As cryptographic protocols continue to become more complex and specialized, their security proofs have grown more complex as well, making manual verification of their correctness more difficult. Formal verification via proof assistants has become a popular approach to solving this, by allowing researchers to write security proofs that can be verified correct by a computer. In this paper we present a new framework of this kind for verifying security proofs, taking a foundational approach...

The Fiat-Shamir (FS) transform is the standard approach to compiling interactive proofs into non-interactive ones. However, the fact that knowledge extraction typically requires rewinding limits its applicability without having to rely on further heuristic conjectures. A better alternative is a transform that guarantees straight-line knowledge extraction. Two such transforms were given by Pass (CRYPTO '03) and Fischlin (CRYPTO '05), respectively, with the latter giving the most practical...

Non-interactive zero-knowledge proofs (NIZK) are essential building blocks in threshold cryptosystems like multiparty signatures, distributed key generation, and verifiable secret sharing, allowing parties to prove correct behavior without revealing secrets. Furthermore, universally composable (UC) NIZKs enable seamless composition in the larger cryptosystems. A popular way to construct NIZKs is to compile interactive protocols using the Fiat-Shamir transform. Unfortunately, Fiat-Shamir...

The theory of compressed $\Sigma$-protocols [AC20, ACF21] provides a standardized framework for creating efficient $\Sigma$-protocols. This method involves two main phases: first, amortization, which combines multiple instances that satisfy a homomorphic relation into a single instance; and second, Bulletproofs compression [BBB+18], which minimizes communication overhead to a logarithmic scale during the verification of the combined instance. For high-degree polynomial (non-homomorphic)...

Cryptographic group actions have gained significant attention in recent years for their application on post-quantum Sigma protocols and digital signatures. In NIST's recent additional call for post-quantum signatures, three relevant proposals are based on group actions: LESS, MEDS, and ALTEQ. This work explores signature optimisations leveraging a group's factorisation. We show that if the group admits a factorisation as a semidirect product of subgroups, the group action can be restricted...

At CRYPTO '94, Cramer, Damgaard, and Schoenmakers introduced a general technique for constructing honest-verifier zero-knowledge proofs of partial knowledge (PPK), where a prover Alice wants to prove to a verifier Bob she knows $\tau$ witnesses for $\tau$ claims out of $k$ claims without revealing the indices of those $\tau$ claims. Their solution starts from a base honest-verifier zero-knowledge proof of knowledge $\Sigma$ and requires to run in parallel $k$ execution of the base...

Proofs of partial knowledge, first considered by Cramer, Damgård and Schoenmakers (CRYPTO'94) and De Santis et al. (FOCS'94), allow for proving the validity of $k$ out of $n$ different statements without revealing which ones those are. In this work, we present a new approach for transforming certain proofs system into new ones that allows for proving partial knowledge. The communication complexity of the resulting proof system only depends logarithmically on the total number of statements...

Sigma protocols are elegant cryptographic proofs that have become a cornerstone of modern cryptography. A notable example is Schnorr's protocol, a zero-knowledge proof-of-knowledge of a discrete logarithm. Despite extensive research, the security of Schnorr's protocol in the standard model is not fully understood. In this paper we study Kilian's protocol, an influential public-coin interactive protocol that, while not a sigma protocol, shares striking similarities with sigma protocols....

Blind signatures have garnered significant attention in recent years, with several efficient constructions in the random oracle model relying on well-understood assumptions. However, this progress does not apply to pairing-free cyclic groups: fully secure constructions over cyclic groups rely on pairings, remain inefficient, or depend on the algebraic group model or strong interactive assumptions. To address this gap, Chairattana-Apirom, Tessaro, and Zhu (CTZ, Crypto 2024) proposed a new...

Foreign field arithmetic often creates significant additional overheads in zero-knowledge proof circuits. Previous work has offloaded foreign arithmetic from proof circuits by using effective and often simple primitives such as Sigma protocols. While these successfully move the foreign field work outside of the circuit, the costs for the Sigma protocol’s verifier still remains high. In use cases where the verifier is constrained computationally this poses a major challenge. One such use case...

Fischlin's transform (CRYPTO 2005) is an alternative to the Fiat-Shamir transform that enables straight-line extraction when proving knowledge. In this work we focus on the problem of using the Fischlin transform to construct UC-secure zero-knowledge from Sigma protocols, since UC security -- that guarantees security under general concurrent composition -- requires straight-line (non-rewinding) simulators. We provide a slightly simplified transform that is much easier to understand, and...

This paper shows novel techniques to reduce the signature size of the code-based signature schemes MEDS and ALTEQ, by a large factor. For both schemes, the signature size is dominated by the responses for rounds with nonzero challenges, and we reduce the signature size by reducing the size of these responses. For MEDS, each of the responses consists of $m^2 + n^2$ field elements,while in our new protocol each response consists of only $2k$ ($k$ is usually chosen to be close to $m$ and $n$)...

