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When parties can trust each other there is no need for the inefficiencies associated with public blockchains in the form of mining and solving the "double spend" problem.
(This occurs if two transactions attempt to spend the the same output, only one of those transactions will be accepted.) Without mining one can just validate the transactions and add to the chain by creating hash functions regardless and forming blocks.
At one extreme of the scale (strictness of zero), any block-adder can add any block meaning it’s very tolerant but also increases the risk that a single block-adder or small group of block-adders can spoof the system.
a small group of entities pretending to be a large group of entities who agree on something in order to spoof the system.
As opposed to a public blockchain network, the transaction validators in a private blockchain are not incentivized in the form of tokens (money) but in having the benefit of being a part of the ledger and being able to read data they consider valuable. What's my threat model for why I might want "consensus" or validation? "These transaction validators play a critical role in the success of the blockchain as they have the ability to "write" to the ledger and send out confirmations of the transactions.
This post will explore this issue further as perhaps there is a role for disinterested/neutral parties to be involved as the transaction validators so there is no conflict of interest. So the first goal is to step back and say: Who needs to see the data? They provide a unique record of truth from which all the parties act.
Most of this information is not public knowledge yet.
However, a few companies have shared how they are doing it and I will list a few examples of such.