Fiat-Shamir for highly sound protocols is instantiable

The Fiat–Shamir (FS) transformation (Fiat and Shamir, Crypto '86) is a popular paradigm for constructing very efficient non-interactive zero-knowledge (NIZK) arguments and signature schemes from a hash function and any three-move interactive protocol satisfying certain properties. Despite its wide-spread applicability both in theory and in practice, the known positive results for proving security of the FS paradigm are in the random oracle model only, i.e., they assume that the hash function is modeled as an external random function accessible to all parties. On the other hand, a sequence of negative results shows that for certain classes of interactive protocols, the FS transform cannot be instantiated in the standard model. We initiate the study of complementary positive results, namely, studying classes of interactive protocols where the FS transform does have standard-model instantiations. In particular, we show that for a class of “highly sound” protocols that we define, instantiating the FS transform via a q-wise independent hash function yields NIZK arguments and secure signature schemes. In the case of NIZK, we obtain a weaker “q-bounded” zero-knowledge flavor where the simulator works for all adversaries asking an a-priori bounded number of queries q; in the case of signatures, we obtain the weaker notion of random-message unforgeability against q-bounded random message attacks. Our main idea is that when the protocol is highly sound, then instead of using random-oracle programming, one can use complexity leveraging. The question is whether such highly sound protocols exist and if so, which protocols lie in this class. We answer this question in the affirmative in the common reference string (CRS) model and under strong assumptions. Namely, assuming indistinguishability obfuscation and puncturable pseudorandom functions we construct a compiler that transforms any 3-move interactive protocol with instance-independent commitments and simulators (a property satisfied by the Lapidot–Shamir protocol, Crypto '90) into a compiled protocol in the CRS model that is highly sound. We also present a second compiler, in order to be able to start from a larger class of protocols, which only requires instance-independent commitments (a property for example satisfied by the classical protocol for quadratic residuosity due to Blum, Crypto '81). For the second compiler we require dual-mode commitments. We hope that our work inspires more research on classes of (efficient) 3-move protocols where Fiat–Shamir is (efficiently) instantiable

Does Fiat-Shamir Require a Cryptographic Hash Function?

The Fiat-Shamir transform is a general method for reducing interaction in public-coin protocols by replacing the random verifier messages with deterministic hashes of the protocol transcript. The soundness of this transformation is usually heuristic and lacks a formal security proof. Instead, to argue security, one can rely on the random oracle methodology, which informally states that whenever a random oracle soundly instantiates Fiat-Shamir, a hash function that is ``sufficiently unstructured\u27\u27 (such as fixed-length SHA-2) should suffice. Finally, for some special interactive protocols, it is known how to (1) isolate a concrete security property of a hash function that suffices to instantiate Fiat-Shamir and (2) build a hash function satisfying this property under a cryptographic assumption such as Learning with Errors. In this work, we abandon this methodology and ask whether Fiat-Shamir truly requires a cryptographic hash function. Perhaps surprisingly, we show that in two of its most common applications --- building signature schemes as well as (general-purpose) non-interactive zero-knowledge arguments --- there are sound Fiat-Shamir instantiations using extremely simple and non-cryptographic hash functions such as sum-mod-p or bit decomposition. In some cases, we make idealized assumptions about the interactive protocol (i.e., we invoke the generic group model), while in others, we argue soundness in the plain model. At a high level, the security of each resulting non-interactive protocol derives from hard problems already implicit in the original interactive protocol. On the other hand, we also identify important cases in which a cryptographic hash function is provably necessary to instantiate Fiat-Shamir. We hope that this work leads to an improved understanding of the precise role of the hash function in the Fiat-Shamir transformation

Efficient NIZKs from LWE via Polynomial Reconstruction and ``MPC in the Head

All existing works building non-interactive zero-knowledge (NIZK) arguments for $\mathsf{NP}$ from the Learning With Errors (LWE) assumption have studied instantiating the Fiat-Shamir paradigm on a parallel repetition of an underlying honest-verifier zero knowledge (HVZK) $\Sigma$ protocol, via an appropriately built correlation-intractable (CI) hash function from LWE. This technique has inherent efficiency losses that arise from parallel repetition. In this work, we show how to make use of the more efficient ``MPC in the Head\u27\u27 technique for building an underlying honest-verifier protocol upon which to apply the Fiat-Shamir paradigm. To make this possible, we provide a new and more efficient construction of CI hash functions from LWE, using efficient algorithms for polynomial reconstruction as the main technical tool. We stress that our work provides a new and more efficient ``base construction\u27\u27 for building LWE-based NIZK arguments for $\mathsf{NP}$. Our protocol can be the building block around which other efficiency-focused bootstrapping techniques can be applied, such as the bootstrapping technique of Gentry et al. (Journal of Cryptology 2015)

