryption.
Our notion relates the ability to accomplish some task when given a ciphertext, to accomplishing the same task without the ciphertext, in the context of \emph{communication complexity}. Thus, we say that a scheme is combinatorially homomorphic if there exists a communication complexity problem $f(x,y)$ (where $x$ is Alice's input and $y$ is Bob's input) which requires communication $c$, but can be solved with communication less than $c$ when Alice is given in addition also an encryption $E_k(y)$ of Bob's input (using Bob's key $k$).
We show that this definition indeed captures pre-existing notions of homomorphic encryption and (suitable variants are) sufficiently strong to derive prior known implications of homomorphic encryption in a conceptually appealing way. These include constructions of (lossy) public-key encryption from homomorphic private-key encryption, as well as collision-resistant hash functions and private information retrieval schemes.

2023

TCC

Communication Lower Bounds of Key-Agreement Protocols via Density Increment Arguments
Abstract

Constructing key-agreement protocols in the random oracle model (ROM) is a viable method to assess the feasibility of developing public-key cryptography within Minicrypt. Unfortunately, as shown by Impagliazzo and Rudich (STOC 1989) and Barak and Mahmoody (Crypto 2009), such protocols can only guarantee limited security: any $\ell$-query protocol can be attacked by an $O(\ell^2)$-query adversary. This quadratic gap matches the key-agreement protocol proposed by Merkle (CACM 78), known as Merkle's Puzzles.
Besides query complexity, the communication complexity of key-agreement protocols in the ROM is also an interesting question in the realm of find-grained cryptography, even though only limited security is achievable. Haitner et al. (ITCS 2019) first observed that in Merkle's Puzzles, to obtain secrecy against an eavesdropper with $O(\ell^2)$ queries, the honest parties must exchange $\Omega(\ell)$ bits.
Therefore, they conjectured that high communication complexity is unavoidable, any $\ell$-query protocols with $c$ bits of communication could be attacked by an $O(c\cdot \ell)$-query adversary.
This, if true, will suggest that Merkle's Puzzle is also optimal regarding communication complexity. Building upon techniques from communication complexity, Haitner et al. (ITCS 2019) confirmed this conjecture for two types of key agreement protocols with certain natural properties.
This work affirms the above conjecture for all non-adaptive protocols with perfect completeness.
Our proof uses a novel idea called \textit{density increment argument}. This method could be of independent interest as it differs from previous communication lower bounds techniques (and bypasses some technical barriers).

2023

TCC

Composable Long-Term Security with Rewinding
Abstract

Long-term security, a variant of Universally Composable (UC) security introduced by Müller-Quade and Unruh (TCC ’07, JoC ’10), allows to analyze the security of protocols in a setting where all hardness assumptions no longer hold after the protocol execution has finished. Such a strict notion is highly desirable when properties such as input privacy need to be guaranteed for a long time, e.g. with zero-knowledge proofs for secure electronic voting. Strong impossibility results rule out so-called long-term-revealing setups, e.g. a common reference string (CRS), to achieve long-term security, with known constructions for long-term security requiring hardware assumptions, e.g. signature cards.
We circumvent these impossibility results with new techniques, enabling rewinding-based simulation in a way that universal composability is achieved. This allows us to construct a long-term-secure composable commitment scheme in the CRS-hybrid model, which is provably impossible in the notion of Müller-Quade and Unruh. We base our construction on a statistically hiding commitment scheme in the CRS-hybrid model with CCA-like properties. To provide a CCA oracle, we cannot rely on super-polynomial extraction techniques and instead extract the value committed to via rewinding. To this end, we incorporate rewinding-based commitment extraction into the UC framework via a helper in analogy to Canetti, Lin and Pass (FOCS 2010), allowing both adversary and environment to extract statistically hiding commitments.
Our new framework provides the first setting in which a commitment scheme that is both statistically hiding and universally composable can be constructed from standard polynomial-time hardness assumptions and a CRS only. We also prove that our CCA oracle is k-robust extractable. This asserts that extraction is possible without rewinding a concurrently executed k-round protocol. Consequently any k-round (standard) UC-secure protocol remains secure in the presence of our helper.
Finally, we prove that building long-term-secure oblivious transfer (and thus general two-party computations) from long-term-revealing setups remains impossible in our setting. Still, our long-term-secure commitment scheme suffices for natural applications, such as long-term secure and composable (commit-and-prove) zero-knowledge arguments of knowledge.

2023

TCC

Concurrent Asynchronous Byzantine Agreement in Expected-Constant Rounds, Revisited
Abstract

It is well known that without randomization, Byzantine agreement (BA) requires a linear number of rounds in the synchronous setting, while it is flat out impossible in the asynchronous setting. The primitive which allows to bypass the above limitation is known as oblivious common coin (OCC). It allows parties to agree with constant probability on a random coin, where agreement is oblivious, i.e., players are not aware whether or not agreement has been achieved.
The starting point of our work is the observation that no known protocol exists for information-theoretic multi-valued OCC with optimal resiliency in the asynchronous setting (with eventual message delivery). This apparent hole in the literature is particularly problematic, as multi-valued OCC is implicitly or explicitly used in several constructions.
In this paper, we present the first information-theoretic multi-valued OCC protocol in the asynchronous setting with optimal resiliency, i.e., tolerating t<n/3 corruptions, thereby filling this important gap. Further, our protocol efficiently implements OCC with an exponential-size domain, a property which is not even achieved by known constructions in the simpler, synchronous setting.
We then turn to the problem of round-preserving parallel composition of asynchronous BA. A protocol for this task was proposed by Ben-Or and El-Yaniv [Distributed Computing ’03]. Their construction, however, is flawed in several ways. Thus, as a second contribution, we provide a simpler, more modular protocol for the above task. Finally, and as a contribution of independent interest, we provide proofs in Canetti's Universal Composability framework; this makes our work the first one offering composability guarantees, which are important as BA is a core building block of secure multi-party computation protocols.

2023

TCC

Counting Unpredictable Bits: A Simple PRG from One-way Functions
Abstract

A central result in the theory of Cryptography, by Hastad, Imagliazzo, Luby and Levin [SICOMP’99], demonstrates that the existence one-way functions (OWF) implies the existence of pseudo-random generators (PRGs). Despite the fundamental importance of this result, and several elegant improvements/simplifications, analyses of constructions of PRGs from OWFs remain complex (both conceptually and technically).
Our goal is to provide a construction of a PRG from OWFs with a simple proof of security; we thus focus on the setting of non-uniform security (i.e., we start off with a OWF secure against non-uniform PPT, and we aim to get a PRG secure against non-uniform PPT).
Our main result is a construction of a PRG from OWFs with a self-contained, simple, proof of security, relying only on the Goldreich-Levin Theorem (and the Chernoff bound). Although our main goal is simplicity, the construction, and a variant there-of, also improves the efficiency—in terms of invocations and seed lengths—of the state-of-the-art constructions due to [Haitner-Reingold-Vadhan, STOC’10] and [Vadhan-Zheng, STOC’12], by a factor O(log2 n).
The key novelty in our analysis is a generalization of the Blum-Micali [FOCS’82] notion of unpredictabilty—rather than requiring that every bit in the output of a function is unpredictable, we count how many unpredictable bits a function has, and we show that any OWF on n input bits (after hashing the input and the output) has n + O(log n) unpredictable output bits. Such unpredictable bits can next be “extracted” into a pseudorandom string using standard techniques.

2023

TCC

Cryptography from Planted Graphs: Security with Logarithmic-Size Messages
Abstract

We study the following broad question about cryptographic primitives: is it possible to achieve security against arbitrary poly(n)-size adversaries with O(log n)-size messages? It is common knowledge that the answer is “no” unless information-theoretic security is possible. In this work, we revisit this question by considering the setting of cryptography with public information and computational security.
We obtain the following main results, assuming variants of well-studied
intractability assumptions:
1. A private simultaneous messages (PSM) protocol for every f : [n] × [n] → {0, 1} with (1 + eps) log n-bit messages, beating the known lower bound on information-theoretic PSM. We apply this towards non-interactive secure 3-party computation with similar message size in the preprocessing model, improving over previous 2-round protocols.
2. A secret-sharing scheme for any “forbidden-graph” access structure on n nodes with O(log n) share size.
3. On the negative side, we show that computational threshold secret-sharing schemes with public information require share size Ω(log log n). For arbitrary access structures, we show that computational security does not help with 1-bit shares.
The above positive results guarantee that any adversary of size n^{o(log n)} achieves an n^{−Ω(1)} distinguishing advantage. We show how to make the advantage negligible by slightly increasing the asymptotic message size, still improving over all known constructions.
The security of our constructions is based on the conjectured hardness of variants of the planted clique problem, which was extensively studied in the algorithms, statistical inference, and complexity-theory communities. Our work provides the first applications of such assumptions to
improving the efficiency of mainstream cryptographic primitives, gives evidence for the necessity of such assumptions, and gives rise to new questions in this domain that may be of independent interest.

2023

TCC

Distributed-Prover Interactive Proofs
Abstract

Interactive proof systems enable a verifier with limited resources to decide an intractable language (or compute a hard function) by communicating with a powerful but untrusted prover. Such systems guarantee soundness: the prover can only convince the verifier of true statements. This is a central notion in computer science with far-reaching implications. One key drawback of the classical model is that the data on which the prover operates must be held by a single machine.
In this work, we initiate the study of distributed-prover interactive proofs (dpIPs):
an untrusted cluster of machines, acting as a distributed prover, interacts with a single verifier. The machines in the cluster jointly store and operate on a massive data-set that no single machine can store. The goal is for the machines in the cluster to convince the verifier of the validity of some statement about its data-set. We formalize the communication and space constraints via the massively parallel computation (MPC) model, a widely accepted analytical framework capturing the computational power of massive data-centers.
Our main result is a compiler that generically augments any verification algorithm in the MPC model with a soundness guarantee. Concretely, for any language $L$ for which there is an MPC algorithm verifying whether $x{\in} L$, we design a new MPC protocol capable of convincing a verifier of the validity of $x\in L$ and where if $x\not\in L$, the verifier will reject almost surely reject, no matter what. The new protocol requires only slightly more rounds, i.e., a $\mathsf{poly}(\log N)$ blowup, and a slightly bigger memory per machine, i.e., $\mathsf{poly}(\lambda)$ blowup, where $N$ is the total size of the dataset and $\lambda$ is a security parameter independent of $N$.
En route, we introduce distributed-prover interactive oracle proofs (dpIOPs), a natural adaptation of the (by now classical) IOP model to the distributed prover setting. We design a dpIOP for algorithms in the MPC model and then tranlate them to ``plain model'' dpIPs via an adaptation of existing polynomial commitment schemes into the distributed prover setting.

