Subversion-Resistant Simulation (Knowledge) Sound NIZKs

In ASIACRYPT 2016, Bellare, Fuchsbauer, and Scafuro studied the security of non-interactive zero-knowledge (NIZK) arguments in the face of parameter subversion. They showed that achieving subversion soundness (soundness without trusting to the third party) and standard zero-knowledge is impossible at the same time. On the positive side, in the best case, they showed that one can achieve subversion zero-knowledge (zero-knowledge without trusting to the third party) and soundness at the same time. In this paper, we show that one can amplify their best positive result and construct NIZK arguments that can achieve subversion zero-knowledge and $\textit{simulation}$ (knowledge) soundness at the same time. Simulation (knowledge) soundness is a stronger notion in comparison with (knowledge) soundness, as it also guarantees non-malleability of proofs. Such a stronger security guarantee is a must in practical systems. To prove the result, we show that given a NIZK argument that achieves Sub-ZK and (knowledge) soundness, one can use an OR-based construction to define a new language and build a NIZK argument that will guarantee Sub-ZK and $\textit{simulation}$ (knowledge) soundness at the same time. We instantiate the construction with the state-of-the-art zk-SNARK proposed by Groth [Eurocrypt 2016] and obtain an efficient SNARK that guarantees Sub-ZK and simulation knowledge soundness

Updatable NIZKs from Non-Interactive Zaps

In ASIACRYPT 2016, Bellare, Fuchsbauer, and Scafuro studied the security of NIZK arguments under subverted Structured Reference String (SRS) and presented some positive and negative results. In their best positive result, they showed that by defining an SRS as a tuple of knowledge assumption in bilinear groups (e.g. $g^a, g^b, g^{ab}$), and then using a Non-Interactive (NI) zap to prove that either there is a witness for the statement $\mathsf{x}$ or one knows the trapdoor of SRS (e.g. $a$ or $b$), one can build NIZK arguments that can achieve soundness and $\textit{subversion zero-knowledge}$ (zero-knowledge without trusting a third party; Sub-ZK). In this paper, we expand their idea and use NI zaps (of knowledge) to build NIZK arguments (of knowledge) with $\textit{updatable}$, $\textit{universal}$, and $\textit{succinct}$ SRS. To this end, we first show that their proposed sound and Sub-ZK NIZK argument can also achieve $\textit{updatable}$ soundness, which is a more desired notion than the plain soundness. Updatable soundness allows the verifier to update the SRS one time and bypass the need for a trusted third party. Then, we show that using a similar OR language, given a NI zap (of knowledge) and a $\textit{key-updatable}$ signature scheme, one can build NIZK arguments that can achieve Sub-ZK and $\textit{updatable}$ simulation soundness (resp. $\textit{updatable}$ simulation extractability). The proposed constructions are the first NIZK arguments that have updatable and succinct SRS, and do not require a random oracle. Our instantiations show that in the resulting NIZK arguments the computational cost for the parties to verify/update the SRS is negligible, namely, a few exponentiations and pairing checks. The run times of the prover and verifier, as well as the size of the proof, are asymptotically the same as those of the underlying NI zap

On QA-NIZK in the BPK Model

Recently, Bellare et al. defined subversion-resistance (security in the case the CRS creator may be malicious) for NIZK. In particular, a Sub-ZK NIZK is zero-knowledge, even in the case of subverted CRS. We study Sub-ZK QA-NIZKs, where the CRS can depend on the language parameter. First, we observe that subversion zero-knowledge (Sub-ZK) in the CRS model corresponds to no-auxiliary-string non-black-box NIZK in the Bare Public Key model, and hence, the use of non-black-box techniques is needed to obtain Sub-ZK. Second, we give a precise definition of Sub-ZK QA-NIZKs that are (knowledge-)sound if the language parameter but not the CRS is subverted and zero-knowledge even if both are subverted. Third, we prove that the most efficient known QA-NIZK for linear subspaces by Kiltz and Wee is Sub-ZK under a new knowledge assumption that by itself is secure in (a weaker version of) the algebraic group model. Depending on the parameter setting, it is (knowledge-)sound under different non-falsifiable assumptions, some of which do not belong to the family of knowledge assumptions

