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D1 but not D2. To distinguish between blocks generated using a small subset of miners from those using all network miners, the target value in the header of each block will be verified. The mining in ρ2 is similar to the mining in the original PoW with the exception that only miners in the correspond- ing M2 can participate, and use the second mining difficulty D2. When a valid block is formed and propagated to the entire network, as shown in Fig. 3, the other active miners in M2 stop the mining of the current block and start new mining epoch. The passive miners (M \ M2) also leave their mining-save mode and join the others in a new epoch. wins the mining race in ρ1 is not allowed to participate in ρ2, and consequently gives the other nodes the chance to win with less competition. The only case when it is still possible that a winner in ρ1 also wins in ρ2, is after timing out ρ2. To better illustrate the impact of Green-PoW on mining shares distribution, we have imported the latest 7437 blocks (50 days) from the Ethereum main network. We plot in Fig. 4 the corresponding shares of each miner as the ratio of its mined blocks relative to the total block count, i.e., 7437. In the same plot, we also include the corresponding shares of each miner when applying Green-PoW to the same blocks. In Green-PoW, two, three, four, or more consecutive blocks are highly improbable to be mined by the same miner. It is worth mentioning that two consecutive blocks could be gen- erated by the same miner under a special condition, where a miner wins the second round block and the next block in the first round of the following epoch. For this reason, we subtracted one block from only 50% of two consecutive blocks cases. As shown in Fig. 4, the corresponding shares of the most powerful miners, i.e., miner numbers 28 and 29, in Green-PoW are reduced, compared to the case of the original PoW. Such an impact limits the dominance that the most powerful miners may have on the network. For example, the most powerful miner had a 5% reduction in its share, which is redistributed among other nodes. The reduction can be more significant if the computing power varies widely among the nodes. Subsequently, better share distribution between miners could be achieved. 5.3 Fork Occurrences As discussed in Section 3.1.3, a fork in blockchain can happen when multiple miners find a block almost simulta- neously. More generally, a fork occurs whenever a miner mj finds a block while another miner mi has already formed a valid block without being aware of it. This situation is likely to occur in a large network having a long propagation delay. To illustrate the effect of propagation delay on the fork occurrences, Decker and Wattenhofer [24] have conducted a theoretical study and presented an approximate model to predict the rate at which forks can occur. For a newly found block bi by mi, the probability of fork occurrence, i.e., conflicting blocks will be found by other miners before being aware of bi, is estimated by (i) determining the number of unaware miners at time t, and (ii) the probability that each unaware miner will find a conflicting block during that time. Given a ratio of unaware miners u(t) about bi during time t, the probability of having a fork (F ) on the network can be expressed as follow: 0∞ u(t) dt Pr[F >0]=1−(1−Pb) (5) Where Pb is the probability of a block being found by the network at a given time t. From the formula, it is clear that the fork rate is proportional to the ratio of unaware miners and thus proportional to the total number of miners in the network. In Green-PoW, during the second round, the ratio of active miners is very small compared to the first round. Subsequently, the ratio of unaware miners u is also very small. 5 5.1 SECURITY ANALYSIS Transaction Censorship The potential of transaction censorship exists in public blockchains because of the limited size of the block and the asynchronous nature of the network, which makes it difficult to verify which set of transactions a particular block must include [18], [22]. Usually, miners select only a subset of transactions from their pool of pending transactions to not exceed the block-size limit and tend to prioritize transactions with higher fees to maximize their profit. Such freedom on selecting transactions to include in a block gives adversary miners the opportunity to censor some transactions from being added to the next block, even those with high fees. By design, PoW extenuates this concern because the cen- sorship time tc is bounded by the average block generation time 1/λ, and restricted by the fact that a malicious miner mc must be the winner. However, a powerful attacker that may successfully mine and win k consecutive blocks will delay the inclusion of some urgent transactions for a longer time, i.e., tc = k/λ. In Green-PoW, a malicious miner mc can still censor transactions for tc during ρ1, yet because the winner miner in ρ1 cannot participate in ρ2, the effect of such an attack is limited to only one mining round. As a result, Green-PoW can reduce the censorship time to nearly 50%, which guarantees users better transaction time and reduces the intensity of a potential denial of service attack launched by powerful miners. The only case when a winner in ρ1 can also participate and win in ρ2, is after timing out ρ2. However, this can only happen under specific conditions, such as separating the set M2 from the rest of the network, which is difficult to be controlled by an attacker. 5.2 Mining Centralization In PoW-based networks, such as Bitcoin or Ethereum, the mining power is concentrated among a relatively small number of miners (pools) which makes the crypto-system highly susceptible to censorship or even 51% attacks [23]. Less powerful miners are usually unfortunate to generate a new valid block in the presence of other superior miners. This will eventually lead to a monopoly based system, where a small percentage of the network earns the highest rewarding shares. By design, Green-PoW can reduce the monopoly of powerful miners, since generating consecutive blocks by the same miner is likely not possible. A miner that 7PDF Image | Efficient Blockchain Proof-of-Work Consensus Algorithm
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