2023

TCC

DORAM revisited: Maliciously secure RAM-MPC with logarithmic overhead
Abstract

Distributed Oblivious Random Access Memory (DORAM) is a secure multiparty protocol that allows a group of participants holding a secret-shared array to read and write to secret-shared locations within the array. The efficiency of a DORAM protocol is measured by the amount of communication and computation required per read/write query into the array. DORAM protocols are a necessary ingredient for executing Secure Multiparty Computation (MPC) in the RAM model.
Although DORAM has been widely studied, all existing DORAM protocols have focused on the setting where the DORAM servers are semi-honest. Generic techniques for upgrading a semi-honest DORAM protocol to the malicious model typically increase the asymptotic communication complexity of the DORAM scheme.
In this work, we present a 3-party DORAM protocol which requires $O((\kappa + D)\log N)$ communication and computation per query, for a database of size $N$ with $D$-bit values, where $\kappa$ is the security parameter. Our hidden constants in the big-O nation are small. We show that our protocol is UC-secure in the presence of a malicious, static adversary. This matches the communication and computation complexity of the best semi-honest DORAM protocols, and is the first malicious DORAM protocol with this complexity.

2023

TCC

Efficiently Testable Circuits without Conductivity
Abstract

The notion of “efficiently testable circuits” (ETC) was recently put forward by Baig et al. (ITCS’23). Informally, an ETC compiler takes as input any Boolean circuit C and outputs a circuit/inputs
tuple (C′, T) where (completeness) C′ is functionally equivalent to C and (security) if C′ is tampered in some restricted way, then this can be detected as C′ will err on at least one input in the small test set T. The compiler of Baig et al. detects tampering even if the adversary can tamper with all wires in the compiled circuit. Unfortunately, the model requires a strong “conductivity” restriction: the compiled circuit has gates with fan-out up to 3, but wires can only be tampered in one way even if the have fan-out greater than one. In this paper, we solve the main open question from their work and construct an ETC compiler without this conductivity restriction. While Baig et al. use gadgets computing the AND and OR of particular subsets of the wires, our compiler computes inner products with random vectors. We slightly relax their security notion and only require that tampering
is detected with high probability over the choice of the randomness. Our compiler increases the size of the circuit by only a small constant factor. For a parameter λ (think λ ≤ 5), the number of additional input and output wires is |C|1/λ, while the number of test queries to detect an error
with constant probability is around 22λ.

2023

TCC

From Polynomial IOP and Commitments to Non-malleable zkSNARKs
Abstract

We study sufficient conditions to compile simulation-extractable zkSNARKs from information-theoretic interactive oracle proofs (IOP) using a simulation-extractable commit-and-prove system for its oracles. Specifically, we define simulation extractability for opening and evaluation proofs of polynomial commitment schemes, which we then employ to prove the security of zkSNARKS obtained from polynomial IOP proof systems. To instantiate our methodology, we additionally prove that KZG commitments satisfy our simulation extractability requirement, despite being naturally malleable. To this end, we design a relaxed notion of simulation extractability that matches how KZG commitments are used and optimized in real-world proof systems. The proof that KZG satisfies this relaxed simulation extractability property relies on the algebraic group model and random oracle model.

2023

TCC

Generalized Special-Sound Interactive Proofs and their Knowledge Soundness
Abstract

A classic result in the theory of interactive proofs shows that a {\em special-sound} $\Sigma$-protocol is automatically a {\em proof of knowledge}. This result is very useful to have, since the latter property is typically tricky to prove from scratch, while the former is often easy to argue\,---\,{\em if} it is satisfied. While classic $\Sigma$-protocols often are special-sound, this is unfortunately not the case for many recently proposed, highly efficient interactive proofs, at least not in this strict sense. Motivated by this, the original result was recently generalized to $k$-special-sound $\Sigma$-protocols (for arbitrary, polynomially bounded $k$), and to multi-round versions thereof. This generalization is sufficient to analyze (e.g.) Bulletproofs-like protocols, but is still insufficient for many other examples.
In this work, we push the relaxation of the special soundness property to the extreme, by allowing an {\em arbitrary} access structure $\Gamma$ to specify for which subsets of challenges it is possible to compute a witness, when given correct answers to these challenges (for a fixed first message). Concretely, for any access structure $\Gamma$, we identify parameters $t_\Gamma$ and $\kappa_\Gamma$, and we show that any $\Gamma$-special-sound $\Sigma$-protocol is a proof of knowledge with knowledge error $\kappa_\Gamma$ if $t_\Gamma$ is polynomially bounded. Similarly for multi-round protocols.
We apply our general result to a couple of simple but important example protocols, where we obtain a tight knowledge error as an immediate corollary. Beyond these simple examples, we analyze the FRI protocol. Here, showing the general special soundness notion is non-trivial, but can be done (for a certain range of parameters) by recycling some of the techniques used to argue ordinary soundness of the protocol (as an IOP). Again as a corollary, we then derive that the FRI protocol, as an interactive proof by using a Merkle-tree commitment, has a knowledge extractor with almost optimal knowledge error, with the caveat that the extractor requires (expected) quasi-polynomial time.
% is a proof of knowledge with almost optimal knowledge error.
Finally, building up on the technique for the parallel repetition of $k$-special-sound $\Sigma$-protocols, we show the same strong parallel repetition result for $\Gamma$-special-sound $\Sigma$-protocol and its multi-round variant.

2023

TCC

Generic-Group Lower Bounds via Reductions Between Geometric-Search Problems: With and Without Preprocessing
Abstract

The generic-group model (GGM) aims to capture algorithms working over groups of prime order that only rely on the group operation, but do not exploit any additional structure given by the concrete implementation of the group. In it, it is possible to prove information-theoretic lower bounds on the hardness of problems like the discrete logarithm (DL) or computational Diffie-Hellman (CDH). Thus, since its introduction, it has served as a valuable tool to assess the concrete security provided by cryptographic schemes based on such problems. A work on the related algebraic-group model (AGM) introduced a method, used by many subsequent works, to adapt GGM lower bounds for one problem to another, by means of conceptually simple reductions.
In this work, we propose an alternative approach to extend GGM bounds from one problem to another. Following an idea by Yun [EC15], we show that, in the GGM, the security of a large class of problems can be reduced to that of geometric search-problems. By reducing the security of the resulting geometric-search problems to variants of the search-by-hypersurface problem, for which information theoretic lower bounds exist, we give alternative proofs of several results that used the AGM approach.
The main advantage of our approach is that our reduction from geometric search-problems works, as well, for the GGM with preprocessing (more precisely the bit-fixing GGM introduced by Coretti, Dodis and Guo [Crypto18]). As a consequence, this opens up the possibility of transferring preprocessing GGM bounds from one problem to another, also by means of simple reductions. Concretely, we prove novel preprocessing bounds on the hardness of the d-strong discrete logarithm, the d-strong Diffie-Hellman inversion, and multi-instance CDH problems, as well as a large class of Uber assumptions. Additionally, our approach applies to Shoup's GGM without additional restrictions on the query behavior of the adversary, while the recent works of Zhang, Zhou, and Katz [AC22] and Zhandry [Crypto22] highlight that this is not the case for the AGM approach.

2023

TCC

Holographic SNARGs for P and Batch-NP from (Polynomially Hard) Learning with Errors
Abstract

A succinct non-interactive argument (SNARG) is called \emph{holographic} if the verifier runs in time sub-linear in the input length when given oracle access to an encoding of the input. We present holographic SNARGs for P and Batch-NP under the learning with errors (LWE) assumption. Our holographic SNARG for P has a verifier that runs in time $\mathsf{poly}(\lambda, \log T, \log n)$ for $T$-time computations and $n$-bit inputs ($\lambda$ is the security parameter), while our holographic SNARG for Batch-NP has a verifier that runs in time $\mathsf{poly}(\lambda, T, \log k)$ for $k$ instances of $T$-time computations. Before this work, constructions with the same asymptotic efficiency were known in the designated-verifier setting or under the sub-exponential hardness of the LWE assumption. We obtain our holographic SNARGs (in the public-verification setting under the polynomial hardness of the LWE assumption) by constructing holographic SNARGs for certain hash computations and then applying known/trivial transformations.
As an application, we use our holographic SNARGs to weaken the assumption needed for a recent public-coin 3-round zero-knowledge (ZK) argument [Kiyoshima, CRYPTO 2022]. Specifically, we use our holographic SNARGs to show that a public-coin 3-round ZK argument exists under the same assumptions as the state-of-the-art private-coin 3-round ZK argument [Bitansky et al., STOC 2018].

2023

TCC

How to Compile Polynomial IOP into Simulation-Extractable SNARKs: A Modular Approach
Abstract

Most succinct arguments (SNARKs) are initially only proven knowledge sound (KS).
We show that the commonly employed compilation strategy from polynomial interactive oracle proofs (PIOP) via polynomial commitments to knowledge sound SNARKS actually also achieves other desirable properties: weak unique response (WUR) and trapdoorless zero-knowledge (TLZK); and that together they imply simulation extractability (SIM-EXT).
The factoring of SIM-EXT into KS + WUR + TLZK is becoming a cornerstone of the analysis of non-malleable SNARK systems. We show how to prove WUR (and TLZK) for PIOP compiled SNARKs under mild falsifiable assumptions on the polynomial commitment scheme. This means that the analysis of knowledge soundness from PIOP properties that inherently relies on non-falsifiable or idealized assumption such as the algebraic group model (AGM) or generic group model (GGM) need not be repeated.
While the proof of WUR requires only mild assumptions on the PIOP, TLZK is a different matter. As perfectly hiding polynomial commitments sometimes come at a substantial performance premium, SNARK designers prefer to employ deterministic commitments with some leakage. This results in the need for a stronger zero-knowledge property for the PIOP.
The modularity of our approach implies that any analysis improvements, e.g. in terms of tightness, credibility of the knowledge assumption and model of the KS analysis, or the precision of capturing real-world optimizations for TLZK also benefits the SIM-EXT guarantees.