In the common random string model, the parties executing a protocol have access to a uniformly random bit string. It is known that under standard intractability assumptions, we can realize any ideal functionality with universally composable (UC) security if a trusted common random string (CrS) setup is available. It was always a question of where this CrS should come from since the parties provably could not compute it themselves. Trust assumptions are required, so minimizing the level of...

We propose protocols in the Compressed Sigma Protocol framework that achieve a succinct verifier. Towards this, we construct a new inner product argument and cast it in the Compressed Sigma Protocol (CSP) framework as a protocol for opening a committed linear form, achieving logarithmic verification. We then use our succinct-verifier CSP to construct a zero-knowledge argument for circuit satisfiability (under the discrete logarithm assumption in bilinear groups) in the updatable...

Zero-Knowledge Proofs (ZKPs), a cryptographic tool known for decades, have gained significant attention in recent years due to advancements that have made them practically applicable in real-world scenarios. ZKPs can provide unique attributes, such as succinctness, non-interactivity, and the ability to prove knowledge without revealing the information itself, making them an attractive solution for a range of applications. This paper aims to critically analyze the applicability of ZKPs in...

Isogeny-based cryptography is an instance of post-quantum cryptography whose fundamental problem consists of finding an isogeny between two (isogenous) elliptic curves $E$ and $E'$. This problem is closely related to that of computing the endomorphism ring of an elliptic curve. Therefore, many isogeny-based protocols require the endomorphism ring of at least one of the curves involved to be unknown. In this paper, we explore the design of isogeny based protocols in a scenario where one...

$\Sigma$-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,...

Ring signature (RS) allows users to demonstrate to verifiers their membership within a specified group (ring) without disclosing their identities. Based on this, RS can be used as a privacy protection technology for users' identities in blockchain. However, there is currently a lack of RS schemes that are fully applicable to the blockchain applications: Firstly, users can only spend a UTXO once, and the current RS schemes are not yet perfect in a one-time manner. At the same time, the...

Isogeny computations in CSIDH (Asiacrypt 2018) are described using a commutative group G acting on the set of supersingular elliptic curves. The commutativity property gives CSIDH enough flexibility to allow the creation of many cryptographic primitives and protocols. Nevertheless, these operations are limited and more complex applications have not yet been proposed. When calling the composition of two group elements of G addition, our goal in this work is to explore exponentiation,...

$\Sigma$-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...

Blind signatures allow the issuing of signatures on messages chosen by the user so that they ensure $\mathit{blindness}$ of the message against the signer. Moreover, a malicious user cannot output $\ell+1$ signatures while only finishing $\ell$ signing sessions. This notion, called $\mathit{one}$-$\mathit{more}$ unforgeability, comes in two flavors supporting either $\mathit{sequential}$ or $\mathit{concurrent}$ sessions. In this paper, we investigate the security of a class of blind...

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...

This paper introduces Sigmabus, a technique designed to enhance the efficiency of zero-knowledge circuits by relocating computationally expensive operations outside the circuit. Specifically, Sigmabus focuses on moving elliptic curve group operations, typically proven with expensive non-native field arithmetic, to external computations. By leveraging Sigma protocols, elliptic curve group operations are proven outside the circuit, while additional constraints are applied to the circuit to...

Sigma protocols are one of the most common and efficient zero-knowledge proofs (ZKPs). Over the decades, a large number of Sigma protocols are proposed, yet few works pay attention to the common design principal. In this work, we propose a generic framework of Sigma protocols for algebraic statements from verifiable secret sharing (VSS) schemes. Our framework provides a general and unified approach to understanding Sigma protocols. It not only neatly explains the classic protocols such as...

We investigate the conditions under which straight-line extractable NIZKs in the random oracle model (i.e. without a CRS) permit multiparty realizations that are black-box in the same random oracle. We show that even in the semi-honest setting, any MPC protocol to compute such a NIZK cannot make black-box use of the random oracle or a hash function instantiating it if security against all-but-one corruptions is desired, unless the number of queries made by the verifier to the oracle grows...

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...

Lyubashevky's signatures are based on the Fiat-Shamir with Aborts paradigm. It transforms an interactive identification protocol that has a non-negligible probability of aborting into a signature by repeating executions until a loop iteration does not trigger an abort. Interaction is removed by replacing the challenge of the verifier by the evaluation of a hash function, modeled as a random oracle in the analysis. The access to the random oracle is classical (ROM), resp. quantum (QROM), if...

We provide a $\Sigma$-protocol for proving that two values committed in different groups are equal. We study our protocol in Lyubashevsky's framework "Fiat-Shamir with aborts" (Asiacrypt’09) and offer concrete parameters for instantiating it. We explain how to use it to compose SNARKs with $\Sigma$-protocols, create efficient proofs of solvency on cryptocurrencies, and join of attributes across different anonymous credentials.

