A Subversion-Resistant SNARK

While succinct non-interactive zero-knowledge arguments of knowledge (zk-SNARKs) are widely studied, the question of what happens when the CRS has been subverted has received little attention. In ASIACRYPT 2016, Bellare, Fuchsbauer and Scafuro showed the first negative and positive results in this direction, proving also that it is impossible to achieve subversion soundness and (even non-subversion) zero knowledge at the same time. On the positive side, they constructed an involved sound and subversion zero-knowledge argument system for NP. We show that Groth\u27s zk-SNARK for \textsc{Circuit-SAT} from EUROCRYPT 2016 can be made computationally knowledge-sound and perfectly composable Sub-ZK with minimal changes. We just require the CRS trapdoor to be extractable and the CRS to be publicly verifiable. To achieve the latter, we add some new elements to the CRS and construct an efficient CRS verification algorithm. We also provide a definitional framework for sound and Sub-ZK SNARKs and describe implementation results of the new Sub-ZK SNARK

On Privacy Preserving Blockchains and zk-SNARKs

Viimastel aastatel on krüptoraha ja plokiahela tehnoloogia leidnud suurt tähelepanu nii kaubanduslikust kui ka teaduslikust vaatenurgast. Krüptoraha kujutab endast digitaalseid münte, mis kasutades krüptograafilisi vahendeid võimaldab turvalisi tehinguid võrdvõrkudes. Bitcoin on kõige tuntum krüptoraha, mis võimaldab otsetehinguid kasutajate pseudonüümide vahel ilma, et oleks vaja kolmandaid osapooli. Paraku kui kasutaja pseudonüüm on seotud tema identiteediga, on kõik tema tehingud jälgitavad ning kaob privaatsus.Selle lahendamiseks on välja pakutud erinevaid privaatsust säilitavaid krüptorahasi, mis kasutavad anonüümsete tehingute saavutamiseks krüptograafilisi tööriistu. Zerocash on üks populaarseimatest privaatsetest krüptorahadest, mis kasutab iga tehingu allika, sihtkoha ja väärtuse varjamiseks nullteadmustõestust.Antud töö koosneb kahest peamisest osast.Esimeses osas kirjeldame, pärast lühikest ülevaadet mõnest privaatsest krüptorahast (Bitcoin, Monero ja Zerocoin), Zerocashi konstruktsiooni ja anname intuitsiivse seletuse selle tööpõhimõttele. Me tutvustame kasutuselevõetud primitiive ja arutleme iga primitiivi rolli üle mündi konstruktsioonis. Erilist tähelepanu pöörame kompaktsetele nullteadmustõestusetele (zk-SNARKidele), millel on peamine roll Zerocashis.Kuna nullteadmustõestus on niivõrd olulisel kohal Zerocashis (ja teistes privaatsetes rakendustes) siis töö teises osas pakume välja uue variatsiooni Grothi 2016. aasta zk-SNARKile, mis on seni kõige tõhusam.Erinevalt Grothi konstruktsioonist, meie variatsioonis ei ole võimalik tõestusi modifitseerida.Muudatused mõjutavad nullteadmustõestuse tõhusust vaid minimaalselt ning meie konstruktsioon on kiirem kui Grothi ja Malleri 2017. nullteadmustõestus, mis samuti välistab muudetavuse.During last few years, along with blockchain technology, cryptocurrencies have found huge attention from both commercial and scientific perspectives. Cryptocurrencies are digital coins which use cryptographic tools to allow secure peer-to-peer monetary transactions. Bitcoin is the most well-known cryptocurrency that allows direct payments between pseudonyms without any third party. If a user's pseudonym is linked to her identity, all her transactions will be traceable, which will violate her privacy. To address this, various privacy-preserving cryptocurrencies have been proposed that use different cryptographic tools to achieve anonymous transactions. Zerocash is one of the most popular ones that uses zero-knowledge proofs to hide the source, destination and value of each transaction. This thesis consists of two main parts. In the first part, after a short overview of some cryptocurrencies (precisely Bitcoin, Monero and Zerocoin), we will explain the construction of Zerocash cryptocurrency and discuss the intuition behind the construction. More precisely, we will introduce the deployed primitives and will discuss the role of each primitive in the construction of the coin. In particular, we explain zero-knowledge Succinct Non-Interactive Arguments of Knowledge (a.k.a. zk-SNARKs) that play the main role in achieving strong privacy in Zerocash. Due to the importance of zk-SNARKs in privacy-preserving applications, in the second part of the thesis, we will present a new variation of Groth's 2016 zk-SNARK that currently is the most efficient pairing-based scheme. The main difference between the proposed variation and the original one is that unlike the original version, new variation guarantees non-malleability of generated proofs. Our analysis shows that the proposed changes have minimal effects on the efficiency of the original scheme and particularly it outperforms Groth and Maller's 2017 zk-SNARK that also guarantees non-malleability of proofs

