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10 Difficulty Over Time 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 10 Bitcoin exchanges, we compare the monetary cost of the energy to the reward for calculating a Bit- coin block. We also consider the likely power con- sumption of the whole Bitcoin mining operation, and show that it is comparable to Ireland’s average electricity consumption. II Bitcoin Mining As we mentioned, a Bitcoin miner is part of Bit- coin’s peer-to-peer network that collects recent transactions and aims to complete a proof of work scheme, based on the ideas of Hashcash[5]. In this scheme, there is a current target value T , which is periodically recalculated by the network (see Sec- tion II.a)). The miner’s aim is to find a nonce value so that H(B.N) < T (1) where B is the string representing the recent trans- actions, N is the nonce value, ‘.’ is the concatena- tion operator and H is the Bitcoin hash function, in this case H(S) := SHA256(SHA256(S)). The proof of work can be achieved by choosing values for N randomly or systematically until eq.1 is satisfied. When an N is found, the resulting block can be sent to the Bitcoin network and added to the Bitcoin blockchain. Finding a block results in a reward of extra Bitcoins for the block’s finder. Thus, the process of finding a suitable N value is referred to as Bitcoin mining. II.a) Difficulty The rate at which Bitcoins can be discovered can be controlled by the Bitcoin Network’s choice of the value of the target, T, in eq.1. However, the target depends on the current number and speed of miners in the Bitcoin network, and is normally quoted in terms of the difficulty, D. The relation- ship between the difficulty and the target T is Tmax D=T where the largest possible value of the target Tmax is (216 − 1)2208 ≈ 2224. The hash function H for Bitcoin has been chosen so that it behaves approximately as a uniformly random value between 0 and 2256−1. Thus, for any given nonce value, the probability of it satisfying eq.1 is p= T =Tmax≈ 1 . 2256 D2256 D232 Each nonce value tested should behave like an in- dependent trial, so the number of trials until a Fig. 1: The change of the difficulty to generate a Bitcoin over time, based on aggregated statistics [6]. block is successfully completed will be geometri- cally distributed, therefore the the expected num- ber of hashes to find a block is D232. If we have a system calculating hashes at a rate R, the expected time to find a block is 1 D232 E[t]=p≈ R . (2) For example, if you can calculate a Bitcoin hash 1 million times a second, and the difficulty is 4, 250, 217, 9201, then E[t] ≈ 1.8 × 1013s. II.b) Change in Difficulty The difficulty, D, is recalculated every 2016 blocks, with the aim of keeping the average time to dis- cover a new block near 10 minutes. At this ideal speed, 2016 blocks will be discovered every two weeks. To calculate the new difficulty, the length of time that it took to calculate the the last 2016 blocks is used to estimate the hash rate of the en- tire Bitcoin network. The new difficulty is selected so that if the same average hash rate is maintained, it will take two weeks to calculate the next 2016 blocks. If the resulting difficulty is more than four times harder (or four times easier) than the cur- rent difficulty, then the result is capped to four times harder (or easier). Restrictions on the range of acceptable difficulties/targets are also applied. The historical values of difficulty to date are shown in Figure 1. The increasing trend in difficulty has been caused by an increase in the resources dedi- cated to calculating hashes in the Bitcoin network. II.c) Change in Reward There are two sources of reward for calculating a new block. First, the block is formed from Bit- coin transactions, and a transaction may choose to include a transaction fee, to be paid to who- ever finds a block containing this transaction. Sec- 1Current as of mid March 2014. Difficulty 2010 2011 2012 2013 2014PDF Image | Bitcoin Mining and its Energy Footprint
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