In the setting of subversion, an adversary tampers with the machines of the honest parties thus leaking the honest parties' secrets through the protocol transcript. The work of Mironov and Stephens-Davidowitz (EUROCRYPT’15) introduced the idea of reverse firewalls (RF) to protect against tampering of honest parties' machines. All known constructions in the RF framework rely on the malleability of the underlying operations in order for the RF to rerandomize/sanitize the transcript. RFs are...

Building on recent disjunctive compilers for zero-knowledge (e.g. Goel et al. [EUROCRYPT'22]) we propose a new compiler that, when applied to sublinear-sized proofs, can result in sublinear-size disjunctive zero-knowledge with sublinear proving times (without meaningfully increasing proof sizes). Our key observation is that simulation in sublinear-size zero-knowledge proof systems can be much faster (both concretely and asymptotically) than the honest prover. We study applying our compiler...

In the light of NIST’s announced reopening of the call for digital signature proposals in 2023 due to lacking diversity, there is a strong need for constructions based on other established hardness assumptions. In this work we construct a new post-quantum secure digital signature scheme based on the $MinRank$ problem, a problem with a long history of applications in cryptanalysis that led to a strong belief in its hardness. Initially following a design by Courtois (Asiacrypt '01) based on...

We formalize security properties of zero-knowledge protocols and their proofs in EasyCrypt. Specifically, we focus on sigma-protocols (three-round protocols). Most importantly, we also cover properties whose security proofs require the use of rewinding; prior work has focused on properties that do not need this more advanced technique. On our way we give generic definitions of the main properties associated with sigma protocols, both in the computational and ...

Recently, there has been great interest towards constructing efficient zero-knowledge proofs for practical languages. In this work, we focus on proofs for threshold relations, in which the prover is required to prove knowledge of witnesses for $k$ out of $\ell$ statements. The main contribution of our work is an efficient and modular transformation that starting from a large class of $\Sigma$-protocols and a corresponding threshold relation $\mathcal{R}_\mathsf{k,\ell}$, provides an...

The goal of this paper is to improve the efficiency and applicability of straightline extraction techniques in the random oracle model. Straightline extraction in the random oracle model refers to the existence of an extractor, which given the random oracle queries made by a prover $P^*(x)$ on some theorem $x$, is able to produce a witness $w$ for $x$ with roughly the same probability that $P^*$ produces a verifying proof. This notion applies to both zero-knowledge protocols and verifiable...

Numerous cryptographic applications require efficient non-interactive zero-knowledge proofs of knowledge (NIZKPoK) as a building block. Typically they rely on the Fiat-Shamir heuristic to do so, as security in the random-oracle model is considered good enough in practice. However, there is a troubling disconnect between the stand-alone security of such a protocol and its security as part of a larger, more complex system where several protocols may be running at the same time. Provable...

Commit-and-open Sigma-protocols are a popular class of protocols for constructing non-interactive zero-knowledge arguments and digital-signature schemes via the Fiat-Shamir transformation. Instantiated with hash-based commitments, the resulting non-interactive schemes enjoy tight online-extractability in the random oracle model. Online extractability improves the tightness of security proofs for the resulting digital-signature schemes by avoiding lossy rewinding or forking-lemma based...

Compressed $\Sigma$-Protocol Theory (CRYPTO 2020) presents an ``alternative'' to Bulletproofs that achieves the same communication complexity while adhering more elegantly to existing $\Sigma$-protocol theory, which enables their techniques to be directly applicable to other widely used settings in the context of ``plug \& play'' algorithmics. Unfortunately, their techniques are restricted to arithmetic circuits over \emph{prime} fields, which rules out the possibility of using more...

The celebrated Fiat-Shamir transformation turns any public-coin interactive proof into a non-interactive one, which inherits the main security properties (in the random oracle model) of the interactive version. While originally considered in the context of 3-move public-coin interactive proofs, i.e., so-called $\Sigma$-protocols, it is now applied to multi-round protocols as well. Unfortunately, the security loss for a $(2\mu + 1)$-move protocol is, in general, $Q^\mu$, where $Q$ is the...

In many occasions, the knowledge error $\kappa$ of an interactive proof is not small enough, and thus needs to be reduced. This can be done generically by repeating the interactive proof in parallel. While there have been many works studying the effect of parallel repetition on the {\em soundness error} of interactive proofs and arguments, the effect of parallel repetition on the {\em knowledge error} has largely remained unstudied. Only recently it was shown that the $t$-fold parallel...

This article presents Lelantus-CLA, an adaptation of Lelantus for use with the Mimblewimble protocol and confidential assets. Whereas Mimblewimble achieves a limited amount of privacy by merging transactions that occur in the same block, Lelantus uses a logarithmic-size proof of membership to effectively enable merging across different blocks. At a high level, this allows value to be added to a common pool and then spent privately, but only once. We explain how to adapt this construction to...

The Schnorr identification and signature schemes have been amongst the most influential cryptographic protocols of the past three decades. Unfortunately, although the best-known attacks on these two schemes are via discrete-logarithm computation, the known approaches for basing their security on the hardness of the discrete logarithm problem encounter the ``square-root barrier''. In particular, in any group of order $p$ where Shoup's generic hardness result for the discrete logarithm problem...