Verifiably-Extractable OWFs and Their Applications to Subversion Zero-Knowledge

An extractable one-way function (EOWF), introduced by Canetti and Dakdouk (ICALP 2008) and generalized by Bitansky et al. (SIAM Journal on Computing vol. 45), is an OWF that allows for efficient extraction of a preimage for the function. We study (generalized) EOWFs that have a public image verification algorithm. We call such OWFs verifiably-extractable and show that several previously known constructions satisfy this notion. We study how such OWFs relate to subversion zero-knowledge (Sub-ZK) NIZKs by using them to generically construct a Sub-ZK NIZK from a NIZK satisfying certain additional properties, and conversely show how to obtain them from any Sub-ZK NIZK. Prior to our work, the Sub-ZK property of NIZKs was achieved using concrete knowledge assumptions

How to Extract Useful Randomness from Unreliable Sources

For more than 30 years, cryptographers have been looking for public sources of uniform randomness in order to use them as a set-up to run appealing cryptographic protocols without relying on trusted third parties. Unfortunately, nowadays it is fair to assess that assuming the existence of physical phenomena producing public uniform randomness is far from reality. It is known that uniform randomness cannot be extracted from a single weak source. A well-studied way to overcome this is to consider several independent weak sources. However, this means we must trust the various sampling processes of weak randomness from physical processes. Motivated by the above state of affairs, this work considers a set-up where players can access multiple potential sources of weak randomness, several of which may be jointly corrupted by a computationally unbounded adversary. We introduce SHELA (Somewhere Honest Entropic Look Ahead) sources to model this situation. We show that there is no hope of extracting uniform randomness from a SHELA source. Instead, we focus on the task of Somewhere-Extraction (i.e., outputting several candidate strings, some of which are uniformly distributed -- yet we do not know which). We give explicit constructions of Somewhere-Extractors for SHELA sources with good parameters. Then, we present applications of the above somewhere-extractor where the public uniform randomness can be replaced by the output of such extraction from corruptible sources, greatly outperforming trivial solutions. The output of somewhere-extraction is also useful in other settings, such as a suitable source of random coins for many randomized algorithms. In another front, we comprehensively study the problem of Somewhere-Extraction from a weak source, resulting in a series of bounds. Our bounds highlight the fact that, in most regimes of parameters (including those relevant for applications), SHELA sources significantly outperform weak sources of comparable parameters both when it comes to the process of Somewhere-Extraction, or in the task of amplification of success probability in randomized algorithms. Moreover, the low quality of somewhere-extraction from weak sources excludes its use in various efficient applications

Short Threshold Dynamic Group Signatures

Traditional group signatures feature a single issuer who can add users to the group of signers and a single opening authority who can reveal the identity of the group member who computed a signature. Interestingly, despite being designed for privacy-preserving applications, they require strong trust in these central authorities who constitute single points of failure for critical security properties. To reduce the trust placed on authorities, we introduce dynamic group signatures which distribute the role of issuer and opener over several entities, and support t_I-out-of-n_I issuance and t_O-out-of-n_O opening. We first define threshold dynamic group signatures and formalize their security. We then give an efficient construction relying on the pairing-based Pointcheval–Sanders (PS) signature scheme (CT-RSA 2018), which yields very short group signatures of two first-group elements and three exponents. We also give a simpler variant of our scheme in which issuance requires the participation of all n_I issuers, but still supports t_O-out-of-n_O opening. It is based on a new multi-signature variant of the PS scheme which allows for efficient proofs of knowledge and is a result of independent inter- est. We prove our schemes secure in the random-oracle model under a non-interactive q-type of assumption