2023

TCC

Ideal-SVP is Hard for Small-Norm Uniform Prime Ideals
Abstract

The presumed hardness of the Shortest Vector Problem for ideal lattices (Ideal-SVP) has been a fruitful assumption to understand other assumptions on algebraic lattices and as a security foundation of cryptosystems. Gentry [CRYPTO’10] proved that Ideal-SVP enjoys a worst-case to average-case reduction, where the average-case distribution is the uniform distribution over the set of inverses of prime ideals of small algebraic norm (below d^O(d) for cyclotomic fields, where d refers to the field degree). De Boer et al. [CRYPTO’20] btained another random self-reducibility result for an average-case distribution involving integral ideals of norm 2^O(d^2).
In this work, we show that Ideal-SVP for the uniform distribution over inverses of small-norm prime ideals reduces to Ideal-SVP for the uniform distribution over small-norm prime ideals. Combined with Gentry’s reduction, this leads to a worst-case to average-case reduction for the uniform distribution over the set of small-norm prime ideals. Using the reduction from Pellet-Mary and Stehlé [ASIACRYPT’21], this notably leads to the first distribution over NTRU instances with a polynomial modulus whose hardness is supported by a worst-case lattice problem.

2023

TCC

Immunizing Backdoored PRGs
Abstract

A backdoored Pseudorandom Generator (PRG) is a PRG
which looks pseudorandom to the outside world, but a saboteur can break
PRG security by planting a backdoor into a seemingly honest choice of
public parameters, pk, for the system. Backdoored PRGs became increas-
ingly important due to revelations about NIST’s backdoored Dual EC
PRG, and later results about its practical exploitability.
Motivated by this, at Eurocrypt’15 Dodis et al. [20] initiated the ques-
tion of immunizing backdoored PRGs. A k-immunization scheme repeat-
edly applies a post-processing function to the output of k backdoored
PRGs, to render any (unknown) backdoors provably useless. For k = 1,
[20] showed that no deterministic immunization is possible, but then
constructed “seeded” 1-immunizer either in the random oracle model,
or under strong non-falsifiable assumptions. As our first result, we show
that no seeded 1-immunization scheme can be black-box reduced to any
efficiently falsifiable assumption.
This motivates studying k-immunizers for k ≥ 2, which have an ad-
ditional advantage of being deterministic (i.e., “seedless”). Indeed, prior
work at CCS’17 [35] and CRYPTO’18 [7] gave supporting evidence that
simple k-immunizers might exist, albeit in slightly different settings. Un-
fortunately, we show that simple standard model proposals of
[35,7]
(including the XOR function [7]) provably do not work in our setting.
On a positive, we confirm the intuition of [35] that a (seedless) random
oracle is a provably secure 2-immunizer. On a negative, no (seedless)
2-immunization scheme can be black-box reduced to any efficiently falsi-
fiable assumption, at least for a large class of natural 2-immunizers which
includes all “cryptographic hash functions.”
In summary, our results show that k-immunizers occupy a peculiar
place in the cryptographic world. While they likely exist, and can be
made practical and efficient, it is unlikely one can reduce their security
to a “clean” standard-model assumption.

2023

TCC

Improved Polynomial Secret-Sharing Schemes
Abstract

Despite active research on secret-sharing schemes for arbitrary access structures for more than 35 years, we do not understand their share size -- the best known upper bound for an arbitrary $n$-party access structure is $2^{O(n)}$, while the best known lower bound is $\Omega(n/\log(n))$. Consistent with our knowledge, the share size can be anywhere between these bounds. To better understand this question, one can study specific families of secret-sharing schemes. For example, linear secret-sharing schemes, in which the sharing and reconstruction functions are linear mappings, have been studied in many papers, e.g., it is known that they require shares of size at least $2^{0.5n}$. Secret-sharing schemes in which the sharing and/or reconstruction are computed by low-degree polynomials have been recently studied by Paskin-Cherniavsky and Radune [ITC 2020] and by Beimel, Othman, and Peter [CRYPTO 2021]. It was shown that secret-sharing schemes with sharing and reconstruction computed by polynomials of degree $2$ are more efficient than linear schemes (i.e., schemes in which the sharing and reconstruction are computed by polynomials of degree one).
Prior to our work, it was not known if using polynomials of higher degree can reduce the share size. We show that this is indeed the case, i.e., we construct secret-sharing schemes for arbitrary access structures with reconstruction by degree-$d$ polynomials, where as the reconstruction degree $d$ increases, the share size decreases. As a step in our construction, we construct conditional disclosure of secrets (CDS) protocols. For example, we construct 2-server CDS protocols for functions $f:[N]\times [N] \to \{0,1\}$ with reconstruction computed by degree-$d$ polynomials with message size $N^{O(\log \log d/\log d)}$. Combining our results with a lower bound of Beimel et al.~[CRYPTO 2021], we show that increasing the degree of the reconstruction function in CDS protocols provably reduces the message size. To construct our schemes, we define \emph{sparse} matching vectors, show constructions of such vectors, and design CDS protocols and secret-sharing schemes with degree-$d$ reconstruction from sparse matching vectors.

2023

TCC

Limits in the Provable Security of ECDSA Signatures
Abstract

Digital Signatures are ubiquitous in modern computing. One of the most widely used digital signature schemes is ECDSA due to its use in TLS, various Blockchains such as Bitcoin and Etherum, and many other applications. Yet the formal analysis of ECDSA is comparatively sparse. In particular, all known security results for ECDSA rely on some idealized model such as the generic group model or the programmable (bijective) random oracle model.
In this work, we study the question whether these strong idealized models are necessary for proving the security of ECDSA. Specifically, we focus on the programmability of ECDSA's ``conversion function'' which maps an elliptic curve point into its $x$-coordinate modulo the group order. Unfortunately, our main results are negative. We establish, by means of a meta reductions, that an algebraic security reduction for ECDSA can only exist if the security reduction is allowed to program the conversion function. As a consequence, a meaningful security proof for ECDSA is unlikely to exist without strong idealization.

2023

TCC

Locally Verifiable Distributed SNARGs
Abstract

The field of distributed certification is concerned with certifying properties of distributed networks, where the communication topology of the network is represented as an arbitrary graph; each node of the graph is a separate processor, with its own internal state. To certify that the network satisfies a given property, a prover assigns each node of the network a certificate, and the nodes then communicate with one another and decide whether to accept or reject. We require soundness and completeness: the property holds if and only if there exists an assignment of certificates to the nodes that causes all nodes to accept. Our goal is to minimize the length of the certificates, as well as the communication between the nodes of the network. Distributed certification has been extensively studied in the distributed computing community, but it has so far only been studied in the information-theoretic setting, where the prover and the network nodes are computationally unbounded.
In this work we introduce and study computationally bounded distributed certification: we define locally verifiable distributed SNARG (LVDSNARGs), which are an analog of SNARGs for distributed networks, and are able to circumvent known hardness results for information-theoretic distributed certification by requiring both the prover and the verifier to be computationally efficient (namely, PPT algorithms).
We give two LVDSNARG constructions: the first allows us to succinctly certify any network property in P, using a global prover that can see the entire network; the second construction gives an efficient distributed prover, which succinctly certifies the execution of any efficient distributed algorithm. Our constructions rely on non-interactive batch arguments for NP (BARGs) and on RAM SNARGs, which have recently been shown to be constructible from standard cryptographic assumptions.

2023

TCC

Lower Bounds on Anonymous Whistleblowing
Abstract

Anonymous transfer, recently introduced by Agrikola, Couteau and Maier [ACM22] (TCC '22), allows a sender to leak a message anonymously by participating in a public non-anonymous discussion where everyone knows who said what. This opens up the intriguing possibility of using cryptography to ensure strong anonymity guarantees in a seemingly non-anonymous environment.
The work of [ACM22] presented a lower bound on anonymous transfer, ruling out constructions with strong anonymity guarantees (where the adversary's advantage in identifying the sender is negligible) against arbitrary polynomial-time adversaries. They also provided a (heuristic) upper bound, giving a scheme with weak anonymity guarantees (the adversary's advantage in identifying the sender is inverse in the number of rounds) against fine-grained adversaries whose run-time is bounded by some fixed polynomial that exceeds the run-time of the honest users. This leaves a large gap between the lower bound and the upper bound, raising the intriguing possibility that one may be able to achieve weak anonymity against arbitrary polynomial time adversaries, or strong anonymity against fine grained adversaries.
In this work, we present improved lower bounds on anonymous transfer, that rule out both of the above possibilities:
- We rule out the existence of anonymous transfer with any non-trivial anonymity guarantees against general polynomial time adversaries.
- Even if we restrict ourselves to fine-grained adversaries whose run-time is essentially equivalent to that of the honest parties, we cannot achieve strong anonymity, or even quantitatively improve over the inverse polynomial anonymity guarantees (heuristically) achieved by [ACM22].
Consequently, constructions of anonymous transfer can only provide security against fine-grained adversaries, and even in that case they achieve at most weak quantitative forms of anonymity.

2023

TCC

Lower Bounds on Assumptions Behind Registration-Based Encryption
Abstract

Registration-based encryption (RBE) is a primitive that aims to offer what identity-based encryption (IBE) offers without the so-called key-escrow problem. In RBE parties who wish to join the system will generate their own secret and public keys and register their public keys to a transparent party called key curator (KC) who does not have any secret state.
The initial constructions of RBE made \emph{non-black-box} use of building block primitives, due to their use of either indistinguishability obfuscation or some garbling scheme. More recently, it was shown how to achieve \emph{black-box} constructions of (variants of) RBE and even stronger primitives based on \emph{bilinear maps} in which the RBE is relaxed to have a CRS whose length can \emph{grow} with the number of registered identities. Making cryptographic constructions in general, and RBE in particular, black-box is an important step as it can play a significant role in its efficiency and potential deployment. Hence, in this work we ask: \emph{what are the minimal assumptions for black-box constructions of RBE?} Particularly, can we black-box construct RBE schemes from the same assumptions used for public-key encryption or simpler algebraic assumptions that hold in the generic group model?
In this work, we prove the first black-box separation results for RBE beyond the separations that follow from the observation that RBE black-box implies public-key encryption. In particular, we answer both of the questions above negatively and prove that neither trapdoor permutations nor (even Shoup's) generic group model can be used as the sole source of hardness for building RBE schemes. More generally, we prove that a relaxation of RBE in which all the keys are registered and compressed at the same time is already too complex to be built from either of the above-mentioned primitives in a black-box way. At a technical level, using compression techniques, we prove lemmas in the TDP and GGM oracle settings that prove the following intuitive yet useful fact: that compact strings cannot signal too many trapdoors, even if their generation algorithm takes exponential time. Due to their generality, our lemmas could be of independent interest and find more applications.