We advance the state-of-the art for zero-knowledge commit-and-prove SNARKs (CP-SNARKs). CP-SNARKs are an important class of SNARKs which, using commitments as ``glue'', allow to efficiently combine proof systems---e.g., general-purpose SNARKs (an efficient way to prove statements about circuits) and $\Sigma$-protocols (an efficient way to prove statements about group operations). Thus, CP-SNARKs allow to efficiently provide zero-knowledge proofs for composite statements such as $h=H(g^{x})$...

Shuffling is one of the most important techniques for privacy-preserving protocols. Its applications are manifold, including, for example, e-voting, anonymous broadcast, or privacy-preserving machine-learning. For many applications, such as secure e-voting, it is crucial that the correctness of the shuffling operation be (publicly) verifiable. To this end, numerous proofs of shuffle have been proposed in the literature. Several of these proofs are actually employed in the real world. In...

Cryptocurrencies have received a lot of research attention in recent years following the release of the first cryptocurrency Bitcoin. With the rise in cryptocurrency transactions, the need for smart contracts has also increased. Smart contracts, in a nutshell, are digitally executed contracts wherein some parties execute a common goal. The main problem with most of the current smart contracts is that there is no privacy for a party's input to the contract from either the blockchain or the...

Proofs of partial knowledge demonstrate the possession of certain subsets of witnesses for a given collection of statements $x_1,\dots,x_n$. Cramer, Damgård, and Schoenmakers (CDS), built proofs of partial knowledge, given ``atomic'' protocols for individual statements $x_i$, by having the prover randomly secret share the verifier's challenge and using the shares as challenges for the atomic protocols. This simple and highly-influential transformation has been used in numerous applications,...

Given two ciphertexts generated with a public-key encryption scheme, the problem of plaintext equality consists in determining whether the ciphertexts hold the same value. Similarly, the problem of plaintext inequality consists in deciding whether they hold a different value. Previous work has focused on building new schemes or extending existing ones to include support for plaintext equality/inequality. We propose generic and simple zero-knowledge proofs for both problems, which can be...

Zero-Knowledge (ZK) Proofs for disjunctive statements have been a focus of a long line of research. Classical results such as Cramer {\em et al.} [CRYPTO'94] and Abe {\em et al.} [AC'02] design generic compilers that transform certain classes of ZK proofs into ZK proofs for disjunctive statements. However, communication complexity of the resulting protocols in these results ends up being proportional to the complexity of proving all clauses in the disjunction. More recently, Heath {\em et...

State-separating proofs (SSP) is a recent methodology for structuring game-based cryptographic proofs in a modular way, by using algebraic laws to exploit the modular structure of composed protocols. While promising, this methodology was previously not fully formalized and came with little tool support. We address this by introducing SSProve, the first general verification framework for machine-checked state-separating proofs. SSProve combines high-level modular proofs about composed...

We show a lattice-based solution for commit-and-prove transparent circuit zero-knowledge (ZK) with polylog-communication, the first not depending on PCPs. We start from compressed $\Sigma$-protocol theory (CRYPTO 2020), which is built around basic $\Sigma$-protocols for opening an arbitrary linear form on a long secret vector that is compactly committed to. These protocols are first compressed using a recursive ``folding-technique'' adapted from Bulletproofs, at the expense of logarithmic...

We show the following generic result. Whenever a quantum query algorithm in the quantum random-oracle model outputs a classical value $t$ that is promised to be in some tight relation with $H(x)$ for some $x$, then $x$ can be efficiently extracted with almost certainty. The extraction is by means of a suitable simulation of the random oracle and works *online*, meaning that it is *straightline*, i.e., without rewinding, and *on-the-fly*, i.e., during the protocol execution and without...

Cramer, Damgård, and Schoenmakers (CDS) built a proof system to demonstrate the possession of subsets of witnesses for a given collection of statements that belong to a prescribed access structure P by composing so-called sigma-protocols for each atomic statement. Their verifier complexity is linear in the size of the monotone span program representation of P. We propose an alternative method for combining sigma-protocols into a single non-interactive system for a compound statement in the...

ElectionGuard is an open source set of software components and specifications from Microsoft designed to allow the modification of a number of different e-voting protocols and products to produce public evidence (transcripts) which anyone can verify. The software uses ElGamal, homomorphic tallying and sigma protocols to enable public scrutiny without adversely affecting privacy. Some components have been formally verified (machine-checked) to be free of certain software bugs but there was no...

Lai et al. (CCS 2019) have shown how Bulletproof’s arithmetic circuit zero-knowledge protocol (Bootle et al., EUROCRYPT 2016 and Bünz et al., S&P 2018) can be generalized to work for bilinear group arithmetic circuits directly, i.e., without requiring these circuits to be translated into arithmetic circuits. In a nutshell, a bilinear group arithmetic circuit is a standard arithmetic circuit augmented with special gates capturing group exponentiations or pairings. Such circuits are highly...

