Proving non-native operations is still a bottleneck in existing incrementally verifiable computations. Prior attempts to solve this issue either simply improve the efficiency of proofs of non-native operations or require folding instances in each curve of a cycle. This paper shows how to avoid altogether in-circuit proofs of non-native operations in the incre- mental steps, and only record them in some auxiliary proof information. These operations are proved natively at the end of the...

This paper investigates the open problem of how to construct non-interactive rational proofs. Rational proofs, introduced by Azar and Micali (STOC 2012), are a model of interactive proofs where a computationally powerful server can be rewarded by a weaker client for running an expensive computation $f(x)$. The honest strategy is enforced by design when the server is rational: any adversary claiming a false output $y \neq f(x)$ will lose money on expectation. Rational proof constructions...

The MPC-in-the-Head framework has been pro- posed as a solution for Non-Interactive Zero-Knowledge Arguments of Knowledge (NIZKAoK) due to its efficient proof generation. However, most existing NIZKAoK constructions using this approach require multiple MPC evaluations to achieve negligible soundness error, resulting in proof size and time that are asymptotically at least λ times the size of the circuit of the NP relation. In this paper, we propose a novel method to eliminate the need for...

In this writeup we discuss the soundness of the Basefold multilinear polynomial commitment scheme [Zeilberger, Chen, Fisch 23] applied to Reed-Solomon codes, and run with proximity parameters up to the Johnson list decoding bound. Our security analysis relies on a generalization of the celebrated correlated agreement theorem from [Ben-Sasson, et al., 20] to linear subcodes of Reed-Solomon codes, which turns out a by-product of the Guruswami-Sudan list decoder analysis. We further highlight...

We present a new approach for constructing non-interactive zero-knowledge (NIZK) proof systems from vector trapdoor hashing (VTDH) -- a generalization of trapdoor hashing [Döttling et al., Crypto'19]. Unlike prior applications of trapdoor hash to NIZKs, we use VTDH to realize the hidden bits model [Feige-Lapidot-Shamir, FOCS'90] leading to black-box constructions of NIZKs. This approach gives us the following new results: - A statistically-sound NIZK proof system based on the hardness of...

We enhance the provable soundness of FRI, an interactive oracle proof of proximity (IOPP) for Reed-Solomon codes introduced by Ben-Sasson et al. in ICALP 2018. More precisely, we prove the soundness error of FRI is less than $\max\left\{O\left(\frac{1}{\eta}\cdot \frac{n}{|\mathbb{F}_q|}\right), (1-\delta)^{t}\right\}$, where $\delta\le 1-\sqrt{\rho}-\eta$ is within the Johnson bound and $\mathbb{F}_q$ is a finite field with characteristic greater than $2$. Previously, the best-known...

We show that every NP relation that can be verified by a bounded-depth polynomial-sized circuit, or a bounded-space polynomial-time algorithm, has a computational zero-knowledge proof (with statistical soundness) with communication that is only additively larger than the witness length. Our construction relies only on the minimal assumption that one-way functions exist. In more detail, assuming one-way functions, we show that every NP relation that can be verified in NC has a...

We revisit the Ligero proximity test, and its logarithmic randomness variant, in the framework of [EA23] and show a simple proof that improves the soundness error of the original logarithmic randomness construction of [DP23] by a factor of two. This note was originally given as a presentation in ZK Summit 11.

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...

A compiler introduced by Kalai et al. (STOC'23) converts any nonlocal game into an interactive protocol with a single computationally-bounded prover. Although the compiler is known to be sound in the case of classical provers, as well as complete in the quantum case, quantum soundness has so far only been established for special classes of games. In this work, we establish a quantum soundness result for all compiled two-player nonlocal games. In particular, we prove that the quantum...

We present Mova, a folding scheme for R1CS instances that does not require committing to error or cross terms, nor makes use of the sumcheck protocol. We compute concrete costs and provide benchmarks showing that, for reasonable parameter choices, Mova's Prover is about $5$ to $10$ times faster than Nova's Prover, and about $1.05$ to $1.3$ times faster than Hypernova's Prover (applied to R1CS instances) -- assuming the R1CS witness vector contains only small elements. Mova's Verifier has a...

Orion (Xie et al. CRYPTO'22) is a recent plausibly post-quantum zero-knowledge argument system with a linear time prover. It improves over Brakedown (Golovnev et al. ePrint'21 and CRYPTO'23) by reducing the proof size and verifier complexity to be polylogarithmic and additionally adds the zero-knowledge property. The argument system is demonstrated to be concretely efficient with a prover time being the fastest among all existing succinct proof systems and a proof size that is an order of...

A dot-product proof (DPP) is a simple probabilistic proof system in which the input statement $\mathbf{x}$ and the proof $\boldsymbol{\pi}$ are vectors over a finite field $\mathbb{F}$, and the proof is verified by making a single dot-product query $\langle \mathbf{q},(\mathbf{x} \| \boldsymbol{\pi}) \rangle$ jointly to $\mathbf{x}$ and $\boldsymbol{\pi}$. A DPP can be viewed as a 1-query fully linear PCP. We study the feasibility and efficiency of DPPs, obtaining the following results: -...