Optimally Sound Sigma Protocols Under DCRA

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 $\Sigma$ protocols that satisfy what we call the \emph{optimal culpable soundness}. Unfortunately, most of the known additively homomorphic cryptosystems (like the Paillier Elgamal cryptosystem that is secure under the standard Decisional Composite Residuosity Assumption) have composite-order plaintext space. We construct optimally culpable sound $\Sigma$ protocols and thus culpably sound non-interactive designated-verifier zero knowledge protocols for NP under standard assumptions given that the least prime divisor of the plaintext space order is large

Non-Interactive Zero Knowledge Proofs in the Random Oracle Model

The Fiat-Shamir (FS) transform is a well known and widely used technique to convert any constant-round public-coin honest-verifier zero-knowledge (HVZK) proof or argument system $CIPC=(Prov,Ver)$ in a non-interactive zero-knowledge (NIZK) argument system $NIZK=(NIZK.Prove, NIZK.Verify)$. The FS transform is secure in the random oracle (RO) model and is extremely efficient: it adds an evaluation of the RO for every message played by $Ver$. While a major effort has been done to attack the soundness of the transform when the RO is instantiated with a ``secure\u27\u27 hash function, here we focus on a different limitation of the FS transform that exists even when there is a secure instantiation of the random oracle: the soundness of $NIZK$ holds against polynomial-time adversarial provers only. Therefore even when $CIPC$ is a proof system, $NIZK$ is only an argument system. In this paper we propose a new transform from 3-round public-coin HVZK proof systems for several practical relations to NIZK proof systems in the RO model. Our transform outperforms the FS transform protecting the honest verifier from unbounded adversarial provers with no restriction on the number of RO queries. The protocols our transform can be applied to are the ones for proving membership to the range of a one-way group homomorphism as defined by [Maurer - Design, Codes and Cryptography 2015] except that we additionally require the function to be endowed with a trapdoor and other natural properties. For instance, we obtain new efficient instantiations of NIZK proofs for relations related to quadratic residuosity and the RSA function. As a byproduct, with our transform we obtain essentially for free the first efficient non-interactive zap (i.e., 1-round non-interactive witness indistinguishable proof system) for several practical languages in the non-programmable RO model and in an ideal-PUF model. Our approach to NIZK proofs can be seen as an abstraction of the celebrated work of [Feige, Lapidot and Shamir - FOCS 1990]

Two-round nn-out-of-nn and Multi-Signatures and Trapdoor Commitment from Lattices

Although they have been studied for a long time, distributed signature protocols have garnered renewed interest in recent years in view of novel applications to topics like blockchains. Most recent works have focused on distributed versions of ECDSA or variants of Schnorr signatures, however, and in particular, little attention has been given to constructions based on post-quantum secure assumptions like the hardness of lattice problems. A few lattice-based threshold signature and multi-signature schemes have been proposed in the literature, but they either rely on hash-and-sign lattice signatures (which tend to be comparatively inefficient), use expensive generic transformations, or only come with incomplete security proofs. In this paper, we construct several lattice-based distributed signing protocols with low round complexity following the Fiat–Shamir with Aborts (FSwA) paradigm of Lyubashevsky (Asiacrypt 2009). Our protocols can be seen as distributed variants of the fast Dilithium-G signature scheme and the full security proof can be made assuming the hardness of module SIS and LWE problems. A key step to achieving security (unexplained in some earlier papers) is to prevent the leakage that can occur when parties abort after their first message—which can inevitably happen in the Fiat–Shamir with Aborts setting. We manage to do so using homomorphic commitments. Exploiting the similarities between FSwA and Schnorr-style signatures, our approach makes the most of observations from recent advancements in the discrete log setting, such as Drijvers et al.’s seminal work on two-round multi-signatures (S&P 2019). In particular, we observe that the use of commitment not only resolves the subtle issue with aborts, but also makes it possible to realize secure two-round $n$-out-of-$n$ distributed signing and multi-signature in the plain public key model, by equipping the commitment with a trapdoor feature. The construction of suitable trapdoor commitment from lattices is a side contribution of this paper

Two results on spontaneous anonymous group signatures.

