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 125× Bitcoin's throughput, and incurs almost no penalty for scaling to more users. 1 INTRODUCTION Cryptographic currencies such as Bitcoin can enable new applications, such as smart contracts [24, 50] and fair protocols [2], can simplify currency conversions [12], and can avoid trusted centralized authorities that regulate transactions. However, current proposals suffer from a trade-off between latency and confidence in a transaction. For example, achieving a high confidence that a transaction has been
Let ES(n) denote the minimum natural number such that every set of ES(n) points in general position in the plane contains n points in convex position. In 1935, Erdős and Szekeres proved that ES(n) ≤ 2n−4 n−2 + 1. In 1961, they obtained the lower bound 2 n−2 + 1 ≤ ES(n), which they conjectured to be optimal. In this paper, we prove that2n − 4 n − 2 .
We introduce compact certificate schemes, which allow any party to take a large number of signatures on a message 𝑀, by many signers of different weights, and compress them to a much shorter certificate. This certificate convinces the verifiers that signers with sufficient total weight signed 𝑀, even though the verifier will not see-let alone verify-all of the signatures. Thus, for example, a compact certificate can be used to prove that parties who jointly have a sufficient total account balance have attested to a given block in a blockchain.After defining compact certificates, we demonstrate an efficient compact certificate scheme. We then show how to implement such a scheme in a decentralized setting over an unreliable network and in the presence of adversarial parties who wish to disrupt certificate creation. Our evaluation shows that compact certificates are 50-280× smaller and 300-4000× cheaper to verify than a natural baseline approach.
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