Proof of work
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A proof-of-work is the verifiable result obtained through a given work. Anyone should be able to make sure that a given share of work was accomplished.
This is used in bitcoin for blocks generation. The block hash itself is a proof-of-work, proving whoever generated that block deserves the associated transaction fees and reward.
Example
Let's say our base work is "Hello world" and our work is to find a hash starting with "000". We are going to increase a value hashes alongside "Hello world" (nonce) until we reach our goal.
Hello world ! 0 : 3f6fc92516327a1cc4d3dca5ab2b27aeedf2d459a77fa06fd3c6b19fb609106a Hello world ! 1 : b5690c48c2d0a09481186aaa99e4e090901ff2ac4d572e6706dfd30eefc22a27 Hello world ! 2 : 5b6fd9c27fcb54ca23404d9428f081b7c9280ba6370e33a6a20b16f40ce76320 Hello world ! 3 : 9c5d769416aa0ca894abf22bd17bd30fbb6959291423ae1903a9f86a1fe7ce78 Hello world ! 4 : 4efc65df7933e4f5cc21947c61d5cc6bd11d644794bfa210603b0547c4b1cc3e Hello world ! 5 : 441b15b67d791620cd50ea537144e3115422e33bdb1b1b9b110d3265f7a9199b Hello world ! 6 : d368331386f0cf773ad53910fefcef4bdceeb526e408d3fbc9408d6f6e481ca4 Hello world ! 7 : 013cc9722f38d2eb6186b75e2e7cbe6e7818e0612a2774d4400416b17ae03b87 Hello world ! 8 : 3a92631799b478c3bcc554df8401b09900fbdb58cc0e58efe711cc475ee097b3 Hello world ! 9 : 66658881696164fcb04f32ec505bb5e515000a85baf691beb63fc9d3f4d0fee2 .... Hello world ! 88 : 80d009db72c6ad35241bb3dbac77cbe177c6a803fe67527c159dbfaf2cbf9f5c Hello world ! 89 : a5b1e789f691f9793f8a84f8ebae3d8e28d49cbe0eeea2da621cd409e3bdee2b Hello world ! 90 : 4eba5b2459caac3d9ff3b787aaa5cac481aaa4a0232fbbe02a8ee4d1101c2ca2 Hello world ! 91 : c811722c68b53614d58d37dcad9d540c2bce9f85b5ccae94424ff4716eea1765 Hello world ! 92 : e30c716fccda22f394a8e80a2670b97968b5416b8b39e2061a7b7d1a9f41e0a9 Hello world ! 93 : 965425c39d4e24c532721d7f7b77a00b31b0c0d0e316d46240c4e6bec9c09f65 Hello world ! 94 : 7090a0e5d88cff635e42ea33fcd6091a058e9cdd58ab8cd5c21c1c70421e35c6 Hello world ! 95 : b74f3b2cf1061895f880a99d1d0249a8cedf223d3ed061150548aa6212c88d43 Hello world ! 96 : 447ca2fa886965af084808d22116edde4383cbaa16fd1fbcf3db61421b9990b9 Hello world ! 97 : 000ba61ca46d1d317684925a0ef070e30193ff5fa6124aff76f513d96f49349d
And here we are. Now we just have to let the network know our block, made of "Hello world" and 97, won. In bitcoin things are a bit more complex, especially since the header contains the merkle hash which depends on the included transactions, which includes the transaction to ourself (which uses a different address each time, thru making the header unique).