Lattice-based Zero-knowledge SNARGs for Arithmetic Circuits

Succinct non-interactive arguments (SNARGs) enable verifying NP computations with substantially lower complexity than that required for classical NP verification. In this work, we construct a zero-knowledge SNARG candidate that relies only on lattice-based assumptions which are claimed to hold even in the presence of quantum computers. Central to this new construction is the notion of linear-targeted malleability introduced by Bitansky et al. (TCC 2013) and the conjecture that variants of Regev encryption satisfy this property. Then, using the efficient characterization of NP languages as Square Arithmetic Programs we build the first quantum-resilient zk-SNARG for arithmetic circuits with a constant-size proof consisting of only 2 lattice-based ciphertexts. Our protocol is designated-verifier, achieves zero-knowledge and has shorter proofs and shorter CRS than the previous such schemes, e.g. Boneh et al. (Eurocrypt 2017)

Lattice-Based zk-SNARKs from Square Span Programs

Zero-knowledge SNARKs (zk-SNARKs) are non-interactive proof systems with short (i.e., independent of the size of the witness) and efficiently verifiable proofs. They elegantly resolve the juxtaposition of individual privacy and public trust, by providing an efficient way of demonstrating knowledge of secret information without actually revealing it. To this day, zk-SNARKs are widely deployed all over the planet and are used to keep alive a system worth billion of euros, namely the cryptocurrency Zcash. However, all current SNARKs implementations rely on so-called pre-quantum assumptions and, for this reason, are not expected to withstand cryptanalitic efforts over the next few decades. In this work, we introduce a new zk-SNARK that can be instantiated from lattice-based assumptions, and which is thus believed to be post-quantum secure. We provide a generalization in the spirit of Gennaro et al. (Eurocrypt'13) to the SNARK of Danezis et al. (Asiacrypt'14) that is based on Square Span Programs (SSP) and relies on weaker computational assumptions. We focus on designated-verifier proofs and propose a protocol in which a proof consists of just 5 LWE encodings. We provide a concrete choice of parameters, showing that our construction is practically instantiable

Declarative design and enforcement for secure cloud applications

The growing demands of users and industry have led to an increase in both size and complexity of deployed software in recent years. This tendency mainly stems from a growing number of interconnected mobile devices and from the huge amounts of data that is collected every day by a growing number of sensors and interfaces. Such increase in complexity imposes various challenges -- not only in terms of software correctness, but also with respect to security. This thesis addresses three complementary approaches to cope with the challenges: (i) appropriate high-level abstractions and verifiable translation methods to executable applications in order to guarantee flawless implementations, (ii) strong cryptographic mechanisms in order to realize the desired security goals, and (iii) convenient methods in order to incentivize the correct usage of existing techniques and tools. In more detail, the thesis presents two frameworks for the declarative specification of functionality and security, together with advanced compilers for the verifiable translation to executable applications. Moreover, the thesis presents two cryptographic primitives for the enforcement of cloud-based security properties: homomorphic message authentication codes ensure the correctness of evaluating functions over data outsourced to unreliable cloud servers; and efficiently verifiable non-interactive zero-knowledge proofs convince verifiers of computation results without the verifiers having access to the computation input.Die wachsenden Anforderungen von Seiten der Industrie und der Endbenutzer verlangen nach immer komplexeren Softwaresystemen -- größtenteils begründet durch die stetig wachsende Zahl mobiler Geräte und die damit wachsende Zahl an Sensoren und erfassten Daten. Mit wachsender Software-Komplexität steigen auch die Herausforderungen an Korrektheit und Sicherheit. Die vorliegende Arbeit widmet sich diesen Herausforderungen in Form dreier komplementärer Ansätze: (i) geeignete Abstraktionen und verifizierbare Übersetzungsmethoden zu ausführbaren Anwendungen, die fehlerfreie Implementierungen garantieren, (ii) starke kryptographische Mechanismen, um die spezifizierten Sicherheitsanforderungen effizient und korrekt umzusetzen, und (iii) zweckmäßige Methoden, die eine korrekte Benutzung existierender Werkzeuge und Techniken begünstigen. Diese Arbeit stellt zwei neuartige Abläufe vor, die verifizierbare Übersetzungen von deklarativen Spezifikationen funktionaler und sicherheitsrelevanter Ziele zu ausführbaren Cloud-Anwendungen ermöglichen. Darüber hinaus präsentiert diese Arbeit zwei kryptographische Primitive für sichere Berechnungen in unzuverlässigen Cloud-Umgebungen. Obwohl die Eingabedaten der Berechnungen zuvor in die Cloud ausgelagert wurden und zur Verifikation der Berechnungen nicht mehr zur Verfügung stehen, ist es möglich, die Korrektheit der Ergebnisse in effizienter Weise zu überprüfen

