Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

Emerging smart contract systems over decentralized cryptocurrencies allow mutually distrustful parties to transact safely without trusted third parties. In the event of contractual breaches or aborts, the decentralized blockchain ensures that honest parties obtain commensurate compensation. Existing systems, however, lack transactional privacy. All transactions, including flow of money between pseudonyms and amount transacted, are exposed on the blockchain. We present Hawk, a decentralized smart contract system that does not store financial transactions in the clear on the blockchain, thus retaining transactional privacy from the public's view. A Hawk programmer can write a private smart contract in an intuitive manner without having to implement cryptography, and our compiler automatically generates an efficient cryptographic protocol where contractual parties interact with the blockchain, using cryptographic primitives such as zero-knowledge proofs. To formally define and reason about the security of our protocols, we are the first to formalize the blockchain model of cryptography. The formal modeling is of independent interest. We advocate the community to adopt such a formal model when designing applications atop decentralized blockchains.

https://www.the-blockchain.com/docs/Hawk%20-%20The%20Blockchain%20Model%20of%20Cryptography%20and%20Privacy-Preserving%20Smart%20Contracts.pdf

Hawk: The Blockchain Model of Cryptography and Privacy-Preserving Smart Contracts

Besides attracting a billion dollar economy, Bitcoin revolutionized the field of digital currencies and influenced many adjacent areas. This also induced significant scientific interest. In this survey, we unroll and structure the manyfold results and research directions. We start by introducing the Bitcoin protocol and its building blocks. From there we continue to explore the design space by discussing existing contributions and results. In the process, we deduce the fundamental structures and insights at the core of the Bitcoin protocol and its applications. As we show and discuss, many key ideas are likewise applicable in various other fields, so that their impact reaches far beyond Bitcoin itself.

Bitcoin and Beyond: A Technical Survey on Decentralized Digital Currencies

Bitcoin is the first and most popular decentralized cryptocurrency to date. In this work, we extract and analyze the core of the Bitcoin protocol, which we term the Bitcoin backbone, and prove two of its fundamental properties which we call common prefix and chain quality in the static setting where the number of players remains fixed. Our proofs hinge on appropriate and novel assumptions on the “hashing power” of the adversary relative to network synchronicity; we show our results to be tight under high synchronization.

https://courses.cs.washington.edu/courses/cse454/15wi/papers/bitcoin-765.pdf

The Bitcoin Backbone Protocol: Analysis and Applications

Algorand is a new cryptocurrency that confirms transactions with latency on the order of a minute while scaling to many users. Algorand ensures that users never have divergent views of confirmed transactions, even if some of the users are malicious and the network is temporarily partitioned. In contrast, existing cryptocurrencies allow for temporary forks and therefore require a long time, on the order of an hour, to confirm transactions with high confidence. Algorand uses a new Byzantine Agreement (BA) protocol to reach consensus among users on the next set of transactions. To scale the consensus to many users, Algorand uses a novel mechanism based on Verifiable Random Functions that allows users to privately check whether they are selected to participate in the BA to agree on the next set of transactions, and to include a proof of their selection in their network messages. In Algorand's BA protocol, users do not keep any private state except for their private keys, which allows Algorand to replace participants immediately after they send a message. This mitigates targeted attacks on chosen participants after their identity is revealed. We implement Algorand and evaluate its performance on 1,000 EC2 virtual machines, simulating up to 500,000 users. Experimental results show that Algorand confirms transactions in under a minute, achieves 125x Bitcoin's throughput, and incurs almost no penalty for scaling to more users.

https://dl.acm.org/doi/pdf/10.1145/3132747.3132757

Algorand: Scaling Byzantine Agreements for Cryptocurrencies

We study a model of fairness in secure computation in which an adversarial party that aborts on receiving output is forced to pay a mutually predefined monetary penalty. We then show how the Bitcoin network can be used to achieve the above notion of fairness in the two-party as well as the multiparty setting (with a dishonest majority). In particular, we propose new ideal functionalities and protocols for fair secure computation and fair lottery in this model.