Proving the security of a Multiparty Computation (MPC) protocol is a difficult task. Under the current simulation-based definition of MPC, a security proof consists of a simulator, which is usually specific to the concrete protocol and requires to be manually constructed, together with a theoretical analysis of the output distribution of the simulator and corrupted parties' views in the real world. This presents an obstacle in verifying the security of a given MPC protocol. Moreover, an...

Interactive proofs and arguments of knowledge can be generalized to the concept of interactive reductions of knowledge, where proving knowledge of a witness for one NP language is reduced to proving knowledge of a witness for another NP language. We take this generalization and specialize it to a class of reductions we refer to as `quirky interactive reductions of knowledge' (or QUIRKs). This name reflects our particular design choices within the broad and diverse world of interactive...

Lipmaa, Parisella, and Siim [Eurocrypt, 2024] proved the extractability of the KZG polynomial commitment scheme under the falsifiable assumption ARSDH. They also showed that variants of real-world zk-SNARKs like Plonk can be made knowledge-sound in the random oracle model (ROM) under the ARSDH assumption. However, their approach did not consider various batching optimizations, resulting in their variant of Plonk having approximately $3.5$ times longer argument. Our contributions are: (1) We...

We construct a succinct non-interactive argument (SNARG) system for every NP language $\mathcal{L}$ that has a propositional proof of non-membership for each $x\notin \mathcal{L}$. The soundness of our SNARG system relies on the hardness of the learning with errors (LWE) problem. The common reference string (CRS) in our construction grows with the space required to verify the propositional proof, and the size of the proof grows poly-logarithmically in the length of the propositional...

Distributed Point Function (DPF) provides a way for a dealer to split a point function $f_{\alpha, \beta}$ into multiple succinctly described function-shares, where the function $f_{\alpha, \beta}$ for a special input $\alpha$, returns a special output value $\beta$, and returns a fixed value $0$ otherwise. As the security requirement, any strict subset of the function-shares reveals nothing about the function $f_{\alpha,\beta}$. However, each function-share can be individually evaluated on...

We construct an adaptively-sound succinct non-interactive argument (SNARG) for NP in the CRS model from sub-exponentially-secure indistinguishability obfuscation ($i\mathcal{O}$) and sub-exponentially-secure one-way functions. Previously, Waters and Wu (STOC 2024), and subsequently, Waters and Zhandry (CRYPTO 2024) showed how to construct adaptively-sound SNARGs for NP by relying on sub-exponentially-secure indistinguishability obfuscation, one-way functions, and an additional algebraic...

A popular way to build post-quantum signature schemes is by first constructing an identification scheme (IDS) and applying the Fiat-Shamir transform to it. In this work we tackle two open questions related to the general applicability of techniques around this approach that together allow for efficient post-quantum signatures with optimal security bounds in the QROM. First we consider a recent work by Aguilar-Melchor, Hülsing, Joseph, Majenz, Ronen, and Yue (Asiacrypt'23) that showed...

Multi-Key Homomorphic Signatures (MKHS) allow one to evaluate a function on data signed by distinct users while producing a succinct and publicly-verifiable certificate of the correctness of the result. All the constructions of MKHS in the state of the art achieve a weak level of succinctness where signatures are succinct in the total number of inputs but grow linearly with the number of users involved in the computation. The only exception is a SNARK-based construction which relies on a...

Interactive proofs are a cornerstone of modern cryptography and as such used in many areas, from digital signatures to multy-party computation. Often the knowledge error $\kappa$ of an interactive proof is not small enough, and thus needs to be reduced. This is usually achieved by repeating the interactive proof in parallel t times. Recently, it was shown that parallel repetition of any $(k_1, \ldots , k_\mu)$-special-sound multi-round public-coin interactive proof reduces the knowledge...

HyperPlonk is a recent SNARK proposal (Eurocrypt'23) that features a linear-time prover and supports custom gates of larger degree than Plonk. For the time being, its instantiations are only proven to be knowledge-sound (meaning that soundness is only guaranteed when the prover runs in isolation) while many applications motivate the stronger notion of simulation-extractability (SE). Unfortunately, the most efficient SE compilers are not immediately applicable to multivariate polynomial...

The significance of succinct zero-knowledge proofs has increased considerably in recent times. However, one of the major challenges that hinder the prover's efficiency is when dealing with Boolean circuits. In particular, the conversion of each bit into a finite field element incurs a blow-up of more than 100x in terms of both memory usage and computation time. This work focuses on data-parallel Boolean circuits that contain numerous identical sub-circuits. These circuits are widely used...