Functional Commitments for All Functions, with Transparent Setup and from SIS

A *functional commitment* scheme enables a user to concisely commit to a function from a specified family, then later concisely and verifiably reveal values of the function at desired inputs. Useful special cases, which have seen applications across cryptography, include vector commitments and polynomial commitments. To date, functional commitments have been constructed (under falsifiable assumptions) only for functions that are essentially *linear*, with one recent exception that works for arbitrarily complex functions. However, that scheme operates in a strong and non-standard model, requiring an online, trusted authority to generate special keys for any opened function inputs. In this work, we give the first functional commitment scheme for nonlinear functions---indeed, for *all functions* of any bounded complexity---under a standard setup and a falsifiable assumption. Specifically, the setup is ``transparent,\u27\u27 requiring only public randomness (and not any trusted entity), and the assumption is the hardness of the standard Short Integer Solution (SIS) lattice problem. Our construction also has other attractive features, including: *stateless updates* via generic composability; excellent *asymptotic efficiency* for the verifier, and also for the committer in important special cases like vector and polynomial commitments, via preprocessing; and *post-quantum security*, since it is based on SIS

Lower Bound on SNARGs in the Random Oracle Model

Succinct non-interactive arguments (SNARGs) have become a fundamental primitive in the cryptographic community. The focus of this work is constructions of SNARGs in the Random Oracle Model (ROM). Such SNARGs enjoy post-quantum security and can be deployed using lightweight cryptography to heuristically instantiate the random oracle. A ROM-SNARG is \emph{$(t,\varepsilon)$-sound} if no $t$-query malicious prover can convince the verifier to accept a false statement with probability larger than $\varepsilon$. Recently, Chiesa-Yogev (CRYPTO \u2721) presented a ROM-SNARG of length ${\Theta}(\log (t/\varepsilon) \cdot \log t)$ (ignoring $\log n$ factors, for $n$ being the instance size). This improvement, however, is still far from the (folklore) lower bound of $\Omega(\log (t/\varepsilon))$. Assuming the \textit{randomized exponential-time hypothesis}, we prove a tight lower bound of ${\Omega}(\log (t/\varepsilon) \cdot \log t)$ for the length of {$(t,\varepsilon)$-sound} ROM-SNARGs. Our lower bound holds for constructions with non-adaptive verifiers and strong soundness notion called \textit{salted soundness}, restrictions that hold for \emph{all} known constructions (ignoring contrived counterexamples). We prove our lower bound by transforming any short ROM-SNARG (of the considered family) into a same length ROM-SNARG in which the verifier asks only a \emph{few} oracles queries, and then apply the recent lower bound of Chiesa-Yogev (TCC \u2720) for such SNARGs

Chainable Functional Commitments for Unbounded-Depth Circuits

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

Indistinguishability Obfuscation via Mathematical Proofs of Equivalence

Over the last decade, indistinguishability obfuscation (iO) has emerged as a seemingly omnipotent primitive in cryptography. Moreover, recent breakthrough work has demonstrated that iO can be realized from well-founded assumptions. A thorn to all this remarkable progress is a limitation of all known constructions of general-purpose iO: the security reduction incurs a loss that is exponential in the input length of the function. This ``input-length barrier\u27\u27 to iO stems from the non-falsifiability of the iO definition and is discussed in folklore as being possibly inherent. It has many negative consequences; notably, constructing iO for programs with inputs of unbounded length remains elusive due to this barrier. We present a new framework aimed towards overcoming the input-length barrier. Our approach relies on short mathematical proofs of functional equivalence of circuits (and Turing machines) to avoid the brute-force ``input-by-input\u27\u27 check employed in prior works. - We show how to obfuscate circuits that have efficient proofs of equivalence in Propositional Logic with a security loss independent of input length. - Next, we show how to obfuscate Turing machines with unbounded length inputs, whose functional equivalence can be proven in Cook\u27s Theory $PV$. - Finally, we demonstrate applications of our results to succinct non-interactive arguments and witness encryption, and provide guidance on using our techniques for building new applications. To realize our approach, we depart from prior work and develop a new gate-by-gate obfuscation template that preserves the topology of the input circuit