2023

TCC

Memory Checking for Parallel RAMs
Abstract

When outsourcing a database to an untrusted remote server, one might want to verify the integrity of contents while accessing it. To solve this, Blum et al. [FOCS `91] propose the notion of \emph{memory checking}. Memory checking allows a user to run a RAM program on a remote server, with the ability to verify integrity of the storage with small private storage.
In this work, we define and initiate the formal study of memory checking for \emph{Parallel RAMs} (PRAMs). The parallel RAM model is very expressive and captures many modern architectures such as multi-core architectures and cloud clusters. When multiple clients run a PRAM algorithm on a shared remote server, it is possible that there are concurrency issues that cause inconsistencies. Therefore, integrity verification is also a desirable property in this setting.
We construct an online memory checker (one that reports faults as soon as they occur) for PRAMs with $O(\log N)$ simulation overhead in both work and depth. Moreover, we construct an offline memory checker (one that reports faults only after a long sequence of operations) with amortized $O(1)$ simulation overhead in both work and depth. As an application of our parallel memory checking constructions, we construct a \emph{maliciously secure oblivious parallel RAM} (OPRAM) with polylogarithmic overhead.

2023

TCC

Multi-Instance Randomness Extraction and Security against Bounded-Storage Mass Surveillance
Abstract

Consider a state-level adversary who observes and stores large amounts of encrypted data from all users on the Internet, but does not have the capacity to store it all. Later, it may target certain "persons of interest" in order to obtain their decryption keys. We would like to guarantee that, if the adversary's storage capacity is only (say) 1% of the total encrypted data size, then even if it can later obtain the decryption keys of arbitrary users, it can only learn something about the contents of (roughly) 1% of the ciphertexts, while the rest will maintain full security. This can be seen as an extension of incompressible cryptography (Dziembowski CRYPTO '06, Guan, Wichs and Zhandry EUROCRYPT '22) to the multi-user setting. We provide solutions in both the symmetric key and public key setting with various trade-offs in terms of computational assumptions and efficiency.
As the core technical tool, we study an information-theoretic problem which we refer to as "multi-instance randomness extraction". Suppose $X_1$, $\ldots$, $X_t$ are correlated random variables whose total joint min-entropy rate is $\alpha$, but we know nothing else about their individual entropies. We choose $t$ random and independent seeds $S_1,\ldots, S_t$ and attempt to individually extract some small amount of randomness $Y_i = Ext(X_i; S_i)$ from each $X_i$. We'd like to say that roughly an $\alpha$-fraction of the extracted outputs $Y_i$ should be indistinguishable from uniform even given all the remaining extracted outputs and all the seeds. We show that this indeed holds for specific extractors based on Hadamard and Reed-Muller codes.

2023

TCC

Multilinear Schwartz-Zippel mod N and Lattice-Based Succinct Arguments
Abstract

We show that for $x <-[0,2^\lambda)^u$ and any integer $N$ the probability that $f(x)=0 mod N$ for any non-zero multilinear polynomial $f \in Z[X_1, ...,X_u]$, co-prime to $N$ is inversely proportional to $N$. As a corollary we show that if $log_2 N≥log_2(2 u)\lambda+8u^2 $ then the probability is bounded by $(u+1)/(2^\lambda)$. We also give tighter numerically derived bounds, showing that if $log_2 N≥418 $, and $u ≤ 20$ the probability is bounded by $u/(2^\lambda)+2^{-120}$.
We then apply this Multilinear Composite Schwartz-Zippel Lemma (LCSZ) to resolve an open problem in the literature on succinct arguments: that the Bulletproofs protocol for linear relations over classical Pedersen commitments in prime-order groups remains knowledge sound when generalized to commitment schemes that are binding only over short integer vectors. In particular, this means that the Bulletproofs protocol can be instantiated with plausibly post-quantum commitments from lattice hardness assumptions (SIS/R-SIS/M-SIS). It can also be instantiated with commitments based on groups of unknown order (GUOs), in which case the verification time becomes logarithmic instead of linear time.
Prior work on lattice-based Bulletproofs (Crypto 2020) and its extensions required modifying the protocol to sample challenges from special sets of polynomial size. This results in a non-negligible knowledge error, necessitating parallel repetition to amplify soundness, which not only impacts efficiency but also poses issues for the Fiat-Shamir transform. Our analysis shows knowledge soundness for the original Bulletproofs protocol with the exponential-size integer challenge set $[0,2^\lambda]$ and thus achieves a negligible soundness error without repetition, circumventing a previous impossibility result (Crypto 2021). Our analysis also closes a critical gap in the original security proof of DARK, a GUO-based polynomial commitment scheme (Eurocrypt 2020).
Along the way to achieving our result we also define Almost Special Soundness (AMSS), a generalization of Special-Soundness. Our main result is divided into two parts: (1) that the Bulletproofs protocol over generalized commitments is AMSS, and (2) that AMSS implies knowledge soundness. This framework serves to simplify the application of our analytical techniques to protocols beyond Bulletproofs in the future.

2023

TCC

Near-Optimal Private Information Retrieval with Preprocessing
Abstract

In Private Information Retrieval (PIR), a client wishes to access an index $i$ from a public $n$-bit database without revealing any information about $i$. Recently, a series of works starting with the seminal paper of Corrigan-Gibbs and Kogan (EUROCRYPT 2020) considered PIR with \emph{client preprocessing} and \emph{no additional server storage}. In this setting, we now have protocols that achieve $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(\sqrt{n})$ (amortized) server time and $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(1)$ (amortized) bandwidth in the two-server model (Shi et al., CRYPTO 2021) as well as $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(\sqrt{n})$ server time and $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(\sqrt{n})$ bandwidth in the single-server model (Corrigan-Gibbs et al., EUROCRYPT 2022). Given existing lower bounds, a single-server PIR scheme with $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(\sqrt{n})$ (amortized) server time and $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(1)$ (amortized) bandwidth is still feasible, however, to date, no known protocol achieves such complexities. In this paper we fill this gap by constructing the first single-server PIR scheme with $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(\sqrt{n})$ (amortized) server time and $\stackrel{\sim}{\smash{O}\rule{0pt}{1.0ex}}$$(1)$ (amortized) bandwidth. Our scheme achieves near-optimal (optimal up to polylogarithmic factors) asymptotics in every relevant dimension.
Central to our approach is a new cryptographic primitive that we call an \emph{adaptable pseudorandom set}: With an adaptable pseudorandom set, one can represent a large pseudorandom set with a succinct fixed-size key $k$, and can both add to and remove from the set a constant number of elements by manipulating the key $k$, while maintaining its concise description as well as its pseudorandomness (under a certain security definition).

2023

TCC

Network Agnostic MPC with Statistical Security
Abstract

In this work, we initiate the study of network agnostic MPC protocols with statistical security. Network agnostic MPC protocols give the best possible security guarantees, irrespective of the behaviour of the underlying network. While network agnostic MPC protocols have been designed earlier with perfect and computational security, nothing is known in the literature regarding their possibility with statistical security. We consider the general-adversary model, where the adversary is characterized by an adversary structure which enumerates all possible candidate subsets of corrupt parties. Known statistically-secure synchronous MPC (SMPC) and asynchronous MPC (AMPC) protocols are secure against adversary structures satisfying the Q^{(2)} and Q^{(3)} conditions respectively, meaning that the union of no two and three subsets from the adversary structure cover the entire set of parties.
Fix adversary structures Z_s and Z_a, satisfying the Q^{(2)} and Q^{(3)} conditions respectively, where
Z_a \subset Z_s. Then given an unconditionally-secure PKI, we ask whether it is possible to design a statistically-secure MPC protocol, which is resilient against Z_s and Z_a in a synchronous and an asynchronous network respectively, even if the parties are unaware of the network type. We show that this is possible iff Z_s and Z_a satisfy the Q^{(2, 1)} condition, meaning that the union of any two subsets from Z_s and any one subset from Z_a is a proper subset of the set of parties. The complexity of our protocol is polynomial in |Z_s|.

2023

TCC

Non-Interactive Anonymous Router with Quasi-Linear Router Computation
Abstract

Anonymous routing is an important cryptographic primitive that allows users to communicate privately on the Internet, without revealing their message contents or their contacts. Until the very recent work of Shi and Wu (Eurocrypt’21), all classical anonymous routing schemes are interactive protocols, and their security rely on a threshold number of the routers being honest. The recent work of Shi and Wu suggested a new abstraction called Non-Interactive Anonymous Router (NIAR), and showed how to achieve anonymous routing non-interactively for the first time. In particular, a single untrusted router receives a token which allows it to obliviously apply a permutation to a set of encrypted messages from the senders. Shi and Wu’s construction suffers from two drawbacks: 1) the router takes time quadratic in the number of senders to obliviously route their messages; and 2) the scheme is proven secure only in the presence of static corruptions.
In this work, we show how to construct a non-interactive anonymous router scheme with sub-quadratic router computation, and achieving security in the presence of adaptive corruptions. To get this result, we assume the existence of indistinguishability obfuscation and one-way functions. Our final result is obtained through a sequence of stepping stones. First, we show how to achieve the desired efficiency, but with security under static corruption and in a selective, single-challenge setting. Then, we go through a sequence of upgrades which eventually get us the final result. We devise various new techniques along the way which lead to some additional results. In particular, our techniques for reasoning about a network of obfuscated programs may be of independent interest.

2023

TCC

On Black-Box Verifiable Outsourcing
Abstract

We study the problem of verifiably outsourcing computation in a model where the verifier has black-box access to the function being computed. We introduce the problem of oracle-aided batch verification of computation (OBVC) for a function class F. This allows a verifier to efficiently verify the correctness of any f \in F evaluated on a batch of n instances x_1, ...., x_n, while only making \lambda calls to an oracle for f (along with O(n \lambda) calls to low-complexity helper oracles), where \lambda denotes a security parameter.
We obtain the following positive and negative results:
1. We build OBVC protocols for the class F of all functions that admit random-self-reductions. Some of our protocols rely on homomorphic encryption schemes.
2. We show that there cannot exist OBVC schemes for the class F of all functions mapping \lambda-bit inputs to \lambda-bit outputs, for any n = \poly(\lambda).