The standard model security of the Fiat-Shamir transform has been an active research area for many years. In breakthrough results, Canetti et al. (STOC'19) and Peikert-Shiehian (Crypto'19) showed that, under the Learning-With-Errors (LWE) assumption, it provides soundness by applying correlation-intractable (CI) hash functions to so-called trapdoor $\Sigma$-protocols. In order to be compatible with CI hash functions based on standard LWE assumptions with polynomial approximation factors,...

Non-interactive zero knowledge (NIZK) enables proving the validity of NP statement without leaking anything else. We study multi-instance NIZKs in the common reference string (CRS) model, against an adversary that adaptively corrupts parties and chooses statements to be proven. We construct the first such $\textit{triply adaptive}$ NIZK that provides full adaptive soundness, as well as adaptive zero-knowledge, assuming either LWE or else LPN and DDH (previous constructions rely on...

We study adaptive security of delayed-input Sigma protocols and non-interactive zero-knowledge (NIZK) proof systems in the common reference string (CRS) model. Our contributions are threefold: - We exhibit a generic compiler taking any delayed-input Sigma protocol and returning a delayed-input Sigma protocol satisfying adaptive-input special honest-verifier zero-knowledge (SHVZK). In case the initial Sigma protocol also satisfies adaptive-input special soundness, our compiler preserves this...

We present a privacy-preserving protocol to anonymously collect information about a social graph. The typical application of our protocol is Bluetooth-enabled ``contact-tracing apps'' which record information about proximity between users to infer the risk of propagation of COVID-19 among them. The main contribution of this work is to enable a central server to construct an anonymous graph of interactions between users. This graph gives the central authority insight on the propagation of...

We present Zerojoin, a privacy-enhancing protocol for UTXO blockchains. Like Zerocoin, our protocol uses zero-knowledge proofs and a pool of participants. However, unlike Zerocoin, our pool size is not monotonically increasing. Thus, our protocol overcomes the major drawback of Zerocoin. Our approach can also be considered a non-interactive variant of CoinJoin, where the interaction is replaced by a public transaction on the blockchain. The security of Zerojoin relies on the...

We revisit recent works by Don, Fehr, Majenz and Schaffner and by Liu and Zhandry on the security of the Fiat-Shamir transformation of $\Sigma$-protocols in the quantum random oracle model (QROM). Two natural questions that arise in this context are: (1) whether the results extend to the Fiat-Shamir transformation of *multi-round* interactive proofs, and (2) whether Don et al.'s $O(q^2)$ loss in security is optimal. Firstly, we answer question (1) in the affirmative. As a byproduct of...

We present new non-interactive zero-knowledge argument systems (NIZK), based on standard assumptions that were previously not known to imply it. In particular, we rely on the hardness of both the learning parity with noise (LPN) assumption, and the existence of trapdoor hash functions (TDH, defined by Döttling et al., Crypto 2019). Such TDH can be based on a number of standard assumptions, including DDH, QR, DCR, and LWE. We revisit the correlation intractability (CI) framework for...

We study the problem of achieving statistical privacy in interactive proof systems and oblivious transfer -- two of the most well studied two-party protocols -- when limited rounds of interaction are available. Statistical Zaps: We give the first construction of statistical Zaps, namely, two-round statistical witness-indistinguishable (WI) protocols with a public-coin verifier. Our construction achieves computational soundness based on the quasi-polynomial hardness of learning with...

We study interactive proof systems (IPSes) in a strong adversarial setting where the machines of *honest parties* might be corrupted and under control of the adversary. Our aim is to answer the following, seemingly paradoxical, questions: - Can Peggy convince Vic of the veracity of an NP statement, without leaking any information about the witness even in case Vic is malicious and Peggy does not trust her computer? - Can we avoid that Peggy fools Vic into accepting false statements, even if...

Sigma-Protocols provide a well-understood basis for secure algorithmics. Recently, Bulletproofs (Bootle et al., EUROCRYPT 2016, and Bünz et al., S&P 2018) have been proposed as a drop-in replacement in case of zero-knowledge (ZK) for arithmetic circuits, achieving logarithmic communication instead of linear. Its pivot is an ingenious, logarithmic-size proof of knowledge BP for certain quadratic relations. However, reducing ZK for general relations to it forces a somewhat cumbersome ...

Ring-SIS based $\Sigma$-protocols require the construction of a challenge set $\mathcal{C}$ in some ring $R$, usually an order in a number field $L$. These protocols impose various requirements on the subset $\mathcal{C}$, and constructing a good or even optimal challenge set is a non-trivial task that involves making various trade-offs. </p> In particular, the set $\mathcal{C}$ should be 'large', elements in $\mathcal{C}$ should be 'small', differences of distinct elements in $\mathcal{C}$...

