To make the constraints on τ step and Tstep precise let us denote the number of

To make the constraints on τ step and tstep precise

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To make the constraints on τ step and T step precise, let us denote the number of honest committee members by д and the malicious ones by b ; in expectation, b + д = τ step , but b + д can vary since it is chosen by sortition. To ensure liveness, as we prove in Appendix C.2 of the technical report [ 27 ], BA requires 1 2 д + b T step · τ step and д > T step · τ step . Due to the probabilistic nature of how committee members are chosen, there is always some small chance that the b and д for some step fail to satisfy the above constraints, and BA ’s goal is to make this probability negligible. Figure 3 plots the expected committee size τ step that is needed to satisfy both constraints, as a function of h , for a probability of violation of 5 × 10 9 ; Appendix B of the technical report [ 27 ] describes this computation in more detail. The figure shows a trade-off: the weaker the assumption on the fraction of money held by honest users ( h ), the larger the committee size needs to be. The results show that, as h approaches 2 3 , the committee size grows quickly. However, at h = 80% , τ step = 2 , 000 can ensure that these constraints hold with probability 1 5 × 10 9 (using T step = 0.685). 11
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76 78 80 82 84 86 88 90 % of Honest Users 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Committee Size 5·10^-9 Figure 3 : The committee size, τ , sufficient to limit the proba- bility of violating safety to 5 × 10 9 . The x-axis specifies h , the weighted fraction of honest users. marks the parameters selected in our implementation. The constraints on τ final and T final are dictated by the proof of safety under weak synchrony; Appendix C.1 of the technical report [ 27 ] shows that τ final = 10 , 000 suffices with T final = 0 . 74 . With these parameters, BA ensures safety even if the lowest-priority block proposer is malicious (proposes differ- ent blocks). Appendix C of the technical report [ 27 ] provides proofs of BA ’s safety under weak synchrony (§C.1), liveness under strong synchrony (§C.2), and efficiency (§C.3). 8 ALGORAND Building Algorand on top of the primitives we have described so far requires Algorand to address a number of higher-level issues, which this section discusses. 8.1 Block format Algorand’s blocks consist of a list of transactions, along with metadata needed by BA . Specifically, the metadata consists of the round number, the proposer’s VRF-based seed (§6), a hash of the previous block in the ledger, and a timestamp indicating when the block was proposed. The list of transactions in a block logically translates to a set of weights for each user’s public key (based on the balance of currency for that key), along with the total weight of all outstanding currency. Once a user receives a block from the highest-priority pro- poser, the user validates the block contents before passing it on to BA . In particular, the user checks that all transactions are valid; that the seed is valid; that the previous block hash is correct; that the block round number is correct; and that the timestamp is greater than that of the previous block and also approximately current (say, within an hour). If any of them are incorrect, the user passes an empty block to BA .
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  • Spring '19
  • NA
  • hash function, Cryptographic hash function, Algorand

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