Assumptions, Efficiency and Trust in Non-Interactive Zero-Knowledge Proofs

Vi lever i en digital verden. En betydelig del av livene våre skjer på nettet, og vi bruker internett for stadig flere formål og er avhengig av stadig mer avansert teknologi. Det er derfor viktig å beskytte seg mot ondsinnede aktører som kan forsøke å utnytte denne avhengigheten for egen vinning. Kryptografi er en sentral del av svaret på hvordan man kan beskytte internettbrukere. Historisk sett har kryptografi hovedsakelig vært opptatt av konfidensiell kommunikasjon, altså at ingen kan lese private meldinger sendt mellom to personer. I de siste tiårene har kryptografi blitt mer opptatt av å lage protokoller som garanterer personvern selv om man kan gjennomføre komplekse handlinger. Et viktig kryptografisk verktøy for å sikre at disse protokollene faktisk følges er kunnskapsløse bevis. Et kunnskapsløst bevis er en prosess hvor to parter, en bevisfører og en attestant, utveksler meldinger for å overbevise attestanten om at bevisføreren fulgte protokollen riktig (hvis dette faktisk er tilfelle) uten å avsløre privat informasjon til attestanten. For de fleste anvendelser er det ønskelig å lage et ikke-interaktivt kunnskapsløst bevis (IIK-bevis), der bevisføreren kun sender én melding til attestanten. IIK-bevis har en rekke ulike bruksområder, som gjør de til attraktive studieobjekter. Et IIK-bevis har en rekke ulike egenskaper og forbedring av noen av disse fremmer vår kollektive kryptografiske kunnskap. I den første artikkelen i denne avhandlingen konstruerer vi et nytt ikke-interaktivt kunnskapsløst bevis for språk basert på algebraiske mengder. Denne artikkelen er basert på arbeid av Couteau og Hartmann (Crypto 2020), som viste hvordan man omformer et bestemt interaktivt kunnskapsløst bevis til et IIK-bevis. Vi følger deres tilnærming, men vi bruker et annet interaktivt kunnskapsløst bevis. Dette fører til en forbedring sammenlignet med arbeidet deres på flere områder, spesielt når det gjelder både formodninger og effektivitet. I den andre artikkelen i denne avhandlingen studerer vi egenskapene til ikke-interaktive kunnskapsløse bevis som er motstandsdyktige mot undergraving. Det er umulig å lage et IIK-bevis uten å stole på en felles referansestreng (FRS) generert av en pålitelig tredjepart. Men det finnes eksempler på IIK-bevis der ingen lærer noe privat informasjon fra beviset selv om den felles referansestrengen ble skapt på en uredelig måte. I denne artikkelen lager vi en ny kryptografisk primitiv (verifiserbart-uttrekkbare enveisfunksjoner) og viser hvordan denne primitiven er relatert til IIK-bevis med den ovennevnte egenskapen.We live in a digital world. A significant part of our lives happens online, and we use the internet for incredibly many different purposes and we rely on increasingly advanced technology. It therefore is important to protect against malicious actors who may try to exploit this reliance for their own gain. Cryptography is a key part of the answer to protecting internet users. Historically, cryptography has mainly been focused on maintaining the confidentiality of communication, ensuring that no one can read private messages sent between people. In recent decades, cryptography has become concerned with creating protocols which guarantee privacy even as they support more complex actions. A crucial cryptographic tool to ensure that these protocols are indeed followed is the zero-knowledge proof. A zero-knowledge proof is a process where two parties, a prover and a verifier, exchange messages to convince the verifier that the prover followed the protocol correctly (if indeed the prover did so) without revealing any private information to the verifier. It is often desirable to create a non-interactive zero-knowledge proof (NIZK), where the prover only sends one message to the verifier. NIZKs have found a number of different applications, which makes them an attractive object of study. A NIZK has a variety of different properties, and improving any of these aspects advances our collective cryptographic knowledge. In the first paper in this thesis, we construct a new non-interactive zero-knowledge proof for languages based on algebraic sets. This paper is based on work by Couteau and Hartmann (Crypto 2020), which showed how to convert a particular interactive zero-knowledge proof to a NIZK. We follow their approach, but we start with a different interactive zero-knowledge proof. This leads to an improvement compared to their work in several ways, in particular in terms of both assumptions and efficiency. In the second paper in this thesis, we study the property of subversion zero-knowledge in non-interactive zero-knowledge proofs. It is impossible to create a NIZK without relying on a common reference string (CRS) generated by a trusted party. However, a NIZK with the subversion zero-knowledge property guarantees that no one learns any private information from the proof even if the CRS was generated dishonestly. In this paper, we create a new cryptographic primitive (verifiably-extractable one-way functions) and show how this primitive relates to NIZKs with subversion zero-knowledge.Doktorgradsavhandlin

