At the heart of the Bitcoin system is a computational process called mining, in which transaction records are added to t

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At The Heart Of The Bitcoin System Is A Computational Process Called Mining In Which Transaction Records Are Added To T 1 (93.24 KiB) Viewed 79 times

At the heart of the Bitcoin system is a computational process called mining, in which transaction records are added to the blockchain in units called blocks via the repeated solution of a computationally-difficult cryptographic problem. Miners receive a reward every time they successfully solve this problem. Currently, this reward is set at 6.25 bitcoins plus any (voluntary) fees that are paid within the transactions. To explain the mining process further, it is necessary to have a brief discussion of how data is stored in computers and how this data can be operated on by a hash function Computers store data in binary (or base 2) form, with each number consisting of a string of ones and zeros. The binary numbers of relevance to Bitcoin are of length 256. Reverting to our normal base 10 counting system, there are approximately 1.16 x 1077 binary numbers of length 256. The important thing about a hash function is that it is easy to calculate its value at a given input, but it is impossible in a practical sense) to reverse the calculation and produce the input if you know only the output number. The process of bitcoin mining involves applying a hash function to the concate- nation of three binary numbers: • the hash number of the block at the head of the current blockchain, • a hash number that summarises information about the transactions that the miner wants to include in the current block, • a random number, called the nonce.
(a) The hash function h used in Bitcoin maps any set of data stored in binary form to an integer in the set (0, 2256 – 1]. Let the random variable X denote the output of h when it is applied to some arbitrary input data. What type of distribution should give a good model for X? What are its parameters?
(b) What type of random experiment is a miner performing when it applies h to the concatenation of the three numbers described above and observes whether the result is less than the threshold l?
(c) Define an appropriate random variable to describe the outcome of this exp ment and give its distribution, including defining any necessary parameter
(d) If the output of the hash function is greater than the threshold l, a miner adjusts the nonce and tries again, thus performing an independent repeat of the random experiment in (b). If the output of the hash function is still greater than l, the miner again adjusts the nonce and tries one more, and so on. Let the random variable N denote the number of times that the miner has to repeat the experiment in (b) before observing an output value that is less than l. Name the distribution of the random variable N and give the values of its parameter(s).
(e) What are the mean and variance of N? (f) Bitcoin miners around the world are currently performing evaluations of the hash function at a total rate of 5/3 x 1018 per second, which means that they perform 6 x 1021 hash function evaluations in an hour. Let M be the number of hash evaluations less than the threshold l that occur in 6 x 1021 hash function calculations. Name the distribution of the random variable M and give the values of its parameter(s).
(g) What are the mean and variance of M? (h) In the original white paper [3] that defined the Bitcoin system, Satoshi Nakamoto (whose name is a pseudonym) recommended that a vendor who has just ac- cepted payment in Bitcoins for some goods should wait until the transaction has been incorporated in a block and then that block and five more are mined before despatching the goods. Let R be the number of times that the random experiment in (b) has to be performed before six blocks are mined worldwide. Name the distribution of the random variable R, give the values of its parameter(s) and specify its mean and variance.