In an Instance-Hiding Interactive Proof (IHIP) [Beaver et al. CRYPTO 90], an efficient verifier with a _private_ input x interacts with an unbounded prover to determine whether x is contained in a language L. In addition to completeness and soundness, the instance-hiding property requires that the prover should not learn anything about x in the course of the interaction. Such proof systems capture natural privacy properties, and may be seen as a generalization of the influential concept of...

Universal verifiability is a must-to-have for electronic voting schemes. It is essential to ensure honest behavior of all the players during the whole process, together with the eligibility. However, it should not endanger the privacy of the individual votes, which is another major requirement. Whereas the first property prevents attacks during the voting process, privacy of the votes should hold forever, which has been called everlasting privacy. A classical approach for universal...

A relativized succinct argument in the random oracle model (ROM) is a succinct argument in the ROM that can prove/verify the correctness of computations that involve queries to the random oracle. We prove that relativized succinct arguments in the ROM do not exist. The impossibility holds even if the succinct argument is interactive, and even if soundness is computational (rather than statistical). This impossibility puts on a formal footing the commonly-held belief that succinct...

This paper presents a comprehensive security analysis of the Adh zero-knowledge proof system, a novel lattice-based, quantum-resistant proof of possession system. The Adh system offers compact key and proof sizes, making it suitable for real-world digital signature and public key agreement protocols. We explore its security by reducing it to the hardness of the Module-ISIS problem and introduce three new variants: Module-ISIS+, Module-ISIS*, and Module-ISIS**. These constructions enhance...

An inner product argument (IPA) is a cryptographic primitive used to construct a zero-knowledge proof system, which is a notable privacy-enhancing technology. We propose a novel efficient IPA called $\mathsf{Cougar}$. $\mathsf{Cougar}$ features cubic root verifier and logarithmic communication under the discrete logarithm (DL) assumption. At Asiacrypt2022, Kim et al. proposed two square root verifier IPAs under the DL assumption. Our main objective is to overcome the limitation of square...

A probabilistically checkable argument (PCA) is a computational relaxation of PCPs, where soundness is guaranteed to hold only for false proofs generated by a computationally bounded adversary. The advantage of PCAs is that they are able to overcome the limitations of PCPs. A succinct PCA has a proof length that is polynomial in the witness length (and is independent of the non-deterministic verification time), which is impossible for PCPs, under standard complexity assumptions. Bronfman and...

Cryptographic protocols are hard to design and prove correct, as witnessed by the ever-growing list of attacks even on protocol standards. Symbolic models of cryptography enable automated formal security proofs of such protocols against an idealized cryptographic model, which abstracts away from the algebraic properties of cryptographic schemes and thus misses attacks. Computational models of cryptography yield rigorous guarantees but support at present only interactive proofs and/or...

Zero-Knowledge Proof (ZKP) technology marks a revolutionary advancement in the field of cryptography, enabling the verification of certain information ownership without revealing any specific details. This technology, with its paradoxical yet powerful characteristics, provides a solid foundation for a wide range of applications, especially in enhancing the privacy and security of blockchain technology and other cryptographic systems. As ZKP technology increasingly becomes a part of the...

A verifiable delay function $\texttt{VDF}(x,T)\rightarrow (y,\pi)$ maps an input $x$ and time parameter $T$ to an output $y$ together with an efficiently verifiable proof $\pi$ certifying that $y$ was correctly computed. The function runs in $T$ sequential steps, and it should not be possible to compute $y$ much faster than that. The only known practical VDFs use sequential squaring in groups of unknown order as the sequential function, i.e., $y=x^{2^T}$. There are two constructions for...

Linkable ring signatures are an important cryptographic primitive for anonymized applications, such as e-voting, e-cash and confidential transactions. To eliminate backdoor and overhead in a trusted setup, transparent setup in the discrete logarithm or pairing settings has received considerable attention in practice. Recent advances have improved the proof sizes and verification efficiency of linkable ring signatures with a transparent setup to achieve logarithmic bounds. Omniring (CCS '19)...

The Fiat-Shamir transform [1] is a well-known and widely employed technique for converting sound public-coin interactive protocols into sound non-interactive protocols. Even though the transformation itself is relatively clear and simple, some implementations choose to deviate from the specifications, for example for performance reasons. In this short note, we present a vulnerability arising from such a deviation in a KZG-based PLONK verifier implementation. This deviation stemmed from the...

Nonlocal games are a foundational tool for understanding entanglement and constructing quantum protocols in settings with multiple spatially separated quantum devices. In this work, we continue the study initiated by Kalai et al. (STOC '23) of compiled nonlocal games, played between a classical verifier and a single cryptographically limited quantum device. Our main result is that the compiler proposed by Kalai et al. is sound for any two-player XOR game. A celebrated theorem of Tsirelson...

Several prior works have suggested to use non-interactive arguments of knowledge with short proofs to aggregate signatures of Falcon, which is part of the first post-quantum signatures selected for standardization by NIST. Especially LaBRADOR, based on standard structured lattice assumptions and published at CRYPTO’23, seems promising to realize this task. However, no prior work has tackled this idea in a rigorous way. In this paper, we thoroughly prove how to aggregate Falcon signatures...