Quasi-Optimal SNARGs via Linear Multi-Prover Interactive Proofs

Succinct non-interactive arguments (SNARGs) enable verifying NP computations with significantly less complexity than that required for classical NP verification. In this work, we focus on simultaneously minimizing the proof size and the prover complexity of SNARGs. Concretely, for a security parameter $\lambda$, we measure the asymptotic cost of achieving soundness error $2^{-\lambda}$ against provers of size $2^\lambda$. We say a SNARG is quasi-optimally succinct if its proof length is $\tilde{O}(\lambda)$, and that it is quasi-optimal, if moreover, its prover complexity is only polylogarithmically greater than the running time of the classical NP prover. We show that this definition is the best we could hope for assuming that NP does not have succinct proofs. Our definition strictly strengthens the previous notion of quasi-optimality introduced in the work of Boneh et al. (Eurocrypt 2017). This work gives the first quasi-optimal SNARG for Boolean circuit satisfiability from a concrete cryptographic assumption. Our construction takes a two-step approach. The first is an information-theoretic construction of a quasi-optimal linear multi-prover interactive proof (linear MIP) for circuit satisfiability. Then, we describe a generic cryptographic compiler that transforms our quasi-optimal linear MIP into a quasi-optimal SNARG by relying on the notion of linear-only vector encryption over rings introduced by Boneh et al. Combining these two primitives yields the first quasi-optimal SNARG based on linear-only vector encryption. Moreover, our linear MIP construction leverages a new robust circuit decomposition primitive that allows us to decompose a circuit satisfiability instance into several smaller circuit satisfiability instances. This primitive may be of independent interest. Finally, we consider (designated-verifier) SNARGs that provide optimal succinctness for a non-negligible soundness error. Concretely, we put forward the notion of 1-bit SNARGs that achieve soundness error 1/2 with only one bit of proof. We first show how to build 1-bit SNARGs from indistinguishability obfuscation, and then show that 1-bit SNARGs also suffice for realizing a form of witness encryption. The latter result highlights a two-way connection between the soundness of very succinct argument systems and powerful forms of encryption

Fiat-Shamir From Simpler Assumptions

We present two new protocols: (1) A succinct publicly verifiable non-interactive argument system for log-space uniform NC computations, under the assumption that any one of a broad class of fully homomorphic encryption (FHE) schemes has almost optimal security against polynomial-time adversaries. The class includes all FHE schemes in the literature that are based on the learning with errors (LWE) problem. (2) A non-interactive zero-knowledge argument system for NP in the common random string model, assuming almost optimal hardness of search-LWE against polynomial-time adversaries. Both results are obtained by applying the Fiat-Shamir transform with explicit, efficiently computable functions (specifically, correlation intractable functions) to certain classes of interactive proofs. We improve over prior work by reducing the security of these protocols to qualitatively weaker computational hardness assumptions. Along the way, we also show that the Fiat-Shamir transform can be soundly applied (in the plain model) to a richer class of protocols than was previously known

Aurora: Transparent Succinct Arguments for R1CS

We design, implement, and evaluate a zkSNARK for Rank-1 Constraint Satisfaction (R1CS), a widely-deployed NP-complete language that is undergoing standardization. Our construction uses a transparent setup, is plausibly post-quantum secure, and uses lightweight cryptography. A proof attesting to the satisfiability of n constraints has size $O(\log^2 n)$; it can be produced with $O(n \log n)$ field operations and verified with $O(n)$. At 128 bits of security, proofs are less than 130kB even for several million constraints, more than 20x shorter than prior zkSNARK with similar features. A key ingredient of our construction is a new Interactive Oracle Proof (IOP) for solving a *univariate* analogue of the classical sumcheck problem [LFKN92], originally studied for *multivariate* polynomials. Our protocol verifies the sum of entries of a Reed--Solomon codeword over any subgroup of a field. We also provide libiop, an open-source library for writing IOP-based arguments, in which a toolchain of transformations enables programmers to write new arguments by writing simple IOP sub-components. We have used this library to specify our construction and prior ones