Chan Kwok Leong.Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.Includes bibliographical references (leaves 72-78).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Preliminaries --- p.4Chapter 2.1 --- Notation --- p.4Chapter 2.2 --- Cryptographic Primitives --- p.5Chapter 2.2.1 --- Symmetric Key Cryptography --- p.5Chapter 2.2.2 --- Asymmetric Key Cryptosystem --- p.6Chapter 2.2.3 --- Secure Hash Function --- p.7Chapter 2.2.4 --- Digital Signature --- p.8Chapter 2.2.5 --- Digital Certificate and Public Key Infrastructure --- p.8Chapter 2.3 --- Provable Security and Security Model --- p.9Chapter 2.3.1 --- Mathematics Background --- p.9Chapter 2.3.2 --- One-Way Function --- p.10Chapter 2.3.3 --- Candidate One-way Functions --- p.12Chapter 2.4 --- Proof Systems --- p.15Chapter 2.4.1 --- Zero-knowledge Protocol --- p.15Chapter 2.4.2 --- Proof-of-Knowledge Protocol --- p.17Chapter 2.4.3 --- Honest-Verifier Zero-Knowledge (HVZK) Proof of Knowl- edge Protocols (PoKs) --- p.18Chapter 2.5 --- Security Model --- p.19Chapter 2.5.1 --- Random Oracle Model --- p.19Chapter 2.5.2 --- Generic group model (GGM) --- p.20Chapter 3 --- Signature Scheme --- p.21Chapter 3.1 --- Introduction --- p.21Chapter 3.2 --- Security Notation for Digital Signature --- p.23Chapter 3.3 --- Security Proof for Digital Signature --- p.24Chapter 3.3.1 --- Random Oracle Model for Signature Scheme --- p.24Chapter 3.3.2 --- Adaptive Chosen Message Attack --- p.24Chapter 3.4 --- Schnorr Identification and Schnorr Signature --- p.25Chapter 3.4.1 --- Schnorr's ROS assumption --- p.26Chapter 3.5 --- Blind Signature --- p.27Chapter 4 --- Spontaneous Anonymous Group (SAG) Signature --- p.30Chapter 4.1 --- Introduction --- p.30Chapter 4.2 --- Background --- p.30Chapter 4.2.1 --- Group Signature --- p.30Chapter 4.2.2 --- Threshold Signature --- p.31Chapter 4.3 --- SAG signatures --- p.33Chapter 4.4 --- Formal Definitions and Constructions --- p.35Chapter 4.4.1 --- Ring-type construction --- p.36Chapter 4.4.2 --- CDS-type construction --- p.36Chapter 4.5 --- Discussion --- p.37Chapter 5 --- Blind Spontaneous Anonymous Signature --- p.39Chapter 5.1 --- Introduction --- p.39Chapter 5.2 --- Definition --- p.40Chapter 5.2.1 --- Security Model --- p.41Chapter 5.2.2 --- Definitions of security notions --- p.41Chapter 5.3 --- Constructing blind SAG signatures --- p.43Chapter 5.3.1 --- Blind SAG signature: CDS-type [1] --- p.43Chapter 5.3.2 --- "Blind SAG signature: ring-type [2, 3]" --- p.44Chapter 5.4 --- Security Analysis --- p.44Chapter 5.4.1 --- Multi-key parallel one-more unforgeability of blind signature --- p.45Chapter 5.4.2 --- Security of our blind SAG signatures --- p.47Chapter 5.5 --- Discussion --- p.49Chapter 6 --- Linkable Spontaneous Anonymous Group Signature --- p.51Chapter 6.1 --- introduction --- p.51Chapter 6.2 --- Related work --- p.51Chapter 6.3 --- Basic Building Blocks --- p.52Chapter 6.3.1 --- Proving the Knowledge of Several Discrete Logarithms --- p.53Chapter 6.3.2 --- Proving the Knowledge of d Out of n Equalities of Discrete Logarithms --- p.55Chapter 6.4 --- Security Model --- p.57Chapter 6.4.1 --- Syntax --- p.57Chapter 6.4.2 --- Notions of Security --- p.59Chapter 6.5 --- Our Construction --- p.63Chapter 6.5.1 --- An Linkable Threshold SAG Signature Scheme --- p.63Chapter 6.5.2 --- Security --- p.65Chapter 6.5.3 --- Discussions --- p.67Chapter 7 --- Conclusion --- p.70Bibliography --- p.7

Delayed-Input Non-Malleable Zero Knowledge and Multi-Party Coin Tossing in Four Rounds