Polynomial commitment scheme allows a prover to commit to a polynomial $f \in \mathcal{R}[X]$ of degree $L$, and later prove that the committed function was correctly evaluated at a specified point $x$; in other words $f(x)=u$ for public $x,u \in\mathcal{R}$. Most applications of polynomial commitments, e.g. succinct non-interactive arguments of knowledge (SNARKs), require that (i) both the commitment and evaluation proof are succinct (i.e., polylogarithmic in the degree $L$) - with the...

Folding is a recent technique for building efficient recursive SNARKs. Several elegant folding protocols have been proposed, such as Nova, Supernova, Hypernova, Protostar, and others. However, all of them rely on an additively homomorphic commitment scheme based on discrete log, and are therefore not post-quantum secure and require a large (256-bit) field. In this work we present LatticeFold, the first lattice-based folding protocol based on the Module SIS problem. This folding protocol...

We study how to construct hash functions that can securely instantiate the Fiat-Shamir transformation against bounded-depth adversaries. The motivation is twofold. First, given the recent fruitful line of research of constructing cryptographic primitives against bounded-depth adversaries under worst-case complexity assumptions, and the rich applications of Fiat-Shamir, instantiating Fiat-Shamir hash functions against bounded-depth adversaries under worst-case complexity assumptions might...

Nova is a new type of recursive proof system that uses a folding scheme as its core building block. This brilliant idea of folding relations can significantly reduce the recursion overhead. In this paper, we study some issues related to Nova’s soundness proof, which relies on the soundness of the folding scheme in a recursive manner. First, due to its recursive nature, the proof strategy inevitably causes the running time of the recursive extractor to expand polynomially for each...

We study succinct non-interactive arguments (SNARGs) and succinct non-interactive arguments of knowledge (SNARKs) for the class $\mathsf{UP}$ in the reusable designated verifier model. $\mathsf{UP}$ is an expressive subclass of $\mathsf{NP}$ consisting of all $\mathsf{NP}$ languages where each instance has at most one witness; a designated verifier SNARG (dvSNARG) is one where verification of the SNARG proof requires a private verification key; and such a dvSNARG is reusable if soundness...

In an (α,β)-weak non-interactive zero knowledge (NIZK), the soundness error is at most α and the zero-knowledge error is at most β. Goyal, Jain, and Sahai (CRYPTO 2019) show that if α+β<1 for some constants α,β, then (α,β)-weak NIZK can be turned into fully-secure NIZK, assuming sub-exponentially-secure public-key encryption. We revisit the problem of NIZK amplification: – We amplify NIZK arguments assuming only polynomially-secure public-key encryption, for any constants α+β<1. – We...

Somewhere statistically binding (SSB) hashing allows us to sample a special hashing key such that the digest statistically binds the input at $m$ secret locations. This hash function is said to be somewhere extractable (SE) if there is an additional trapdoor that allows the extraction of the input bits at the $m$ locations from the digest. Devadas, Goyal, Kalai, and Vaikuntanathan (FOCS 2022) introduced a variant of somewhere extractable hashing called rate-1 fully local SE hash...

Trusted setup is commonly used for non-interactive proof and argument systems. However, there is no guarantee that the setup parameters in these systems are generated in a trustworthy manner. Building upon previous works, we conduct a systematic study of non-interactive zero-knowledge arguments in the common reference string model where the authority running the trusted setup might be corrupted. We explore both zero-knowledge and soundness properties in this setting.  - We consider a new...

A list polynomial commitment scheme (LPC) is a polynomial commitment scheme with a relaxed binding property. Namely, in an LPC setting, a commitment to a function $f(X)$ can be opened to a list of low-degree polynomials close to $f(X)$ (w.r.t. the relative Hamming distance and over a domain $D$). The scheme also allows opening one of the polynomials of the list at an arbitrary point $x$ and convincing a verifier that one of the polynomials in the list evaluates to the purported...

We prove that the seminal KZG polynomial commitment scheme (PCS) is black-box extractable under a simple falsifiable assumption ARSDH. To create an interactive argument, we construct a compiler that combines a black-box extractable non-interactive PCS and a polynomial IOP (PIOP). The compiler incurs a minor cost per every committed polynomial. Applying the Fiat-Shamir transformation, we obtain slightly less efficient variants of well-known PIOP-based zk-SNARKs, such as Plonk, that are...

A succinct non-interactive argument (SNARG) for $\mathsf{NP}$ allows a prover to convince a verifier that an $\mathsf{NP}$ statement $x$ is true with a proof of size $o(|x| + |w|)$, where $w$ is the associated $\mathsf{NP}$ witness. A SNARG satisfies adaptive soundness if the malicious prover can choose the statement to prove after seeing the scheme parameters. In this work, we provide the first adaptively-sound SNARG for $\mathsf{NP}$ in the plain model assuming sub-exponentially-hard...