Machine-checked proofs of security are important to increase the rigour of provable security. In this work we present a formalised theory of two fundamental two party cryptographic primitives: $\Sigma$-protocols and Commitment Schemes. $\Sigma$-protocols allow a prover to convince a verifier that they possess some knowledge without leaking information about the knowledge. Commitment schemes allow a committer to commit to a message and keep it secret until revealing it at a later time. We...

This work presents sigma protocols to prove knowledge of: -a solution to a system of quadratic polynomials, -a solution to an instance of the Permuted Kernel Problem and -a witness for a variety of lattice statements (including SIS). Our sigma protocols have soundness error 1/q', where q' is any number bounded by the size of the underlying finite field. This is much better than existing proofs, which have soundness error 2/3 or (q'+1)/2q'. The prover and verifier time of our proofs are...

Modern cryptocurrencies such as Bitcoin and Ethereum achieve decentralization by replacing a trusted center with a distributed and append-only ledger (known as blockchain). However, removing this trusted center comes at significant cost of privacy due to the public nature of blockchain. Many existing cryptocurrencies fail to provide transaction anonymity and confidentiality, meaning that addresses of sender, receiver and transfer amount are publicly accessible. As the privacy concerns...

Blockchain-based smart contract platforms like Ethereum have become quite popular as a way to remove trust and add transparency to distributed applications. While different types of important applications can be easily built on such platforms, there does not seem to be an easy way to add a meaningful level of privacy to them. In this paper, we propose Zether, a fully-decentralized, confidential payment mechanism that is compatible with Ethereum and other smart contract platforms. We take an...

A proof system is publicly verifiable, if anyone, by looking at the transcript of the proof, can be convinced that the corresponding theorem is true. Public verifiability is important in many applications since it allows to compute a proof only once while convincing an unlimited number of verifiers. Popular interactive proof systems (e.g., $\Sigma$-protocols) protect the witness through various properties (e.g., witness indistinguishability (WI) and zero knowledge (ZK)) but typically...

We construct non-interactive zero-knowledge (NIZK) arguments for $\mathsf{NP}$ from any circular-secure fully homomorphic encryption (FHE) scheme. In particular, we obtain such NIZKs under a circular-secure variant of the learning with errors (LWE) problem while only assuming a standard (poly/negligible) level of security. Our construction can be modified to obtain NIZKs which are either: (1) statistically zero-knowledge arguments in the common random string model or (2) statistically sound...

We propose a generic construction of a $\Sigma$-protocol of commit-and-prove type, which is an AND-composition of $\Sigma$-protocols on statements that include a common commitment. Our protocol enables a prover to convince a verifier that the prover knows a bundle of witnesses that have a common component which we call a base witness point. When the component $\Sigma$-protocols are of witness-indistinguishable argument systems, our $\Sigma$-protocol is also a witness-indistinguishable...

The two most common ways to design non-interactive zero-knowledge (NIZK) proofs are based on Sigma protocols and QAP-based SNARKs. The former is highly efficient for proving algebraic statements while the latter is superior for arithmetic representations. Motivated by applications such as privacy-preserving credentials and privacy-preserving audits in cryptocurrencies, we study the design of NIZKs for composite statements that compose algebraic and arithmetic statements in arbitrary ways....

Higher-level cryptographic privacy-enhancing protocols such as anonymous credentials, voting schemes, and e-cash are often constructed by suitably combining signature, commitment, and encryption schemes with zero-knowledge proofs. Indeed, a large body of protocols have been constructed in that manner from Camenisch-Lysyanskaya signatures and generalized Schnorr proofs. In this paper, we build a similar framework for lattice-based schemes by presenting a signature and commitment scheme that...

Given a well-chosen additively homomorphic cryptosystem and a $\Sigma$ protocol with a linear answer, Damgård, Fazio, and Nicolosi proposed a non-interactive designated-verifier zero knowledge argument in the registered public key model that is sound under non-standard complexity-leveraging assumptions. In 2015, Chaidos and Groth showed how to achieve the weaker yet reasonable culpable soundness notion under standard assumptions but only if the plaintext space order is prime. It makes use of...

The Fiat-Shamir construction (Crypto 1986) is an efficient transformation in the random oracle model for creating non-interactive proof systems and signatures from sigma-protocols. In classical cryptography, Fiat-Shamir is a zero-knowledge proof of knowledge assuming that the underlying sigma-protocol has the zero-knowledge and special soundness properties. Unfortunately, Ambainis, Rosmanis, and Unruh (FOCS 2014) ruled out non-relativizing proofs under those conditions in the quantum...

Practical anonymous credential systems are generally built around sigma-protocol ZK proofs. This requires that credentials be based on specially formed signatures. Here we ask whether we can instead use a standard (say, RSA, or (EC)DSA) signature that includes formatting and hashing messages, as a credential, and still provide privacy. Existing techniques do not provide efficient solutions for proving knowledge of such a signature: On the one hand, ZK proofs based on garbled circuits...