2023

TCC

On One-way Functions and Sparse Languages
Abstract

We show equivalence between the existence of one-way
functions and the existence of a \emph{sparse} language that is
hard-on-average w.r.t. some efficiently samplable ``high-entropy''
distribution.
In more detail, the following are equivalent:
- The existence of a $S(\cdot)$-sparse language $L$ that is
hard-on-average with respect to some samplable distribution with
Shannon entropy $h(\cdot)$ such that $h(n)-\log(S(n)) \geq 4\log n$;
- The existence of a $S(\cdot)$-sparse language $L \in
\NP$, that is
hard-on-average with respect to some samplable distribution with
Shannon entropy $h(\cdot)$ such that $h(n)-\log(S(n)) \geq n/3$;
- The existence of one-way functions.
Our results are inspired by, and generalize, results from the elegant
recent paper by Ilango, Ren and Santhanam (IRS, STOC'22), which presents
similar connections for \emph{specific} sparse languages.

2023

TCC

On Secure Computation of Solitary Output Functionalities With and Without Broadcast
Abstract

Solitary output secure computation models scenarios, where a single entity wishes to compute a function over an input that is distributed among several mutually distrusting parties. The computation should guarantee some security properties, such as correctness, privacy, and guaranteed output delivery. Full security captures all these properties together. This setting is becoming very important, as it is relevant to many real-world scenarios, such as service providers wishing to learn some statistics on the private data of their users.
In this paper, we study full security for solitary output three-party functionalities in the point-to-point model (without broadcast) assuming at most a single party is corrupted. We give a characterization of the set of three-party Boolean functionalities and functionalities with up to three possible outputs (over a polynomial-size domain) that are computable with full security in the point-to-point model against a single corrupted party. We also characterize the set of three-party functionalities (over a polynomial-size domain) where the output receiving party has no input. Using this characterization, we identify the set of parameters that allow certain functionalities related to private set intersection to be securely computable in this model. Our characterization in particular implies that, even in the solitary output setting, without broadcast not many ``interesting'' three-party functionalities can be computed with full security.
Our main technical contribution is a reinterpretation of the hexagon argument due to Fischer et al. [Distributed Computing '86]. While the original argument relies on the agreement property (i.e., all parties output the same value) to construct an attack, we extend the argument to the solitary output setting, where there is no agreement. Furthermore, using our techniques, we were also able to advance our understanding of the set of solitary output three-party functionalities that can be computed with full security, assuming broadcast but where two parties may be corrupted. Specifically, we extend the set of such functionalities that were known to be computable, due to Halevi et al. [TCC '19].

2023

TCC

On the Cost of Post-Compromise Security in Concurrent Continuous Group-Key Agreement
Abstract

Continuous Group-Key Agreement (CGKA) allows a group of users to maintain a shared key.
It is the fundamental cryptographic primitive underlying group messaging schemes and related protocols, most notably TreeKEM, the underlying key agreement protocol of the Messaging Layer Security (MLS) protocol, a standard for group messaging by the IETF.
CKGA works in an asynchronous setting where parties only occasionally must come online, and their messages are relayed by an untrusted server.
The most expensive operation provided by CKGA is that which allows for a user to refresh their key material in order to achieve forward secrecy (old messages are secure when a user is compromised) and post-compromise security (users can heal from compromise).
One caveat of early CGKA protocols is that these update operations had to be performed sequentially, with any user wanting to update their key material having had to receive and process all previous updates.
Late versions of TreeKEM do allow for concurrent updates at the cost of a communication overhead per update message that is linear in the number of updating parties.
This was shown to be indeed necessary when achieving PCS in just two rounds of communication by [Bienstock et al. TCC'20].
The recently proposed protocol CoCoA [Alwen et al. Eurocrypt'22], however, shows that this overhead can be reduced if PCS requirements are relaxed, and only a logarithmic number of rounds is required.
The natural question, thus, is whether CoCoA is optimal in this setting.
In this work we answer this question, providing a lower bound on the cost (concretely, the amount of data to be uploaded to the server) for CGKA protocols that heal in an arbitrary k number of rounds, that shows that CoCoA is very close to optimal.
Additionally, we extend CoCoA to heal in an arbitrary number of rounds, and propose a modification of it, with a reduced communication cost for certain k.
We prove our bound in a combinatorial setting where the state of the protocol progresses in rounds, and the state of the protocol in each round is captured by a set system, each set specifying a set of users who share a secret key.
We show this combinatorial model is equivalent to a symbolic model capturing building blocks including PRFs and public-key encryption, related to the one used by Bienstock et al.
Our lower bound is of order k * n^{1+1/(k-1)} / log(k), where 2 < k < log(n) is the number of updates per user the protocol requires to heal.
This generalizes the n^2 bound for k=2 from Bienstock et al.
This bound almost matches the k * n^{1+2/(k-1)} or k^2 * n^{1+1/(k-1)} efficiency we get for the variants of the CoCoA protocol also introduced in this paper.

2023

TCC

On the Impossibility of Surviving (Iterated) Deletion of Weakly Dominated Strategies in Rational MPC
Abstract

Rational multiparty computation (rational MPC) provides a framework for analyzing MPC protocols through the lens of game theory. One way to judge whether an MPC protocol is rational is through weak domination: Rational players would not adhere to an MPC protocol if deviating never decreases their utility, but sometimes increases it.
Secret reconstruction protocols are of particular importance in this setting because they represent the last phase of most (rational) MPC protocols. We show that most secret reconstruction protocols from the literature are not, in fact, stable with respect to weak domination. Furthermore, we formally prove that (under certain assumptions) it is impossible to design a secret reconstruction protocol which is a Nash equlibrium but not weakly dominated if (1) shares are authenticated or (2) half of all players may form a coalition.

2023

TCC

On the Multi-User Security of LWE-based NIKE
Abstract

Non-interactive key exchange (NIKE) schemes like the Diffie-Hellman key exchange are a widespread building block in several cryptographic protocols. Since the Diffie-Hellman key exchange is not post-quantum secure, it is important to investigate post-quantum alternatives.
We analyze the security of the LWE-based NIKE by Ding et al. (ePrint 2012) and Peikert (PQCrypt 2014) in a multi-user setting where the same public key is used to generate shared keys with multiple other users. The Diffie-Hellman key exchange achieves this security notion. The mentioned LWE-based NIKE scheme comes with an inherent correctness error (Guo et al., PKC 2020) and this has significant implications for the multi-user security, necessitating a closer examination.
Single-user security generically implies multi-user security when all users generate their keys honestly for NIKE schemes with negligible correctness error. However, the LWE-based NIKE requires a super-polynomial modulus to achieve a negligible correctness error, which makes the scheme less efficient. We show that
- generically, single-user security does not imply multi-user security when the correctness error is non-negligible, but despite this
- the LWE-based NIKE with polynomial modulus is multi-user secure for honest users when the number of users is fixed in advance. This result leverages the leakage-resilience properties of LWE.
We then turn to a stronger model of multi-user security that allows adversarially generated public keys. For this model, we consider a variant of the LWE-based NIKE where each public key is equipped with a NIZKPoK of the secret key. Adding NIZKPoKs is a standard technique for this stronger model and Hesse et al. (Crypto 2018) showed that this is sufficient to achieve security in the stronger multi-user security model for perfectly correct NIKEs (which the LWE-based NIKE is not). We show that
- for certain parameters that include all parameters with polynomial modulus, the LWE-based NIKE can be efficiently attacked with adversarially generated public keys, despite the use NIZKPoKs, but
- for suitable parameters (that require a super-polynomial modulus), this security notion is achieved by the LWE-based NIKE with NIZKPoKs.
This stronger security notion has been achieved for the LWE-based NIKE previously only in the QROM, while all our results are in the standard model.

2023

TCC

On the Round Complexity of Fully Secure Solitary MPC with Honest Majority
Abstract

We study the problem of secure multiparty computation for functionalities where only one
party receives the output, to which we refer as solitary MPC. Recently, Halevi et al. (TCC 2019) studied fully secure (i.e., with guaranteed output delivery) solitary MPC and showed impossibility of such protocols for certain functionalities when there is no honest majority among the parties.
In this work, we study the round complexity of fully secure solitary MPC in the honest majority setting and with computational security. We note that a broadcast channel or public key infrastructure (PKI) setup is necessary for an n-party protocol against malicious adversaries
corrupting up to t parties where n/3 ≤ t < n/2. Therefore, we study the following settings and
ask the question: Can fully secure solitary MPC be achieved in fewer rounds than fully secure
standard MPC in which all parties receive the output?
• When there is a broadcast channel and no PKI:
– We start with a negative answer to the above question. In particular, we show that
the exact round complexity of fully secure solitary MPC is 3, which is the same as
fully secure standard MPC.
– We then study the minimal number of broadcast rounds needed to design round optimal fully secure solitary MPC. We show that both the first and second rounds of broadcast are necessary when $2 \lceil n/5 \rceil \leq t < n/2$, whereas pairwise-private channels suffice in the last round. Notably, this result also applies to fully secure standard MPC in which all parties receive the output.
• When there is a PKI and no broadcast channel, nevertheless, we show more positive results:
– We show an upper bound of 5 rounds for any honest majority. This is superior to the
super-constant lower bound for fully secure standard MPC in the exact same setting.
– We complement this by showing a lower bound of 4 rounds when $3\lceil n/7 \rceil \leq t < n/2$.
– For the special case of t = 1, n = 3, when the output receiving party does not have an input to the function, we show an upper bound of 2 rounds, which is optimal. When the output receiving party has an input to the function, we show a lower bound of 3, which matches an upper bound from prior work.
– For the special case of t = 2, n = 5, we show a lower bound of 3 rounds (an upper bound of 4 follows from prior work).
All our results also assume the existence of a common reference string (CRS) and pairwise private channels. Our upper bounds use a decentralized threshold fully homomorphic encryption (dTFHE) scheme (which can be built from the learning with errors (LWE) assumption) as the
main building block.

