Note: Some would argue that such a system cannot be defined as a blockchain. Also, Blockchain is still in it’s early stages. It is unclear how the technology will pan out and will be adopted. Many argue that private or federated Blockchains might suffer the fate of Intranets in the 1990’s, when private companies built their own private LANs or WANs instead of using the public Internet and all the services, but has more or less become obsolete especially with the advent of SAAS in the Web2.

It may sound nitpicky, but I think that description leaves something to be desired in terms of presenting the “correct” mental model. First, there is no such thing as “a” bitcoin, as I am sure the author would agree. Speaking of spending or moving bitcoins perpetuates the notion of bitcoins as “things”. It might be preferable to say that you are spending or moving “units of the bitcoin protocol”. There is something similar going on here with dollars. The dollars in your bank account aren’t things either, they are units of demand or claim on a currency. The fact that printed dollars have serial numbers tends to confuse this notion. Treating something as a “thing’ which is not a thing is sometimes referred to as the reification fallacy.

A typical use case for a private blockchain is intra-business: when a company decides to implement blockchain as a business solution, they may opt for a chain to which only company members have access. This is useful if there’s no need for anybody outside of the company to become part of the chain, because private blockchains are more efficient than public and consortium chains. Also, because they are smaller and contained, it is easier for a consensus process or other technical stipulation to be altered on a blockchain. So, for example, if the developers or proprietors want to change the cryptographic method which runs its consensus process, it is much easier to do this on a private blockchain than a public or consortium chain.

A consortium blockchain is part public, part private. This split works at the level of the consensus process: on a consortium chain, a pre-selected group of nodes control the consensus process, but other nodes may be allowed to participate in creating new transactions and/or reviewing it. The specific configuration of each consortium chain (i.e., which nodes have the power to authorize transactions via the consensus process, which can review the history of the chain, which can create new transactions, and more) is the decision of each individual consortium.
“Blockchain could significantly reduce time delays and human mistakes, and monitor cost, labor, waste and emissions at every point in the supply chain. In the food sector, a manufacturer could automatically identify contaminated products in a matter of seconds and wouldn’t need to pull an entire product line from store shelves in the case of contamination.”
In some cases, these advantages are unneeded, but in others they are quite powerful - powerful enough to be worth 3x longer confirmation times and paying $0.03 for a transaction (or, once scalability technology comes into play, $0.0003 for a transaction). Note that by creating privately administered smart contracts on public blockchains, or cross-chain exchange layers between public and private blockchains, one can achieve many kinds of hybrid combinations of these properties. The solution that is optimal for a particular industry depends very heavily on what your exact industry is. In some cases, public is clearly better; in others, some degree of private control is simply necessary. As is often the case in the real world, it depends.

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A private blockchain network requires an invitation and must be validated by either the network starter or by a set of rules put in place by the network starter. Businesses who set up a private blockchain, will generally set up a permissioned network. This places restrictions on who is allowed to participate in the network, and only in certain transactions. Participants need to obtain an invitation or permission to join. The access control mechanism could vary: existing participants could decide future entrants; a regulatory authority could issue licenses for participation; or a consortium could make the decisions instead. Once an entity has joined the network, it will play a role in maintaining the blockchain in a decentralized manner.
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