Previous studies on deep-learning-based side-channel attacks (DL-SCAs) have shown that traditional performance evaluation metrics commonly used in DL, like accuracy and F1 score, are not effective in evaluating DL-SCA performance. Therefore, some previous studies have proposed new alternative metrics for evaluating the performance of DL-SCAs. Notably, perceived information (PI) and effective perceived information (EPI) are major metrics based on information theory. While it has been...

We study the complexity of memory checkers with computational security and prove the first general tight lower bound. Memory checkers, first introduced over 30 years ago by Blum, Evans, Gemmel, Kannan, and Naor (FOCS '91, Algorithmica '94), allow a user to store and maintain a large memory on a remote and unreliable server by using small trusted local storage. The user can issue instructions to the server and after every instruction, obtain either the correct value or a failure (but not...

Key Transparency (KT) systems enable service providers of end-to-end encrypted communication (E2EE) platforms to maintain a Verifiable Key Directory (VKD) that maps each user's identifier, such as a username or email address, to their identity public key(s). Users periodically monitor the directory to ensure their own identifier maps to the correct keys, thus detecting any attempt to register a fake key on their behalf to Meddler-in-the-Middle (MitM) their communications. We introduce and...

Dual-receiver encryption (DRE) is a special form of public key encryption (PKE) that allows a sender to encrypt a message for two recipients. Without further properties, the difference between DRE and PKE is only syntactical. One such important property is soundness, which requires that no ciphertext can be constructed such that the recipients decrypt to different plaintexts. Many applications rely on this property in order to realize more complex protocols or primitives. In addition, many...

In this paper, we focus on the common prime RSA variant and introduces a novel investigation into the partial key exposure attack targeting it. We explore the vulnerability of this RSA variant, which employs two common primes $p$ and $q$ defined as $p=2ga+1$ and $q=2gb+1$ for a large prime $g$. Previous cryptanalysis of common prime RSA has primarily focused on the small private key attack. In our work, we delve deeper into the realm of partial key exposure attacks by categorizing them into...

Differential privacy is a fundamental concept for protecting individual privacy in databases while enabling data analysis. Conceptually, it is assumed that the adversary has no direct access to the database, and therefore, encryption is not necessary. However, with the emergence of cloud computing and the «on-cloud» storage of vast databases potentially contributed by multiple parties, it is becoming increasingly necessary to consider the possibility of the adversary having (at least...

We present a general method to simplify soundness proofs under certain conditions. Given an adversary $\mathcal{A}$ able to break a scheme $S$ with non-negligible probability $t$, we define the concept of $\textit{trace}$ of a $\textit{winning configuration}$, which is already implicitly used in soundness proofs. If a scheme can be constructed that (1) takes a random configuration $e$, being the inputs and execution environment of $\mathcal{A}$, (2) "guesses" a trace, (3) modifies $e$ based...

We investigate proof systems where security holds against rational parties instead of malicious ones. Our starting point is the notion of rational arguments, a variant of rational proofs (Azar and Micali, STOC 2012) where security holds against rational adversaries that are also computationally bounded. Rational arguments are an interesting primitive because they generally allow for very efficient protocols, and in particular sublinear verification (i.e. where the Verifier does not have...

The Fiat-Shamir transformation is a general principle to turn any public-coin interactive proof into non-interactive one (with security then typically analyzed in the random oracle model). While initially used for 3-round protocols, many recent constructions use it for multi-round protocols. However, in general the soundness error of the Fiat-Shamir transformed protocol degrades exponentially in the number of rounds. On the positive side, it was shown that for the special class of...

Succinctness and zero-knowledge are two fundamental properties in the study of cryptographic proof systems. Several recent works have formalized the connections between these two notions by showing how to realize non-interactive zero-knowledge (NIZK) arguments from succinct non-interactive arguments. Specifically, Champion and Wu (CRYPTO 2023) as well as Bitansky, Kamath, Paneth, Rothblum, and Vasudevan (ePrint 2023) recently showed how to construct a NIZK argument for NP from a...

This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture...

After the pioneering results proposed by Bellare et al in ASIACRYPT 2016, there have been lots of efforts to construct zero-knowledge succinct non-interactive arguments of knowledge protocols (zk-SNARKs) that satisfy subversion zero knowledge (S-ZK) and standard soundness from the zk-SNARK in the common reference string (CRS) model. The various constructions could be regarded secure in the bare public key (BPK) model because of the equivalence between S-ZK in the CRS model, and uniform...

Parallel repetition refers to a set of valuable techniques used to reduce soundness error of probabilistic proofs while saving on certain efficiency measures. Parallel repetition has been studied for interactive proofs (IPs) and multi-prover interactive proofs (MIPs). In this paper we initiate the study of parallel repetition for probabilistically checkable proofs (PCPs). We show that, perhaps surprisingly, parallel repetition of a PCP can increase soundness error, in fact bringing the...