In this work we start from the following two results in the state-of-the art: 1.4-round non-malleable zero knowledge (NMZK): Goyal et al. in FOCS 2014 showed the first 4-round one-one NMZK argument from one-way functions (OWFs). Their construction requires the prover to know the instance and the witness already at the 2nd round.2.4-round multi-party coin tossing (MPCT): Garg et al. in Eurocrypt 2016 showed the first 4-round protocol for MPCT. Their result crucially relies on 3-round 3-robust parallel non-malleable commitments. So far there is no candidate construction for such a commitment scheme under standard polynomial-time hardness assumptions. We improve the state-of-the art on NMZK and MPCT by presenting the following two results: 1.a delayed-input 4-round one-many NMZK argument IINMZKfrom OWFs; moreover IINMZKis also a delayed-input many-many synchronous NMZK argument.2.a 4-round MPCT protocol IIMPCTfrom one-to-one OWFs; IIMPCTuses IINMZKas subprotocol and exploits the special properties (e.g., delayed input, many-many synchronous) of IINMZK. Both IINMZKand IIMPCTmake use of a special proof of knowledge that offers additional security guarantees when played in parallel with other protocols. The new technique behind such a proof of knowledge is an additional contribution of this work and is of independent interest

Practical Round-Optimal Blind Signatures in the ROM from Standard Assumptions

Blind signatures serve as a foundational tool for privacy-preserving applications and have recently seen renewed interest due to new applications in blockchains and privacy-authentication tokens. With this, constructing practical round-optimal (i.e., signing consists of the minimum two rounds) blind signatures in the random oracle model (ROM) has been an active area of research, where several impossibility results indicate that either the ROM or a trusted setup is inherent. In this work, we present two round-optimal blind signatures under standard assumptions in the ROM with different approaches: one achieves the smallest sum of the signature and communication sizes, while the other achieves the smallest signature size. Both of our instantiations are based on standard assumptions over asymmetric pairing groups, i.e., CDH, DDH, and/or SXDH. Our first construction is a highly optimized variant of the generic blind signature construction by Fischlin (CRYPTO\u2706) and has signature and communication sizes 447 B and 303 B, respectively. We progressively weaken the building blocks required by Fischlin and we result in the first blind signature where the sum of the signature and communication sizes fit below 1 KB based on standard assumptions. Our second construction is a semi-generic construction from a specific class of randomizable signature schemes that admits an all-but-one reduction. The signature size is only 96 B while the communication size is 2.2 KB. This matches the previously known smallest signature size while improving the communication size by several orders of magnitude. Finally, both of our constructions rely on a (non-black box) fine-grained analysis of the forking lemma that may be of independent interest

Efficient Perfectly Sound One-message Zero-Knowledge Proofs via Oracle-aided Simulation

In this paper we put forth new efficient one-message proof systems for several practical applications, like proving that an El Gamal ciphertext (over a multiplicative group) decrypts to a given value and correctness of a shuffle. Our proof systems are built from multiplicative groups of hidden order, are not based on any setup/trust assumption like the RO or the common reference string model and are perfectly sound, that is they are written proofs in the sense of mathematics. Our proof systems satisfy a generalization of zero-knowledge (ZK) that we call harmless zero-knowledge (HZK). The simulator of an $O$-HZK proof for a relation over a language $L$ is given the additional capability of invoking an oracle $O$ relative to which $L$ is hard to decide. That is, the proof does not leak any knowledge that an adversary might not compute by itself interacting with an oracle $O$ that does not help to decide the language. Unlike ZK, non-interactivity and perfect soundness do not contradict HZK and HZK can replace ZK in any application in which, basically, the computational assumptions used in the application hold even against adversaries with access to $O$. An $O$-HZK proof is witness hiding (WH) for distributions hard against adversaries with access to $O$, and strong-WI when quantifying over distributions that are indistinguishable by adversaries with access to $O$. Moreover, an $O$-HZK proof is witness indistinguishable (and the property does not depend on the oracle). We provide a specific oracle DHInvO that is enough powerful to make our main proof systems DHInvO-HZK but not trivial: indeed, we show concrete and practical cryptographic protocols that can be proven secure employing a DHInvO-HZK proof in the reduction and that are instead not achievable using traditional ZK (unless resorting to the CRS/RO models). Efficient one-message proof systems with perfect soundness were only known for relations over bilinear groups and were proven only witness indistinguishable. As byproduct, we also obtain a perfectly sound non-interactive ZAP, WH and HZK proof system for $NP$ relations from number-theoretic assumptions over multiplicative groups of hidden order. No non-interactive WH proof system for $NP$ (neither for simpler non-trivial relations) was previously known