2023

TCC

On Time-Space Lower Bounds for Finding Short Collisions in Sponge Hash Functions
Abstract

Sponge paradigm, used in the design of SHA-3, is an alternative hashing technique to the popular Merkle-Damg\r ard paradigm. We revisit the problem of finding $B$-block-long collisions in sponge hash functions in the auxiliary-input random permutation model, in which an attacker gets a piece of $S$-bit advice about the random permutation and makes $T$ (forward or inverse) oracle queries to the random permutation.
Recently, significant progress has been made in the Merkle-Damg\r ard setting and optimal bounds are known for a large range of parameters, including all constant values of $B$. However, the sponge setting is widely open: there exist significant gaps between known attacks and security bounds even for $B=1$.
Freitag, Ghoshal and Komargodski (CRYPTO 2022) showed a novel attack for $B=1$ that takes advantage of the inverse queries and achieves advantage $\Omega(\min(S^2T^2/2^{2c}$, $ (S^2T/2^{2c})^{2/3})+T^2/2^r)$, where $r$ is bit-rate and $c$ is the capacity of the random permutation. However, they only showed an $O(ST/2^c+T^2/2^r)$ security bound, leaving open an intriguing quadratic gap. For $B=2$, they beat the general security bound %$O(ST^2/2^c+T^2/2^r)$,
by Coretti, Dodis,
Guo (CRYPTO 2018) for arbitrary values of $B$. However, their highly non-trivial argument is quite laborious, and no better (than the general) bounds are known for $B\geq 3$.
In this work, we study the possibility of proving better security bounds in the sponge setting. To this end,
\begin{itemize}
\item For $B=1$, we prove an improved $O(S^2T^2/2^{2c}+S/2^c+T/2^c+T^2/2^r)$ bound. Our bound strictly improves the bound by Freitag et al., %Ghoshal and Komargodski,
and is optimal for $ST^2\leq 2^c$. %and is optimal up to a factor of $(ST^2/2^c)^{2/3}$ for $ST^2>2^c$.
\item For $B=2$, we give a considerably simpler and more modular proof, recovering the bound obtained by Freitag et al. %, Ghoshal and Komargodski.
\item We obtain our bounds by adapting the recent multi-instance technique of Akshima, Guo and Liu (CRYPTO 2022) which bypasses limitations of prior techniques in the Merkle-Damg\r ard setting. To complement our results, we provably show that the recent multi-instance technique cannot further improve our bounds for $B=1,2$, and the general %$O(ST^2/2^c+T^2/2^r)$
bound by Correti et al., for $B\geq 3$.
\end{itemize}
Overall, our results yield the state-of-the-art security bounds for finding short collisions, and fully characterize the power of the multi-instance technique in the sponge setting.
\keywords{Collision \and hash functions \and Sponge \and multi-instance \and pre-computation \and auxiliary input}

2023

TCC

One-out-of-Many Unclonable Cryptography: Definitions, Constructions, and More
Abstract

The no-cloning principle of quantum mechanics enables us to achieve amazing unclonable cryptographic primitives, which is impossible in classical cryptography. However, the security definitions for unclonable cryptography are tricky. Achieving desirable security notions for unclonability is a challenging task. In particular, there is no indistinguishable-secure unclonable encryption and quantum copy-protection for single-bit output point functions in the standard model. To tackle this problem, we introduce and study relaxed but meaningful security notions for unclonable cryptography in this work. We call the new security notion one-out-of-many unclonable security.
We obtain the following results.
- We show that one-time strong anti-piracy secure secret key single-decryptor encryption (SDE) implies one-out-of-many indistinguishable-secure unclonable encryption.
- We construct a one-time strong anti-piracy secure secret key SDE scheme in the standard model from the LWE assumption. This scheme can encrypt multi-bit messages.
- We construct one-out-of-many copy-protection for single-bit output point functions from one-out-of-many indistinguishable-secure unclonable encryption and the LWE assumption.
- We construct one-out-of-many unclonable predicate encryption (PE) from one-out-of-many indistinguishable-secure unclonable encryption and the LWE assumption.
Thus, we obtain one-out-of-many indistinguishable-secure unclonable encryption, one-out-of-many copy-protection for single-bit output point functions, and one-out-of-many unclonable PE in the standard model from the LWE assumption. In addition, our one-time SDE scheme is the first multi-bit SDE scheme that does not rely on any oracle heuristics and strong assumptions such as indistinguishability obfuscation and witness encryption.

2023

TCC

Proactive Secret Sharing with Constant Communication
Abstract

This paper presents the first protocols for Proactive Secret Sharing (PSS) that only require constant (in the number of parties, n) communication per party per epoch. By harnessing the power of expander graphs, we are able to obtain strong guarantees about the security of the system. We present the following PSS protocols:
– A PSS protocol that provides privacy (but no robustness) against an adversary controlling O(n) parties per epoch.
– A PSS protocol that provides robustness (but no privacy) against an adversary controlling O(n) parties per epoch.
– A PSS protocol that provides privacy against an adversary controlling O(n^a) ) parties per epoch and provides robustness against an adversary controlling O(n^(1−a)) parties per epoch, for any constant 0 ≤ a ≤ 1. Instantiating this with a = 1/2 gives a PSS protocol that is proactively secure (private and robust) against an adversary controlling O(√n) parties per epoch.
Additionally, we discuss how secure channels, whose existence is usually assumed by PSS protocols, are challenging to create in the mobile adversary setting, and we present a method to instantiate them from a weaker assumption.

2023

TCC

Pseudorandomness with Proof of Destruction and Applications
Abstract

Two fundamental properties of quantum states that quantum information theory explores are pseudorandomness and provability of destruction. We introduce the notion of quantum pseudorandom states with proofs of destruction (PRSPD) that combines both these properties. Like standard pseudorandom states (PRS), these are efficiently generated quantum states that are indistinguishable from random, but they can also be measured to create a classical string. This string is verifiable (given the secret key) and certifies that the state has been destructed. We show that, similarly to PRS, PRSPD can be constructed from any post-quantum one-way function. As far as the authors are aware, this is the first construction of a family of states that satisfies both pseudorandomness and provability of destruction.
We show that many cryptographic applications that were shown based on PRS variants using quantum communication can be based on (variants of) PRSPD using only classical communication. This includes symmetric encryption, message authentication, one-time signatures, commitments, and classically verifiable private quantum coins.

2023

TCC

Public-Key Encryption with Quantum Keys
Abstract

In the framework of Impagliazzo's five worlds, a distinction is often made between two worlds, one where public-key encryption exists (Cryptomania), and one in which only one-way functions exist (MiniCrypt). However, the boundaries between these worlds can change when quantum information is taken into account. Recent work has shown that quantum variants of oblivious transfer and multi-party computation, both primitives that are classically in Cryptomania, can be constructed from one-way functions, placing them in the realm of quantum MiniCrypt (the so-called MiniQCrypt). This naturally raises the following question:
Is it possible to construct a quantum variant of public-key encryption, which is at the heart of Cryptomania, from one-way functions or potentially weaker assumptions?
In this work, we initiate the formal study of the notion of quantum public-key encryption (qPKE), i.e., public-key encryption where keys are allowed to be quantum states. We propose new definitions of security and several constructions of qPKE based on the existence of one-way functions (OWF), or even weaker assumptions, such as pseudorandom function-like states (PRFS) and pseudorandom function-like states with proof of destruction (PRFSPD). Finally, to give a tight characterization of this primitive, we show that computational assumptions are necessary to build quantum public-key encryption. That is, we give a self-contained proof that no quantum public-key encryption scheme can provide information-theoretic security.

2023

TCC

Public-Key Encryption, Local Pseudorandom Generators, and the Low-Degree Method
Abstract

The low-degree method postulates that no efficient algorithm outperforms low-degree polynomials in certain hypothesis-testing tasks. It has been used to understand computational indistinguishability in high-dimensional statistics.
We explore the use of the low-degree method in the context of cryptography. To this end, we apply it in the design and analysis of a new public-key encryption scheme whose security is based on Goldreich's pseudorandom generator. The scheme is a combination of two proposals of Applebaum, Barak, and Wigderson, and inherits desirable features from both.

2023

TCC

Publicly Verifiable Deletion from Minimal Assumptions
Abstract

We present a general compiler to add the publicly verifiable deletion property for various cryptographic primitives including public key encryption, attribute-based encryption, and quantum fully homomorphic encryption. Our compiler only uses one-way functions, or more generally hard quantum planted problems for NP, which are implied by one-way functions.
It relies on minimal assumptions and enables us to add the publicly verifiable deletion property with no additional assumption for the above primitives. Previously, such a compiler needs additional assumptions such as injective trapdoor one-way functions or pseudorandom group actions [Bartusek-Khurana-Poremba, CRYPTO 2023]. Technically, we upgrade an existing compiler for privately verifiable deletion [Bartusek-Khurana, CRYPTO 2023] to achieve publicly verifiable deletion by using digital signatures.

2023

TCC

Randomized Functions with High Round Complexity
Abstract

Consider two-party secure function evaluation against an honest-but-curious adversary in the information-theoretic plain model.
We study the round complexity of securely realizing a given secure function evaluation functionality.
Chor-Kushilevitz-Beaver (1989) proved that the round complexity of securely evaluating a deterministic function depends solely on the cardinality of its domain and range.
A folklore conjecture asserts that this phenomenon extends to functions with randomized output.
Our work falsifies this folklore conjecture -- revealing intricate subtleties even for this elementary security notion.
For every $r$, there is a function $f_r$ with binary inputs and five output alphabets that has round complexity $r$.
Previously, such a construction was known using $(r+1)$ output symbols.
This counter-example is optimal -- we prove that any securely realizable function with binary inputs and four output alphabets has round complexity at most four.
We work in the geometric framework Basu-Khorasgani-Maji-Nguyen (FOCS--2022) introduced to investigate randomized functions' round complexity.
Our work establishes a connection between secure computation and the lamination hull (geometric object originally motivated by applications in hydrodynamics).
Our counterexample constructions are related to the ``tartan square'' construction in the lamination hull literature.

2023

TCC

Revisiting Updatable Encryption: Controlled Forward Security, Constructions and a Puncturable Perspective
Abstract

Updatable encryption (UE) allows a third party to periodically rotate encryption keys from one epoch to another without the need to download, decrypt, re-encrypt, and upload already encrypted data by a client. Updating those outsourced ciphertexts is carried out via the use of so-called update tokens which in turn are generated during key rotation and can be sent (publicly) to the third party. The arguably most efficient variant of UE is ciphertext-independent UE as the key rotation does not depend on the outsourced ciphertexts which makes it particularly interesting in scenarios where access to (information of the) ciphertexts is not possible during key rotation.
Available security notions cannot guarantee any form of _forward security_ (i.e., old ciphertexts are in danger after key leakage). Counter-intuitively, forward security would violate correctness, as ciphertexts should be updatable ad-infinitum given the update token. In this work, we investigate if we can have at least some form of "controlled" forward security to mitigate the following shortcoming: an adversary would record available information (i.e., some ciphertexts, all update tokens) and simply would wait for a single key leakage to decrypt all data ever encrypted.
Our threefold contribution is as follows:
a) First, we introduce an epoch-based UE CPA security notion to allow fine-grained updatability. It covers the concept of expiry epochs, i.e., ciphertexts can lose the ability of being updatable via a token after a certain epoch has passed. This captures the above mentioned shortcoming as the encrypting party can decide how long a ciphertext can be updatable (and, hence, decryptable).
b) Second, we introduce a novel approach of constructing UE which significantly departs from previous ones and in particular views UE from the perspective of puncturable encryption (Green and Miers, S&P'15). We define tag-inverse puncturable encryption as a new variant that generalizes UE and may be of independent interest.
c) Lastly, we present and prove secure the first UE scheme with the aforementioned properties. It is constructed via tag-inverse puncturable encryption and instantiated from standard assumptions. As it turned out, constructing such puncturing schemes is not straightforward and we require adapted proof techniques. Surprisingly, as a special case, this yields the first backwards-leak UE scheme with sub-linear ciphertexts from standard assumptions (an open problem posted in two recent works by Jiang Galteland and Pan & Miao et al., PKC'23).