Non-interactive zero-knowledge (NIZK) proof systems are a cornerstone of modern cryptography, but their security has received little attention in the quantum settings. Motivated by improving our understanding of this fundamental primitive against quantum adversaries, we propose a new definition of security against quantum adversary. Specifically, we define the notion of quantum simulation soundness (SS-NIZK), that allows the adversary to access the simulator in superposition. We show a...

$\Sigma$-protocols are a widely utilized, relatively simple and well understood type of zero-knowledge proofs. However, the well known Schnorr $\Sigma$-protocol for proving knowledge of discrete logarithm in a cyclic group of known prime order, and similar protocols working over this type of groups, are hard to generalize to dealing with other groups. In particular with hidden order groups, due to the inability of the knowledge extractor to invert elements modulo the order. In this paper,...

Proof-carrying data (PCD) is a powerful cryptographic primitive that allows mutually distrustful parties to perform distributed computation in an efficiently verifiable manner. Real-world deployments of PCD have sparked keen interest within the applied community and industry. Known constructions of PCD are obtained by recursively-composing SNARKs or related primitives. Unfortunately, known security analyses incur expensive blowups, which practitioners have disregarded as the analyses...

Many of the three-round blind signatures based on identification protocols are only proven to be $\ell$-concurrently unforgeable for $\ell = \mathsf{polylog}(\lambda)$. It was only recently shown in a seminal work by Benhamouda et al. (EUROCRYPT'21) that this is not just a limitation of the proof technique. They proposed an elegant polynomial time attack against the $\ell$-concurrently unforgeability of the classical blind Schnorr protocol for $\ell = \mathsf{poly}(\lambda)$. However,...

The generic-group model (GGM) and the algebraic-group model (AGM) have been exceptionally successful in proving the security of many classical and modern cryptosystems. These models, however, come with standard-model uninstantiability results, raising the question whether the schemes analyzed under them can be based on firmer standard-model footing. We formulate the uber-knowledge (UK) assumption, a standard-model assumption that naturally extends the uber-assumption family to...

$\Sigma$-protocols, a class of interactive two-party protocols, which are used as a framework to instantiate many other authentication schemes, are automatically a proof of knowledge (PoK) given that they satisfy the "special-soundness" property. This fact provides a convenient method to compose $\Sigma$-protocols and PoKs for complex relations. However, composing in this manner can be error-prone. While they must satisfy special-soundness, this is unfortunately not the case for many...

The MPC-in-the-Head paradigm is instrumental in building zero-knowledge proof systems and post-quantum signatures using techniques from secure multi-party computation. In this work, we extend and improve the recently proposed framework of MPC-in-the-Head based on threshold secret sharing, here called Threshold Computation in the Head. We first address some limitations of this framework, namely its overhead in the communication cost, its constraint on the number of parties and its degradation...

In remote state preparation with verifiability (RSPV), a client would like to prepare a quantum state (sampled from a state family) on the server side, such that ideally the client knows its full description, while the server holds and only holds the state itself. A closely related notion called self-testing, which is recently generalized to the single-server computationally-secure setting [MV21, aims at certifying the server's operation. These notions have been widely studied in various...

This work introduces the notion of naysayer proofs. We observe that in numerous (zero-knowledge) proof systems, it is significantly more efficient for the verifier to be convinced by a so-called naysayer that a false proof is invalid than it is to check that a genuine proof is valid. We show that every NP language has constant-size and constant-time naysayer proofs. We also show practical constructions for several example proof systems, including FRI polynomial commitments, post-quantum...

Recent works on lattice-based extractable polynomial commitments can be grouped into two classes: (i) non-interactive constructions that stem from the functional commitment by Albrecht, Cini, Lai, Malavolta and Thyagarajan (CRYPTO 2022), and (ii) lattice adaptations of the Bulletproofs protocol (S&P 2018). The former class enjoys security in the standard model, albeit a knowledge assumption is desired. In contrast, Bulletproof-like protocols can be made secure under falsifiable assumptions,...

We propose a new notion of knowledge soundness, denoted rogue-instance security, for interactive and non-interactive 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...

Gentry and Wichs proved that adaptively sound SNARGs for hard languages need non-falsifiable assumptions. Lipmaa and Pavlyk claimed Gentry-Wichs is tight by constructing a non-adaptively sound zk-SNARG FANA for NP from falsifiable assumptions. We show that FANA is flawed. We define and construct a fully algebraic $F$-position-binding vector commitment scheme VCF. We construct a concretely efficient commit-and-prove zk-SNARK Punic, a version of FANA with an additional VCF commitment to the...

Succinct arguments allow a prover to convince a verifier of the validity of any statement in a language, with minimal communication and verifier's work. Among other approaches, lattice-based protocols offer solid theoretical foundations, post-quantum security, and a rich algebraic structure. In this work, we present some new approaches to constructing efficient lattice-based succinct arguments. Our main technical ingredient is a new commitment scheme based on vanishing polynomials, a notion...

