Exploring Constructions of Compact NIZKs from Various Assumptions

A non-interactive zero-knowledge (NIZK) protocol allows a prover to non-interactively convince a verifier of the truth of the statement without leaking any other information. In this study, we explore shorter NIZK proofs for all NP languages. Our primary interest is NIZK proofs from falsifiable pairing/pairing-free group-based assumptions. Thus far, NIZKs in the common reference string model (CRS-NIZKs) for NP based on falsifiable pairing-based assumptions all require a proof size at least as large as $O(|C| k)$, where $C$ is a circuit computing the NP relation and $k$ is the security parameter. This holds true even for the weaker designated-verifier NIZKs (DV-NIZKs). Notably, constructing a (CRS, DV)-NIZK with proof size achieving an additive-overhead $O(|C|) + poly(k)$, rather than a multiplicative-overhead $|C| \cdot poly(k)$, based on any falsifiable pairing-based assumptions is an open problem. In this work, we present various techniques for constructing NIZKs with compact proofs, i.e., proofs smaller than $O(|C|) + poly(k)$, and make progress regarding the above situation. Our result is summarized below. - We construct CRS-NIZK for all NP with proof size $|C| + poly(k)$ from a (non-static) falsifiable Diffie-Hellman (DH) type assumption over pairing groups. This is the first CRS-NIZK to achieve a compact proof without relying on either lattice-based assumptions or non-falsifiable assumptions. Moreover, a variant of our CRS-NIZK satisfies universal composability (UC) in the erasure-free adaptive setting. Although it is limited to NP relations in NC1, the proof size is $|w| \cdot poly(k)$ where $w$ is the witness, and in particular, it matches the state-of-the-art UC-NIZK proposed by Cohen, shelat, and Wichs (EPRINT\u2718) based on lattices. - We construct (multi-theorem) DV-NIZKs for NP with proof size $|C|+poly(k)$ from the computational DH assumption over pairing-free groups. This is the first DV-NIZK that achieves a compact proof from a standard DH type assumption. Moreover, if we further assume the NP relation to be computable in NC1 and assume hardness of a (non-static) falsifiable DH type assumption over pairing-free groups, the proof size can be made as small as $|w| + poly(k)$. Another related but independent issue is that all (CRS, DV)-NIZKs require the running time of the prover to be at least $|C|\cdot poly(k)$. Considering that there exists NIZKs with efficient verifiers whose running time is strictly smaller than $|C|$, it is an interesting problem whether we can construct prover-efficient NIZKs. To this end, we construct prover-efficient CRS-NIZKs for NP with compact proof through a generic construction using laconic functional evaluation schemes (Quach, Wee, and Wichs (FOCS\u2718)). This is the first NIZK in any model where the running time of the prover is strictly smaller than the time it takes to compute the circuit $C$ computing the NP relation. Finally, perhaps of an independent interest, we formalize the notion of homomorphic equivocal commitments, which we use as building blocks to obtain the first result, and show how to construct them from pairing-based assumptions

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