2023

TCC

Revocable Cryptography from Learning with Errors
Abstract

Quantum cryptography leverages many unique features of quantum information in order to
construct cryptographic primitives that are oftentimes impossible classically. In this work, we
build on the no-cloning principle of quantum mechanics and design cryptographic schemes with
key-revocation capabilities. We consider schemes where secret keys are represented as quantum
states with the guarantee that, once the secret key is successfully revoked from a user, they no
longer have the ability to perform the same functionality as before.
We define and construct several fundamental cryptographic primitives with key-revocation
capabilities, namely pseudorandom functions, secret-key and public-key encryption, and even
fully homomorphic encryption. Our constructions either assume the quantum subexponential
hardness of the learning with errors problem or are based on new conjectures. Central to all our
constructions is our approach for making the Dual-Regev encryption scheme (Gentry, Peikert
and Vaikuntanathan, STOC 2008) revocable.

2023

TCC

Rigorous Foundations for Dual Attacks in Coding Theory
Abstract

Dual attacks aiming at decoding generic linear codes have been found recently to outperform for certain parameters information set decoding techniques
which have been for $60$ years the dominant tool for solving this problem and choosing the parameters of code-based cryptosystems. However, the analysis of
the complexity of these dual attacks relies on some unproven assumptions that are not even fully backed up with experimental evidence.
These dual attacks can actually be viewed as the code-based analogue of dual attacks in lattice based cryptography. Here too, dual attacks have been found out
those past years to be strong competitors to primal attacks and a controversy has emerged whether similar heuristics made for instance on the independence of certain
random variables really hold. We will show that the dual attacks in coding theory can be studied by providing in a first step
a simple alternative expression of the fundamental quantity used in these dual attacks. We then show that this expression can be studied without relying on independence
assumptions whatsoever. This study leads us to discover that there is indeed a problem with the latest and most powerful dual attack proposed in \cite{CDMT22} and that
for the parameters chosen in this algorithm there are indeed false candidates which are produced and which are not predicted by the analysis provided there which relies on
independence assumptions. We then suggest a slight modification of this algorithm consisting in a further verification step, analyze it thoroughly, provide experimental evidence
that our analysis is accurate and show that the complexity claims made in \cite{CDMT22} are indeed valid for this modified algorithm. This approach provides a simple methodology
for studying rigorously dual attacks which could turn out to be useful for further developing the subject.

2023

TCC

Rogue-Instance Security for Batch Knowledge Proofs
Abstract

We propose a new notion of knowledge soundness, denoted \emph{rogue-instance security}, for interactive and non-interactive \emph{batch} knowledge proofs. Our notion, inspired by the standard notion of rogue-key security for multi-signature schemes, considers a setting in which a malicious prover is provided with an honestly-generated instance $x_1$, and may then be able to maliciously generate related ``rogue'' instances $x_2,\ldots,x_k$ for convincing a verifier in a batch knowledge proof of corresponding witnesses $w_1,\ldots,w_k$ for all $k$ instances -- without actually having knowledge of the witness $w_1$ corresponding to the honestly-generated instance. This setting provides a powerful security guarantee for batch versions of a wide variety of practically-relevant protocols, such as Schnorr's protocol and similar ones.
We present a highly-efficient generic construction of a batch proof-of-knowledge applicable to any \emph{algebraic} Sigma protocols. The algebraic property refers to a homomorphic structure of the underlying group and includes Schnorr's protocol and others. We provide an almost tight security analysis for our generic batch protocol, which significantly improves upon the previously known security bounds even for the specific case of batch Schnorr protocol. We extend our results beyond algebraic Sigma protocols. We analyze the rogue-instance security of a general batch protocol with plus-one special soundness (a generalization of standard special soundness) and achieve improved security bounds in the generic case.
Our results use a particular type of \emph{high-moment} assumptions introduced by Rotem and Segev (CRYPTO 2021). These assumptions consider the hardness of a relation against algorithms with bounded \emph{expected} running time. Although Rotem and Segev introduced these assumptions, they did not provide evidence to support their hardness. To substantiate and validate the high-moment assumptions, we present a new framework for assessing the concrete hardness of cryptographic problems against oracle algorithms with bounded expected runtime. Our framework covers generic models, including the generic group model, random oracle model, and more. Utilizing our framework, we achieve the first hardness result for these high-moment assumptions. In particular, we establish the second-moment hardness of the discrete-logarithm problem against expected-time algorithms in the generic group model.

2023

TCC

Round-Robin is Optimal: Lower Bounds for Group Action Based Protocols
Abstract

An hard homogeneous space (HHS) is a finite group acting on a set with
the group action being hard to invert and the set lacking any algebraic
structure.
As such HHS could potentially replace finite groups where the discrete logarithm is hard for building cryptographic primitives and protocols in a post-quantum world.
Threshold HHS-based primitives typically require parties to compute the group action of a secret-shared input on a public set element.
On one hand this could be done through generic MPC techniques, although they incur in prohibitive costs due to the high complexity of circuits evaluating group actions known to date.
On the other hand round-robin protocols only require black box usage of the HHS.
However these are highly sequential procedures, lasting as many rounds as parties involved.
The high round complexity appears to be inherent due the lack of homomorphic properties in HHS, yet no lower bounds were known so far.
In this work we formally show that round-robin protocols are optimal.
In other words, any \textit{at least passively secure} distributed computation of a group action making black-box use of an HHS must take a number of rounds greater or equal to the threshold parameter.
We furthermore study \textit{fair} protocols in which all users receive the output in the same round (unlike plain round-robin), and prove communication and computation lower bounds of $\Omega(n \log_2 n)$ for $n$ parties.
Our results are proven in Shoup's Generic Action Model (GAM), and hold regardless of the underlying computational assumptions.

2023

TCC

Searching for ELFs in the Cryptographic Forest
Abstract

Extremely Lossy Functions (ELFs) are families of functions that, depending on the choice during key generation, either operate in injective mode or instead have only a polynomial image size. The choice of the mode is indistinguishable to an outsider. ELFs were introduced by Zhandry (Crypto 2016) and have been shown to be very useful in replacing random oracles in a number of applications.
One open question is to determine the minimal assumption needed to instantiate ELFs. While all constructions of ELFs depend on some form of exponentially-secure public-key primitive, it was conjectured that exponentially-secure secret-key primitives, such as one-way functions, hash functions or one-way product functions, might be sufficient to build ELFs. In this work we answer this conjecture mostly negative: We show that no primitive, which can be derived from a random oracle (which includes all secret-key primitives mentioned above), is enough to construct even moderately lossy functions in a black-box manner. However, we also show that (extremely) lossy functions themselves do not imply public-key cryptography, leaving open the option to build ELFs from some intermediate primitive between the classical categories of secret-key and public-key cryptography.

2023

TCC

Security Proofs for Key-Alternating Ciphers with Non-Independent Round Permutations
Abstract

This work studies the key-alternating ciphers (KACs) whose round permutations are not necessarily independent. We revisit existing security proofs for key-alternating ciphers with a single permutation (KACSPs), and extend their method to an arbitrary number of rounds. In particular, we propose new techniques that can significantly simplify the proofs, and also remove two unnatural restrictions in the known security bound of 3-round KACSP (Wu et al., Asiacrypt 2020). With these techniques, we prove the first tight security bound for t-round KACSP, which was an open problem. We stress that our techniques apply to all variants of KACs with non-independent round permutations, as well as to the standard KACs.

2023

TCC

Security with Functional Re-Encryption from CPA
Abstract

The notion of functional re-encryption security (funcCPA) for public-key encryption schemes was recently introduced by Akavia et al. (TCC'22), in the context of homomorphic encryption.
This notion lies in between CPA security and CCA security: we give the attacker a *functional re-encryption oracle* instead of the decryption oracle of CCA security. This oracle takes a ciphertext ct and a function f, and returns fresh encryption of f applied to the decryption of ct; in symbols, ct'=Enc(f(Dec(ct))).
In this work we observe that funcCPA security may have applications beyond homomorphic encryption, and set out to study its properties. As our main contribution, we prove that
funcCPA is "closer to CPA than to CCA"; that is, funcCPA secure encryption can be constructed in a black-box manner from CPA-secure encryption. We stress that, prior to our work, this was not known even for regular re-encryption queries corresponding to identity function f.
At the core of our result is a new technique, showing how to handle *adaptive* functional re-encryption queries using tools previously developed in the context of non-malleable encryption, which roughly corresponds to a single *non-adaptive* parallel decryption query.

2023

TCC

Semi-Quantum Copy-Protection and More
Abstract

Properties of quantum mechanics have enabled the emergence of quantum cryptographic protocols achieving important goals which are proven to be impossible classically. Unfortunately, this usually comes at the cost of needing quantum power from every party in the protocol, while arguably a more realistic scenario would be a network of classical clients, classically interacting with a quantum server.
In this paper, we focus on copy-protection, which is a quantum primitive that allows a program to be evaluated, but not copied, and has shown interest especially due to its links to other unclonable cryptographic primitives. Our main contribution is to show how to dequantize quantum copy-protection schemes constructed from hidden coset states, by giving a construction for classically-instructed remote state preparation for coset states, which *preserves hardness properties of hidden coset states*. We then apply this dequantizer to obtain semi-quantum cryptographic protocols for copy-protection and tokenized signatures with strong unforgeability. In the process, we present the first secure copy-protection scheme for point functions in the plain model and a new direct product hardness property of coset states which immediately implies a strongly unforgeable tokenized signature scheme.