Correlation intractability is an emerging cryptographic paradigm that enabled several recent breakthroughs in establishing soundness of the Fiat-Shamir transform and, consequently, basing non-interactive zero-knowledge proofs and succinct arguments on standard cryptographic assumptions. In a nutshell, a hash family is said to be \emph{correlation intractable} for a class of relations $\mathcal{R}$ if, for any relation $R\in\mathcal{R}$, it is hard given a random hash function $h\gets H$ to...

Despite the notable advances in the development of high-assurance, verified implementations of cryptographic protocols, such implementations typically face significant performance overheads, particularly due to the penalties induced by formal verification and automated extraction of executable code. In this paper, we address some core performance challenges facing computer-aided cryptography by presenting a formal treatment for accelerating such verified implementations based on multiple...

Threshold ECDSA receives interest lately due to its widespread adoption in blockchain applications. A common building block of all leading constructions involves a secure conversion of multiplicative shares into additive ones, which is called the multiplicative-to-additive (MtA) function. MtA dominates the overall complexity of all existing threshold ECDSA constructions. Specifically, $O(n^2)$ invocations of MtA are required in the case of $n$ active signers. Hence, improvement of MtA leads...

We initiate the study of witness authenticating NIZK proof systems (waNIZKs), in which one can use a witness $w$ of a statement $x$ to identify whether a valid proof for $x$ is indeed generated using $w$. Such a new identification functionality enables more diverse applications, and it also puts new requirements on soundness that: (1) no adversary can generate a valid proof that will not be identified by any witness; (2) or forge a proof using some valid witness to frame others. To work...

Interactive oracle proofs (IOPs) (Ben-Sasson et al., TCC 2016; Reingold et al., SICOMP 2021) have emerged as a powerful model for proof systems combining IP and PCP. While IOPs are not any more powerful than PCPs from a complexity theory perspective, their potential to create succinct proofs and arguments has been demonstrated by many recent constructions achieving better parameters such as total proof length, alphabet size, and query complexity. In this work, we establish new results on the...

Modern SNARK designs typically follow a frontend-backend paradigm: The frontend compiles a user's program into some equivalent circuit representation, while the backend calls for a SNARK specifically made for proving circuit satisfiability. While these circuits are often defined over small fields, the backend prover always needs to lift the computation to much larger fields to ensure soundness. This gap introduces concrete overheads for ZK applications like zkRollups, where group-based...

This paper presents a provably secure, higher-order, and leakage-resilient (LR) rekeying scheme named LR Rekeying with Random oracle Repetition (LR4), along with a quantitative security evaluation methodology. Many existing LR primitives are based on a concept of leveled implementation, which still essentially require a leak-free sanctuary (i.e., differential power analysis (DPA)-resistant component(s)) for some parts. In addition, although several LR pseudorandom functions (PRFs) based on...

We establish new results on the Fiat-Shamir (FS) security of several protocols that are widely used in practice, and we provide general tools for establishing similar results for others. More precisely, we: (1) prove the FS security of the FRI and batched FRI protocols; (2) analyze a general class of protocols, which we call $\delta$-correlated, that use low-degree proximity testing as a subroutine (this includes many "Plonk-like" protocols (e.g., Plonky2 and Redshift), ethSTARK, RISC Zero,...

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...

We show that every language in NP has an Interactive Oracle Proof (IOP) with inverse polynomial soundness error and small query complexity. This achieves parameters that surpass all previously known PCPs and IOPs. Specifically, we construct an IOP with perfect completeness, soundness error $1/n$, round complexity $O(\log \log n)$, proof length $poly(n)$ over an alphabet of size $O(n)$, and query complexity $O(\log \log n)$. This is a step forward in the quest to establish the sliding-scale...

We introduce a new candidate post-quantum digital signature scheme from the regular syndrome decoding (RSD) assumption, an established variant of the syndrome decoding assumption which asserts that it is hard to find $w$-regular solutions to systems of linear equations over $\mathbb{F}_2$ (a vector is regular if it is a concatenation of $w$ unit vectors). Our signature is obtained by introducing and compiling a new 5-round zero-knowledge proof system constructed using the MPC-in-the-head...

In this paper we provide the implementation details and performance analysis of the lattice-based post-quantum commitment scheme introduced by Martínez and Morillo in their work titled «RLWE-Based Zero-Knowledge Proofs for Linear and Multiplicative Relations» together with the corresponding Zero-Knowledge Proofs of Knowledge (ZKPoK) of valid openings, linear and multiplicative relations among committed elements. We bridge the gap between the existing theoretical proposals and practical...

This paper introduces arithmetic sketching, an abstraction of a primitive that several previous works use to achieve lightweight, low-communication zero-knowledge verification of secret-shared vectors. An arithmetic sketching scheme for a language $\mathcal{L} \in \mathbb{F}^n$ consists of (1) a randomized linear function compressing a long input x to a short “sketch,” and (2) a small arithmetic circuit that accepts the sketch if and only if $x \in \mathcal{L}$, up to some small error. If...