Universally Composable Verifiable Random Oracles

Random Oracles werden häufig in der Kryptographie eingesetzt um sehr effiziente Instanziierungen mächtiger kryptographischer Primitive zu konstruieren. Jedoch ist diese Praxis im Allgemeinen nicht zulässig wie verschiedene Nicht-Instanziierungs-Ergebnisse für Random Oracles mittels lokal berechenbarer Familien von Funktionen durch Halevi et al. (JACM ’04) zeigt. Die Random Oracle Modell kann sicher eingesetzt werden, indem Random Oracles nicht mit einer lokal berechenbaren Hashfunktion, sondern stattdessen mit einem interaktiven Protokoll instanziiert werden. In der realen Welt könnte solch ein interaktives Protokoll beispielsweise aus einem vertrauenswürdigen Server, welcher über das Internet erreichbar ist, bestehen. Dieser Server würde sodann eine der bekannten Techniken wie lazy sampling oder das Auswerten einer Pseudo-Zufälligen Funktion verwenden, um die Funktionalität eines Random Oracle bereitzustellen. Ein klarer Nachteil dieses Ansatzes ist die große Menge an Interaktion, die bei jeder Berechnung, die eine Auswertung des Random Oracle beinhaltet, nötig ist. Wir wollen diese Interaktion auf ein Minimum reduzieren. Um obiges Unmöglichkeitsresultat zu umgehen, muss die Auswertung des Random Oracle auf einer frischen Eingabe Interaktion der auswertenden Partei mit einer anderen Partei beinhalten. Dies ist jedoch nicht der einzige Verwendungszweck von Random Oracles, der häufig in kryptographischen Protokollen auftritt. Bei einem weiteren solchen Zweck wertet zunächst eine Partei A das Orakel auf einer Eingabe aus und erhält einen Hashwert. Im Anschluss sendet A Eingabe und Ausgabe (im Kontext eines Protokolls) an eine zweite Partei B und möchte B davon überzeugen, dass das Random Oracle korrekt ausgewertet wurde. Eine einfache Möglichkeit dies zu prüfen besteht darin, dass B selbst eine Auswertung des Random Oracle auf der erhaltenen Eingabe tätigt und die beiden Ausgaben vergleicht. In unserem Kontext benötigt dies jedoch erneut Interaktion. Der Wunsch diesen zweiten Verwendungszweck nicht-interaktiv zu machen führt uns zum Begriff eines Verifiable Random Oracle (VRO) als Erweiterung eines Random Oracle. Abstrakt besteht ein VRO aus zwei Orakeln. Das erste Orakel verhält sich wie ein Random Oracle dessen Ausgabe um einen Korrektheitsbeweis erweitert wurde. Mit Hilfe dieses Beweises kann das zweite Orakel dazu verwendet werden öffentlich die korrekte Auswertung des Random Oracle zu verifizieren. Obwohl diese Orakel-basierte Formulierung nicht notwendigerweise nicht-interaktive Verifikation besitzt, so erlaubt jedoch die Einführung expliziter Korrektheitsbeweise dies. In dieser Masterarbeit formalisieren wir zunächst den Begriff eines VRO im Universal Composability Framework von Canetti (FOCS ’01). Danach wenden wir VROs auf zwei kryptographische Anwendungen an, die in ihrer ursprünglichen Formulierung das Random Oracle Modell verwenden, und zeigen, das deren Sicherheitseigenschaften erhalten bleiben. Um zu zeigen, dass unsere Definition realisierbar ist, konstruieren wir mehrere Protokolle, die die ideale VRO Funktionalität realisieren. Diese reichen von Protokollen für eine einzelne vertrauenswürdige Partei bis hin zu verteilten Protokollen, die eine gewisse Menge an böswilliger Korruption erlauben. Wir vergleichen weiterhin VROs mit ähnlichen existierenden Primitiven

Non-interactive Zaps of Knowledge

International audienceWhile non-interactive zero-knowledge (NIZK) proofs require trusted parameters, Groth, Ostrovsky and Sahai constructed non-interactive witness-indistinguishable (NIWI) proofs without any setup; they called their scheme a non-interactive zap. More recently, Bellare, Fuchsbauer and Scafuro investigated the security of NIZK in the face of parameter subversion and observe that NI zaps provide subversion-resistant soundness and WI.Arguments of knowledge prove that not only the statement is true, but also that the prover knows a witness for it, which is essential for anonymous identification. We present the first NIWI argument of knowledge without parameters, i.e., a NI zap of knowledge. Consequently, our scheme is also the first subversion-resistant knowledge-sound proof system, a notion recently proposed by Fuchsbauer