/pdf/how-to-use-bitcoin-to-design-fair-protocols-4p4i057g1r.pdf

How to Use Bitcoin to Design Fair Protocols

We describe a lightweight protocol for oblivious evaluation of a pseudorandom function (OPRF) in the presence of semihonest adversaries. In an OPRF protocol a receiver has an input r; the sender gets output s and the receiver gets output F(s; r), where F is a pseudorandom function and s is a random seed. Our protocol uses a novel adaptation of 1-out-of-2 OT-extension protocols, and is particularly efficient when used to generate a large batch of OPRF instances. The cost to realize m OPRF instances is roughly the cost to realize 3:5m instances of standard 1-out-of-2 OTs (using state-of-the-art OT extension). We explore in detail our protocol's application to semihonest secure private set intersection (PSI). The fastest state-of- the-art PSI protocol (Pinkas et al., Usenix 2015) is based on efficient OT extension. We observe that our OPRF can be used to remove their PSI protocol's dependence on the bit-length of the parties' items. We implemented both PSI protocol variants and found ours to be 3.1{3.6 faster than Pinkas et al. for PSI of 128-bit strings and sufficiently large sets. Concretely, ours requires only 3.8 seconds to securely compute the intersection of 220-size sets, regardless of the bitlength of the items. For very large sets, our protocol is only 4:3 slower than the insecure naive hashing approach for PSI.

/pdf/efficient-batched-oblivious-prf-with-applications-to-private-3qpry2p4kg.pdf

Efficient Batched Oblivious PRF with Applications to Private Set Intersection

We study a model of incentivizing correct computations in a variety of cryptographic tasks. For each of these tasks we propose a formal model and design protocols satisfying our model's constraints in a hybrid model where parties have access to special ideal functionalities that enable monetary transactions. We summarize our results: Verifiable computation. We consider a setting where a delegator outsources computation to a worker who expects to get paid in return for delivering correct outputs. We design protocols that compile both public and private verification schemes to support incentivizations described above. Secure computation with restricted leakage. Building on the recent work of Huang et al. (Security and Privacy 2012), we show an efficient secure computation protocol that monetarily penalizes an adversary that attempts to learn one bit of information but gets detected in the process. Fair secure computation. Inspired by recent work, we consider a model of secure computation where a party that aborts after learning the output is monetarily penalized. We then propose an ideal transaction functionality FML and show a constant-round realization on the Bitcoin network. Then, in the FML-hybrid world we design a constant round protocol for secure computation in this model. Noninteractive bounties. We provide formal definitions and candidate realizations of noninteractive bounty mechanisms on the Bitcoin network which (1) allow a bounty maker to place a bounty for the solution of a hard problem by sending a single message, and (2) allow a bounty collector (unknown at the time of bounty creation) with the solution to claim the bounty, while (3) ensuring that the bounty maker can learn the solution whenever its bounty is collected, and (4) preventing malicious eavesdropping parties from both claiming the bounty as well as learning the solution. All our protocol realizations (except those realizing fair secure computation) rely on a special ideal functionality that is not currently supported in Bitcoin due to limitations imposed on Bitcoin scripts. Motivated by this, we propose validation complexity of a protocol, a formal complexity measure that captures the amount of computational effort required to validate Bitcoin transactions required to implement it in Bitcoin. Our protocols are also designed to take advantage of optimistic scenarios where participating parties behave honestly.

/pdf/how-to-use-bitcoin-to-incentivize-correct-computations-2jxs8vyg4f.pdf

How to Use Bitcoin to Incentivize Correct Computations

Bitcoin, Ethereum and other blockchain-based cryptocurrencies, as deployed today, cannot support more than several transactions per second. Off-chain payment channels, a “layer 2” solution, are a leading approach for cryptocurrency scaling. They enable two mutually distrustful parties to rapidly send payments between each other and can be linked together to form a payment network, such that payments between any two parties can be routed through the network along a path that connects them.

Sprites and State Channels: Payment Networks that Go Faster Than Lightning

It is well known that Bitcoin, Ethereum, and other blockchain-based cryptocurrencies are facing hurdles in scaling to meet user demand. One of the most promising approaches is to form a network of “off-chain payment channels,” which are backed by on-chain currency but support rapid, optimistic transactions and use the blockchain only in case of disputes. We develop a novel construction for payment channels that reduces the worst-case “collateral cost” for offchain payments. In existing proposals, particularly the Lightning Network, a payment across a path of ` channels requires locking up collateral for O(`∆) time, where ∆ is the time to commit a on-chain transaction. Our construction reduces this cost to O(`+∆). We formalize our construction in the simulation-based security model, and provide an implementation as an Ethereum smart contract. Our construction relies on a general purpose primitive called a “state channel,” which is of independent interest.

Ranjit Kumaresan

Papers

How to Use Bitcoin to Design Fair Protocols

Efficient Batched Oblivious PRF with Applications to Private Set Intersection

How to Use Bitcoin to Incentivize Correct Computations

Sprites and State Channels: Payment Networks that Go Faster Than Lightning

Sprites: Payment Channels that Go Faster than Lightning.