In the threshold version of Paillier's encryption scheme, a set of parties collectively holds the secret decryption key through a secret sharing scheme. Whenever a ciphertext is to be decrypted, the parties send their decryption shares, which are then verified for correctness and combined into the plaintext. The scheme has been widely adopted in various applications, from secure voting to general purpose MPC protocols. However, among the handful existing proposals for a maliciously...

We present a new method for transforming zero-knowledge protocols in the designated verifier setting into public-coin protocols, which can be made non-interactive and publicly verifiable. Our transformation applies to a large class of ZK protocols based on oblivious transfer. In particular, we show that it can be applied to recent, fast protocols based on vector oblivious linear evaluation (VOLE), with a technique we call VOLE-in-the-head, upgrading these protocols to support public...

We build non-interactive zero-knowledge (NIZK) and ZAP arguments for all $\mathsf{NP}$ where soundness holds for infinitely-many security parameters, and against uniform adversaries, assuming the subexponential hardness of the Computational Diffie-Hellman (CDH) assumption. We additionally prove the existence of NIZK arguments with these same properties assuming the polynomial hardness of both CDH and the Learning Parity with Noise (LPN) assumption. In both cases, the CDH assumption does not...

Nova is an efficient recursive proof system built from an elegant folding scheme for (relaxed) R1CS statements. The original Nova paper (CRYPTO'22) presented Nova using a single elliptic curve group of order $p$. However, for improved efficiency, the implementation of Nova alters the scheme to use a 2-cycle of elliptic curves. This altered scheme is only described in the code and has not been proven secure. In this work, we point out a soundness vulnerability in the original implementation...

Polynomial commitments schemes are a powerful tool that enables one party to commit to a polynomial $p$ of degree $d$, and prove that the committed function evaluates to a certain value $z$ at a specified point $u$, i.e. $p(u) = z$, without revealing any additional information about the polynomial. Recently, polynomial commitments have been extensively used as a cryptographic building block to transform polynomial interactive oracle proofs (PIOPs) into efficient succinct arguments. In...

A classic result in the theory of interactive proofs shows that a special-sound $\Sigma$-protocol is automatically a 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 -- 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...

Compressed oracles (Zhandry, Crypto 2019) are a powerful technique to reason about quantum random oracles, enabling a sort of lazy sampling in the presence of superposition queries. A long-standing open question is whether a similar technique can also be used to reason about random (efficiently invertible) permutations. In this work, we make a step towards answering this question. We first define the compressed permutation oracle and illustrate its use. While the soundness of this...

Batch proofs are proof systems that convince a verifier that $x_1,\dots,x_t \in \mathcal{L}$, for some $\mathsf{NP}$ language $\mathcal{L}$, with communication that is much shorter than sending the $t$ witnesses. In the case of *statistical soundness* (where the cheating prover is unbounded but the honest prover is efficient given the witnesses), interactive batch proofs are known for $\mathsf{UP}$, the class of *unique-witness* $\mathsf{NP}$ languages. In the case of computational soundness...

A flurry of excitement amongst researchers and practitioners has produced modern proof systems built using novel technical ideas and seeing rapid deployment, especially in cryptocurrencies. Most of these modern proof systems use the Fiat-Shamir (F-S) transformation, a seminal method of removing interaction from a protocol with a public-coin verifier. Some prior work has shown that incorrectly applying F-S (i.e., using the so-called "weak" F-S transformation) can lead to breaks of classic...

This technical paper explores two solutions for arithmetization of computational integrity statements in STARKs, namely the algebraic intermediate representation, AIR, and is preprocessed variant, PAIR. The work then focuses on their soundness implications for Reed-Solomon proximity testing. It proceeds by presenting a comparative study of these methods, providing their theoretical foundations and deriving the degree bounds for low-degree proximity testing. The study shows that using PAIR...

Succinct Non-interactive Arguments of Knowledge (SNARKs) have seen interest and development from the cryptographic community over recent years, and there are now constructions with very small proof size designed to work well in practice. A SNARK protocol can only be widely accepted as secure, however, if a rigorous proof of its security properties has been vetted by the community. Even then, it is sometimes the case that these security proofs are flawed, and it is then necessary for further...

Fair Exchange (FE) protocols are a class of cryptographic protocol in which two parties, X and Y , exchange some secret data, where the ability of each party to receive secret data is contingent on having sent secret data of their own. When exchanging secret data without the support of FE protocols, whoever sends their secret first makes themselves vulnerable to the possibility that the other participant will cheat and won’t send their secret in return. It is widely believed that FE...

Increasing deployment of advanced zero-knowledge proof systems, especially zkSNARKs, has raised critical questions about their security against real-world attacks. Two classes of attacks of concern in practice are adaptive soundness attacks, where an attacker can prove false statements by choosing its public input after generating a proof, and malleability attacks, where an attacker can use a valid proof to create another valid proof it could not have created itself. Prior work has shown...