2023

TCC

Simplex Consensus: A Fast and Simple Consensus Protocol
Abstract

We present a theoretical framework for analyzing the efficiency of consensus protocols, and apply it to analyze the optimistic and pessimistic confirmation times of state-of-the-art partially-synchronous protocols in the so-called ``rotating leader/random leader'' model of consensus (recently popularized in the blockchain setting).
We next present a new and simple consensus protocol in the partially synchronous setting, tolerating $f < n/3$ byzantine faults; in our eyes, this protocol is essentially as simple to describe as the simplest known protocols, but it also enjoys an even simpler security proof, while matching and, even improving, the efficiency of the state-of-the-art (according to our theoretical framework).
As with the state-of-the-art protocols, our protocol assumes a (bare) PKI, a digital signature scheme, collision-resistant hash functions, and a random leader election oracle, which may be instantiated with a random oracle (or a CRS).

2023

TCC

Synchronizable Fair Exchange
Abstract

Fitzi, Garay, Maurer, and Ostrovsky (J.\ Cryptology 2005) showed that in the presence of a dishonest majority, no primitive of cardinality $n - 1$ is complete for realizing an arbitrary $n$-party functionality with {\em guaranteed output delivery}. In this work, we introduce a new $2$-party primitive $\mathcal{F}_{\mathsf{SyX}}$ (``synchronizable fair exchange'') and show that it is complete for realizing any $n$-party functionality with {\em fairness} in a setting where all parties are pairwise connected by instances of $\mathcal{F}_{\mathsf{SyX}}$.
In the $\mathcal{F}_{\mathsf{SyX}}$-hybrid model, the two parties {\em load} $\mathcal{F}_{\mathsf{SyX}}$ with some input, and following this, either party can {\em trigger} $\mathcal{F}_{\mathsf{SyX}}$ with a ``witness'' at a later time to receive the output from $\mathcal{F}_{\mathsf{SyX}}$. Crucially the other party also receives output from $\mathcal{F}_{\mathsf{SyX}}$ when $\mathcal{F}_{\mathsf{SyX}}$ is triggered. The trigger witnesses allow us to {\em synchronize} the trigger phases of multiple instances of $\mathcal{F}_{\mathsf{SyX}}$, thereby aiding in the design of fair multiparty protocols. Additionally, a pair of parties may {\em reuse} a single {\em a priori} loaded instance of $\mathcal{F}_{\mathsf{SyX}}$ in any number of multiparty protocols (involving different sets of parties).

2023

TCC

Taming Adaptivity in YOSO Protocols: The Modular Way
Abstract

YOSO-style MPC protocols (Gentry et al., Crypto’21), is a promising framework where the overall computation is partitioned into small, short-lived pieces, delegated to subsets of one-time stateless parties. Such protocols enable gaining from the security benefits provided by using a large community of participants where “mass corruption” of a large fraction of participants is considered unlikely, while keeping the computational and communication costs manageable. However, fully realizing and analyzing YOSO-style protocols has proven to be challenging: While different components have been defined and realized in various works, there is a dearth of protocols that have reasonable efficiency and enjoy full end to end security against adaptive adversaries.
The YOSO model separates the protocol design, specifying the short-lived responsibilities, from the mechanisms assigning these responsibilities to machines participating in the computation. These protocol designs must then be translated to run directly on the machines, while preserving security guarantees. We provide a versatile and modular framework for analyzing the security of YOSO-style protocols, and show how to use it to compile any protocol design that is secure against static corruptions of t out of c parties, into protocols that withstand adaptive corruption of T out of N machines (where T/N is closely related to t/c, specifically when t/c < 0.5, we tolerate T/N ≤ 0.29) at overall communication cost that is comparable to that of the traditional protocol even when c << N.
Furthermore, we demonstrate how to minimize the use of costly non-committing encryption,
thereby keeping the computational and communication overhead manageable even in practical terms, while still providing end to end security analysis. Combined with existing approaches for transforming stateful protocols into stateless ones while preserving static security (e.g. Gentry et al. 21, Kolby et al. 22), we obtain end to end security.

2023

TCC

Three Party Secure Computation with Friends and Foes
Abstract

In secure multiparty computation (MPC), the goal is to allow a set of mutually distrustful parties to compute some function of their private inputs in a way that preserves some security properties, even in the face of adversarial behavior by some of the parties. However, classical security definitions do not pose any privacy restrictions on the view of honest parties. Thus, if an attacker adversarially leaks private information to \emph{honest} parties, it does not count as a violation of privacy. Moreover, even the protocol itself may instruct all parties to send their inputs to other honest parties (if, say, all corrupted parties have been previously revealed). This is arguably undesirable, and in real-life scenarios, it is hard to imagine that possible users would agree to have their private information revealed, even if only to other honest parties.
To address this issue, Alon et al.~[CRYPTO 20] introduced the notion of \emph{security with friends and foes} (FaF security). In essence, $(t,h)$-FaF security requires that a malicious adversary corrupting up to $t$ parties cannot help a coalition of $h$ semi-honest parties to learn anything beyond what they can learn from their inputs and outputs (combined with the input and outputs of the malicious parties). They further showed that $(t,h)$-FaF full security with $n$ parties is achievable for any functionality if and only if $2t+h<n$. A remaining important open problem is to characterize the set of $n$-party functionalities that can be computed with $(t,h)$-FaF full security assuming $2t+h\geq n$.
In this paper, we focus on the special, yet already challenging, case of $(1,1)$-FaF full security for three-party, 2-ary (two inputs), symmetric (all parties output the same value) functionalities. We provide several positive results, a lower bound on the round complexity, and an impossibility result. In particular, we prove the following.
1. We identify a large class of three-party Boolean symmetric 2-ary functionalities that can be computed with $(1,1)$-FaF full security.
2. We identify a large class of three-party (possibly non-Boolean) symmetric 2-ary functionalities, for which no $O(\log\secParam)$-round protocol computes them with $(1,1)$-FaF full security. This matches the round complexity of our positive results for various interesting functionalities, such as equality of strings.

2023

TCC

Towards Topology-Hiding Computation from Oblivious Transfer
Abstract

Topology-Hiding Computation (THC) enables parties to securely compute a function on an incomplete network without revealing the network topology. It is known that secure computation on a complete network can be based on oblivious transfer (OT), even if a majority of the participating parties are corrupt. In contrast, THC in the dishonest majority setting is only known from assumptions that imply (additively) homomorphic encryption, such as Quadratic Residuosity, Decisional Diffie-Hellman, or Learning With Errors.
In this work we move towards closing the gap between MPC and THC by presenting a protocol for THC on general graphs secure against all-but-one semi-honest corruptions from constant-round constant-overhead secure two-party computation. Our protocol is therefore the first to achieve THC on arbitrary networks without relying on assumptions with rich algebraic structure. As a technical tool, we introduce the notion of locally simulatable MPC, which we believe to be of independent interest.

2023

TCC

Weakening Assumptions for Publicly-Verifiable Deletion
Abstract

We develop a simple compiler that generically adds publicly-verifiable deletion to a variety of cryptosystems. Our compiler only makes use of one-way functions (or one-way state generators, if we allow the public verification key to be quantum). Previously, similar compilers either relied on indistinguishability obfuscation along with any one-way function (Bartusek et. al., ePrint:2023/265), or on almost-regular one-way functions (Bartusek, Khurana and Poremba, CRYPTO 2023).

2023

TCC

Your Reputation's Safe with Me: Framing-Free Distributed Zero-Knowledge Proofs
Abstract

{\em Distributed Zero-Knowledge (dZK) proofs}, recently introduced by Boneh et al. (CYPTO`19), allow a prover $\prover$ to prove NP statements on an input $x$ which is {\em distributed} between $k$ verifiers $\verifier_1,\ldots,\verifier_k$, where each $\verifier_i$ holds only a piece of $x$. As in standard ZK proofs, dZK proofs guarantee {\em Completeness} when all parties are honest; {\em Soundness} against a malicious prover colluding with $t$ verifiers; and {\em Zero Knowledge} against a subset of $t$ malicious verifiers, in the sense that they learn nothing about the NP witness and the input pieces of the honest verifiers. Unfortunately, dZK proofs provide no correctness guarantee for an honest prover against a subset of maliciously corrupted verifiers. In particular, such verifiers might be able to ``frame'' the prover, causing honest verifiers to reject a true claim. This is a significant limitation, since such scenarios arise naturally in dZK applications, e.g., for proving honest behavior, and such attacks are indeed possible in existing dZKs (Boneh et al., CRYPTO`19).
We put forth and study the notion of {\em strong completeness} for dZKs, guaranteeing that true claims are accepted even when $t$ verifiers are maliciously corrupted. We then design strongly-complete dZK proofs using the ``MPC-in-the-head'' paradigm of Ishai et al. (STOC`07), providing a novel analysis that exploits the unique properties of the distributed setting.
To demonstrate the usefulness of strong completeness, we present several applications in which it is instrumental in obtaining security. First, we construct a certifiable version of Verifiable Secret Sharing (VSS), which is a VSS in which the dealer additionally proves that the shared secret satisfies a given NP relation. Our construction withstands a constant fraction of corruptions, whereas a previous construction of Ishai et al. (TCC`14) required $k=\poly\left(t\right)$. We also design a {\em reusable} version of certifiable VSS that we introduce, in which the dealer can prove an {\em unlimited} number of predicates on the {\em same} shared secret.
Finally, we extend a compiler of Boneh et al. (CRYPTO`19), who used dZKs to transform a class of ``natural'' semi-honest protocols in the honest-majority setting into maliciously secure ones with abort. Our compiler uses {\em strongly-complete} dZKs to obtain {\em identifiable} abort.

2023

TCC

Zombies and Ghosts: Optimal Byzantine Agreement in the Presence of Omission Faults
Abstract

Studying the feasibility of Byzantine Agreement (BA) in realistic fault models is an important question in the area of distributed computing and cryptography. In this work, we revisit the mixed fault model with Byzantine (malicious) faults and omission faults put forth by Hauser, Maurer, and Zikas (TCC 2009), who showed that BA (and MPC) is possible with t Byzantine faults, s send faults (whose outgoing messages may be dropped) and r receive faults (whose incoming messages may be lost) if n>3t+r+s. We generalize their techniques and results by showing that BA is possible if n>2t+r+s, given the availability of a cryptographic setup. Our protocol is the first to match the recent lower bound of Eldefrawy, Loss, and Terner (ACNS 2022) for this setting.