One of the most promising innovations offered by the cryptographic currencies (like Bitcoin) are the so-called \emph{smart contracts}, which can be viewed as financial agreements between mutually distrusting participants. Their execution is enforced by the mechanics of the currency, and typically has monetary consequences for the parties. The rules of these contracts are written in the form of so-called ``scripts'', which are pieces of code in some ``scripting language''. Although smart...

Proofs of partial knowledge allow a prover to prove knowledge of witnesses for k out of n instances of NP languages. Cramer, Schoenmakers and Damg\aa rd [CDS94] provided an efficient construction of a 3-round public-coin witness-indistinguishable (k, n)-proof of partial knowledge for any NP language, by cleverly combining n executions of Sigma-protocols for that language. This transform assumes that all n instances are fully specified before the proof starts, and thus directly rules out the...

In this paper we describe ZKBoo, a proposal for practically efficient zero-knowledge arguments especially tailored for Boolean circuits and report on a proof-of-concept implementation. As an highlight, we can generate (resp. verify) a non-interactive proof for the SHA-1 circuit in approximately 13ms (resp. 5ms), with a proof size of 444KB. Our techniques are based on the “MPC-in-the-head” approach to zero-knowledge of Ishai et al. (IKOS), which has been successfully used to achieve...

We explore a new man-in-the-middle adversarial model for multi-prover interactive proofs (MIPs), and construct round-optimal, unconditionally secure, non-malleable MIPs. We compile from a large sub-class of Sigma protocols to a non-malleable MIP, avoiding the use of expensive NP-reductions to Graph Hamiltonicity or other NP-complete problems. Our compiler makes novel use of non-malleable codes - in particular, we rely on many-many non-malleable codes constructed recently by Chattopadhyay,...

In [CDS94] Cramer, Damgård and Schoenmakers (CDS) devise an OR-composition technique for Sigma-protocols that allows to construct highly-efficient proofs for compound statements. Since then, such technique has found countless applications as building block for designing efficient protocols. Unfortunately, the CDS OR-composition technique works only if both statements are fixed before the proof starts. This limitation restricts its usability in those protocols where the theorems to be...

The Fiat-Shamir (FS) transform uses a hash function to generate, without any further overhead, non-interactive zero-knowledge (NIZK) argument systems from constant-round public-coin honest-verifier zero-knowledge (public-coin HVZK) proof systems. In the proof of zero knowledge, the hash function is modeled as a programmable random oracle (PRO). In TCC 2015, Lindell embarked on the challenging task of obtaining a similar transform with improved heuristic security. Lindell showed that, for...

Abstract. The concept of adaptive security for proofs of knowledge was recently studied by Bernhard et al. They formalised adaptive security in the ROM and showed that the non-interactive version of the Schnorr protocol obtained using the Fiat-Shamir transformation is not adaptively secure unless the one-more discrete logarithm problem is easy. Their only construction for adaptively secure protocols used the Fischlin transformation [3] which yields protocols with straight-line extractors. In...

The well-known Signed ElGamal scheme consists of ElGamal encryption with a non-interactive Schnorr proof of knowledge. While this scheme should be intuitively secure against chosen-ciphertext attacks in the random oracle model, its security has not yet been proven nor disproven so far, without relying on further non-standard assumptions like the generic group model. Currently, the best known positive result is that Signed ElGamal is non-malleable under chosen-plaintext attacks. In this paper...

We formalise the notion of adaptive proofs of knowledge in the random oracle model, where the extractor has to recover witnesses for multiple, possibly adaptively chosen statements and proofs. We also discuss extensions to simulation soundness, as typically required for the ``encrypt-then-prove'' construction of strongly secure encryption from IND-CPA schemes. Utilizing our model we show three results: (1) Simulation-sound adaptive proofs exist. (2) The ``encrypt-then-prove'' construction...

We propose a general technique that allows improving the complexity of zero-knowledge protocols for a large class of problems where previously the best known solution was a simple cut-and-choose style protocol, i.e., where the size of a proof for problem instance $x$ and error probability $2^{-n}$ was $O(|x| n)$ bits. By using our technique to prove $n$ instances simultaneously, we can bring down the proof size per instance to $O(|x| + n)$ bits for the same error probability while using no...

In a sanitizable signature scheme the signer allows a designated third party, called the sanitizer, to modify certain parts of the message and adapt the signature accordingly. Ateniese et al. (ESORICS 2005) introduced this primitive and proposed five security properties which were formalized by Brzuska et al.~(PKC 2009). Subsequently, Brzuska et al. (PKC 2010) suggested an additional security notion, called unlinkability which says that one cannot link sanitized message-signature pairs of...

We design an efficient commitment scheme, and companion zero-knowledge proofs of knowledge, based on the learning with errors over rings (RLWE) problem. In particular, for rings in which almost all elements have inverses, we construct a perfectly binding commitment scheme whose hiding property relies on the RLWE assumption. Our scheme maps elements from the ring (or equivalently, n elements from F_q) to a small constant number of ring elements. We then construct Sigma-protocols for...

