The system is completely open source, which means anyone can view, suggest changes to their codebase. You can also read more about blockchain via https://www.supercolony.net/.
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Although it has become increasingly popular thanks to the growth of Bitcoin, it has actually been around since 2008, making it about a decade old (computationally ancient).
The most important thing about blockchain is that it is designed to build applications that not require central data processing services. This means that if you use a system based on it (i.e. Bitcoin), your data will be stored on 1000's of "independent" servers around the world (which no central service has).
The way this service works is by creating a "book". This book allows users to create "transactions" with each other – by storing the contents of those transactions in new "blocks" in each "blockchain" database.
Depending on the application making the transactions, they must be encrypted with a different algorithm. Since this encryption uses cryptography to "encrypt" the data stored in each new "block", the term "crypto" means the process of cryptographic delivery of new blockchain data that an application may create.
Of course, if we remove the problem with the basic idea of Bitcoin, etc., the basis of the service is hat it is basically a system that works on a network of processing machines (called "miners"). They all run blockchain software – and work to "compile" new transactions into "blocks" that keep the Bitcoin database up to date.