Mining for Privacy: How to Bootstrap a Snarky Blockchain

Non-interactive zero-knowledge proofs, and more specifically succinct non-interactive zero-knowledge arguments (zk-SNARKs), have been proven to be the “swiss army knife” of the blockchain and distributed ledger space, with a variety of applications in privacy, interoperability and scalability. Many commonly used SNARK systems rely on a structured reference string, the secure generation of which turns out to be their Achilles heel: If the randomness used for the generation is known, the soundness of the proof system can be broken with devastating consequences for the underlying blockchain system that utilises them. In this work we describe and analyze, for the first time, a blockchain mechanism that produces a secure SRS with the characteristic that security is shown for the exact same conditions under which the blockchain protocol is proven to be secure. Our mechanism makes use of the recent discovery of updateable structure reference strings to perform this secure generation in a fully distributed manner. In this way, the SRS emanates from the normal operation of the blockchain protocol itself without the need of additional security assumptions or off-chain computation and/or verification. We provide concrete guidelines for the parameterisation of this system which allows for the completion of a secure setup in a reasonable period of time. We also provide an incentive scheme that, when paired with the update mechanism, properly incentivises participants into contributing to secure reference string generation

Benchmarking the Setup of Updatable zk-SNARKs

Subversion-resistant zk-SNARKs allow the provers to verify the Structured Reference String (SRS), via an SRS Verification (SV) algorithm and bypass the need for a Trusted Third Party (TTP). Pairing-based zk-SNARKs with $updatable$ and $universal$ SRS are an extension of subversion-resistant ones which additionally allow the verifiers to update the SRS, via an SRS Updating (SU) algorithm, and similarly bypass the need for a TTP. In this paper, we examine the setup of these zk-SNARKs by benchmarking the efficiency of the SV and SU algorithms within the $\textsf{Arkworks}$ library. The benchmarking covers a range of updatable zk-SNARKs, including Sonic, Plonk, Marlin, Lunar, and Basilisk. Our analysis reveals that relying solely on the standard Algebraic Group Model (AGM) may not be sufficient in practice, and we may need a model with weaker assumptions. Specifically, we find that while Marlin is secure in the AGM, additional elements need to be added to its SRS to formally prove certain security properties in the updatable CRS model. We demonstrate that the SV algorithms become inefficient for mid-sized circuits with over 20,000 multiplication gates and 100 updates. To address this, we introduce Batched SV algorithms (BSV) that leverage standard batching techniques and offer significantly improved performance. As a tool, we propose an efficient verification approach that allows the parties to identify a malicious SRS updater with logarithmic verification in the number of updates. In the case of Basilisk, for a circuit with $2^{20}$ multiplication gates, a $1000$-time updated SRS can be verified in less than 30 sec, a malicious updater can be identified in less than 4 min (improvable by pre-computation), and each update takes less than 6 min