In this short paper, we present a Fiat-Shamir type transform that takes any Sigma protocol for a relation $R$ and outputs a non-interactive zero-knowledge proof (not of knowledge) for the associated language $L_R$, in the common reference string model. As in the Fiat-Shamir transform, we use a hash function $H$. However, zero-knowledge is achieved under standard assumptions in the common reference string model (without any random oracle), and soundness is achieved in the...

We present a construction for non-interactive zero-knowledge proofs of knowledge in the random oracle model from general sigma-protocols. Our construction is secure against quantum adversaries. Prior constructions (by Fiat-Shamir and by Fischlin) are only known to be secure against classical adversaries, and Ambainis, Rosmanis, Unruh (FOCS 2014) gave evidence that those constructions might not be secure against quantum adversaries in general. To prove security of our constructions, we...

In their seminal work on non-malleable cryptography, Dolev, Dwork and Naor, showed how to construct a non-malleable commitment with logarithmically-many "rounds"/"slots", the idea being that any adversary may successfully maul in some slots but would fail in at least one. Since then new ideas have been introduced, ultimately resulting in constant-round protocols based on any one-way function. Yet, in spite of this remarkable progress, each of the known constructions of non-malleable...

Quantum zero-knowledge proofs and quantum proofs of knowledge are inherently difficult to analyze because their security analysis uses rewinding. Certain cases of quantum rewinding are handled by the results by Watrous (SIAM J Comput, 2009) and Unruh (Eurocrypt 2012), yet in general the problem remains elusive. We show that this is not only due to a lack of proof techniques: relative to an oracle, we show that classically secure proofs and proofs of knowledge are insecure in the quantum...

Related key attacks (RKAs) are powerful cryptanalytic attacks where an adversary can change the secret key and observe the effect of such changes at the output. The state of the art in RKA security protects against an a-priori unbounded number of certain algebraic induced key relations, e.g., affine functions or polynomials of bounded degree. In this work, we show that it is possible to go beyond the algebraic barrier and achieve security against arbitrary key relations, by restricting the...

All recent implementations of two-party secure computation protocols require specific complexity assumptions for their correctness and/or efficiency (e.g., DDH, homomorphic encryption, Sigma protocols for specific languages). We propose and implement a Yao-based protocol for secure two-party computation against malicious adver- saries that enjoys the following benefits: (1) it assumes the minimal hardness assumption, that is, oblivious transfers; (2) it has constant round complexity; (3) its...

We initiate a general study of schemes resilient to both tampering and leakage attacks. Tampering attacks are powerful cryptanalytic attacks where an adversary can change the secret state and observes the effect of such changes at the output. Our contributions are outlined below: (1) We propose a general construction showing that any cryptographic primitive where the secret key can be chosen as a uniformly random string can be made secure against bounded tampering and leakage. This holds ...

In this paper we introduce the notion of {\em Algebraic (Trapdoor) One Way Functions}, which, roughly speaking, captures and formalizes many of the properties of number-theoretic one-way functions. Informally, a (trapdoor) one way function $F: X \to Y$ is said to be algebraic if $X$ and $Y$ are (finite) abelian cyclic groups, the function is {\em homomorphic} i.e. $F(x)\cdot F(y) = F(x \cdot y)$, and is {\em ring-homomorphic}, meaning that it is possible to compute linear operations ``in the...

In this paper, we investigate on threshold proofs, a framework for distributing the prover’s side of interactive proofs of knowledge over multiple parties. Interactive proofs of knowledge (PoK) are widely used primitives of cryptographic protocols, including important user-centric protocols, such as identification schemes, electronic cash (e-cash), and anonymous credentials. We present a security model for threshold proofs of knowledge and develop threshold versions of well-known primitives...

Digital signature is one of the basic primitives in cryptography. A common paradigm of obtaining signatures, known as the Fiat-Shamir (FS) paradigm, is to collapse any &#931;-protocol (which is 3-round public-coin honest-verifier zero-knowledge) into a non-interactive scheme with hash functions that are modeled to be random oracles (RO). The Digital Signature Standard (DSS) and Schnorr’s signature schemes are two salient examples following the FS-paradigm. In this work, we present a...

We present a method to compile Yao's two-player garbled circuit protocol into one that is secure against malicious adversaries that relies on witness indistinguishability. Our approach can enjoy lower communication and computation overhead than methods based on cut-and-choose and lower overhead than methods based on zero-knowledge proofs (or sigma-protocols). To do so, we develop and analyze new solutions to issues arising with this transformation: - How to guarantee the generator’s input...

Zero-knowledge proofs of knowledge (ZK-PoK) for discrete logarithms and related problems are indispensable for practical cryptographic protocols. At \emph{Eurocrypt 2009}, Camenisch, Kiayias, and Yung provided a specification language (the \emph{CKY-language}) for such protocols, which allows one to modularly design and analyze cryptographic protocols: protocol designers just need to specify the statement they want to prove in zero-knowledge and are ensured that an efficient proof protocol...