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I realise that somebody who manages to find a new block in the blockchain gets 25 bitcoins as a "reward".
Where do these bitcoins actually come from? How are they minted? How are they actually generated? What's the "everyman's" explanation of what happens, in the blockchain, so that these "new" 25 bitcoins are actually created?
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The 25 BTC are actually generated out of thin-air. Essentially every peer in the network knows to accept blocks where the first transaction in every block should be of maximum 25 BTC. If it's more, then the block is considered invalid. Note that these are the coins given to the miner since the miner is the one creating the block and can therefore set to which address these 25 BTC go to. Also note that 25 is not an hardcoded value, it changes with time. Again, there's another rule "programmed" into each peer where they can calculate the current reward. Currently the maximum allowed is 25, but every certain amounts of blocks, peers know that the value needs to be halved - it was originally 50 BTC, now it's 25, next it will be 12.5.
So to answer your question, peers collectively know that the rule is to generate coins out of thin air every so often, and the person that gets them is the person that mined the block. You might be asking: what if a dishonest peer changes the 25 BTC transaction to go to himself rather than the miner. They can't do this because they would need to re-mine that block (changing contents of a block also changes its hash).
It is actually allowed to create a block which claims less reward than allowed. In that case, the coins are just lost. Nobody does this of course, as it is silly. But not invalid. – Pieter Wuille – 2015-06-14T10:15:17.477
Blocks with lower than maximum rewards do exist though. – Anonymous – 2015-06-14T10:27:02.693
ah sorry you're right. edited to say "should be of maximum 25 BTC" – Luca Matteis – 2015-06-14T10:28:15.783
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When somebody is trying to find a block, they ratify a set of transactions.
The protocol has a rule asserting that this transaction set must include a Coinbase Transaction as the first transaction. The Coinbase Transaction is a transaction that sends money from "nowhere" to the miners address. Its output is limited to be smaller or equal to the sum of the transaction fees and the current block reward (currently 25 BTC).
As every mining entity is working on their individual data set, including their own coinbase transaction, whoever finds the block automatically creates the new coins and credits them to their own address.
possible duplicate: How exactly do new bitcoins come into existence?, Who issues bitcoins and how?, related: How do Bitcoin clients know how many new bitcoins to mint when creating a block?, What is the coinbase?
– Murch – 2015-06-14T13:31:03.027I've changed "after" in the title to "when" because generating the reward is an effect of finding a block. – Murch – 2015-06-14T16:18:56.463