Ardor is a blockchain platform predicated on childchains (sidechains) that use proof of stake (PoS) consensus. It uses the primary chain as a security chain and the childchains for processing transactions to increase scalability. Their design is specifically focused on speed and efficiency through PoS consensus and removing blockchain bloat through pruning.
A private blockchain on the other hand provides only the owner to have the rights on any changes that have to be done. This could be seen as a similar version to the existing infrastructure wherein the owner (a centralized authority) would have the power to change the rules, revert transactions, etc. based on the need. This could be a concept with huge interest from FI’s and large companies. It could find use cases to build proprietary systems and reduce the costs, while at the same time increase their efficiency. Some of the examples could be:
“Further, contribution is weighted by computational power rather than one threshold signature contribution per party, which allows anonymous membership without risk of a Sybil attack (when one party joins many times and has disproportionate input into the signature). For this reason, the DMMS has also been described as a solution to the Byzantine Generals Problem[AJK05].”

In general, so far there has been little emphasis on the distinction between consortium blockchains and fully private blockchains, although it is important: the former provides a hybrid between the “low-trust” provided by public blockchains and the “single highly-trusted entity” model of private blockchains, whereas the latter can be more accurately described as a traditional centralized system with a degree of cryptographic auditability attached. However, to some degree there is good reason for the focus on consortium over private: the fundamental value of blockchains in a fully private context, aside from the replicated state machine functionality, is cryptographic authentication, and there is no reason to believe that the optimal format of such authentication provision should consist of a series of hash-linked data packets containing Merkle tree roots; generalized zero knowledge proof technology provides a much broader array of exciting possibilities about the kinds of cryptographic assurances that applications can provide their users. In general, I would even argue that generalized zero-knowledge-proofs are, in the corporate financial world, greatly underhyped compared to private blockchains.
I said above that you can build sophisticated rules into Bitcoin transactions to specify how ownership is proved. However, the Bitcoin scripting language is deliberately limited and many ideas in the Smart Contracts space are difficult or impossible to implement. So projects such as Ethereum are building an entirely new infrastructure to explore these ideas
The idea emerged that the Bitcoin blockchain could be in fact used for any kind of value transaction or any kind of agreement such as P2P insurance, P2P energy trading, P2P ride sharing, etc. Colored Coins and Mastercoin tried to solve that problem based on the Bitcoin Blockchain Protocol. The Ethereum project decided to create their own blockchain, with very different properties than Bitcoin, decoupling the smart contract layer from the core blockchain protocol, offering a radical new way to create online markets and programmable transactions known as Smart Contracts.

Every node in a decentralized system has a copy of the blockchain. Data quality is maintained by massive database replication[8] and computational trust. No centralized "official" copy exists and no user is "trusted" more than any other.[4] Transactions are broadcast to the network using software. Messages are delivered on a best-effort basis. Mining nodes validate transactions,[22] add them to the block they are building, and then broadcast the completed block to other nodes.[24]:ch. 08 Blockchains use various time-stamping schemes, such as proof-of-work, to serialize changes.[34] Alternative consensus methods include proof-of-stake.[22] Growth of a decentralized blockchain is accompanied by the risk of centralization because the computer resources required to process larger amounts of data become more expensive.[35]
Decentralized web. The sidechain technology holds premises to expand one of the main values of the blockchains – the decentralization of confidence. There is no need for central structure behind the transactions - the holders of cryptocurrencies are free to use their assets the way they want. The sidechains make their deals even more protected and reliable.
“We believe that public blockchains with censorship resistance have the potential to disrupt society, when private blockchains are merely a cost-efficiency tool for banking back offices. One can measure its potential in trillions of dollars, the other in billions. But as they are totally orthogonal, both can coexist in the same time, and therefore there is no need to oppose them as we can often see it.” 
Step back from the details for moment and consider what’s been described.  We now have a way to move coins from Bitcoin onto another platform (a sidechain) and move them back again.   That’s pretty much what we do when we move them to a wallet platform or an exchange.  The difference is that the “platform” they’ve been moved to is also a blockchain… so it has the possibility of decentralised security, visibility and to gain from other innovation in this space.
Further, despite sidechains being independent of each other, they are responsible for their individual security and need the requisite mining power to remain secure. Bitcoin’s blockchain has sufficient PoW mining power to remain secure even from the most coordinated of attacks, but many more nascent sidechains lack the necessary network effects and mining power to guarantee security to users.

The public blockchain is open to anyone who wants to deploy smart contracts and have their executions performed by public mining nodes. Bitcoin is one of the largest public blockchain networks today. As such, there is limited privacy in the public blockchain. Mining nodes in the public blockchain requires a substantial amount of computational power to maintain the distributed ledger at a large scale. In the Ethereum public blockchain, smart contract codes can be viewed openly.
A Sidechain, in simplest terms, is just a separate blockchain but is attached to the parent through the use of a two-way peg which allows for assets to be interchangeable and moved across the chain at a fixed deterministic exchange rate. This two-way peg works by utilizing simple payment verification or SPV as it's otherwise known. To show and prove ownership of the assets on the parent chain.
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