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And now for the second clever part. The logic above is symmetric. So, at any point, whoever is holding these coins on the sidechain can send them back to the Bitcoin network by creating a special transaction on the sidechain that immobilises the bitcoins on the sidechain. They’ll disappear from the sidechain and become available again on the Bitcoin network, under the control of whoever last owned them on the sidechain.
Public blockchains are just that, public. Anyone that wants to read, write, or join a public blockchain can do so. Public chains are decentralized meaning no one body has control over the network, ensuring the data can’t be changed once validated on the blockchain. Simply meaning, anyone, anywhere, can use a public blockchain to input transactions and data as long as they are connected to the network.
Sidechains with specific purposes could be formed with specific features while still enjoying the widespread adoption and value that Bitcoin holds.  Most importantly it can add these features without consensus from the Bitcoin community. Sidechains have the potential to replace many Cryptocurrencies as it allows features that were previously unique to these currencies to be available on Bitcoin. It also allows developers to experiment with sidechains and scope its full potential while still keeping coins linked to Bitcoin.

Elements Alpha functions as a sidechain to Bitcoin’s testnet, though the peg mechanism currently works through a centralized protocol adapter, as described in the Sidechains whitepaper. It relies on an auditable federation of signers to manage the testnet coins transferred into the sidechain via the “Deterministic Pegs” Element, and to produce blocks via the “Signed Blocks” Element. This makes it possible to immediately explore the new chain’s possibilities, using different security trade-offs. They plan to, in a later release, upgrade the protocol adapter to support fully decentralized merge-mining of the sidechain, and ultimately to phase in the full 2-way peg mechanism.

“Given all of this, it may seem like private blockchains are unquestionably a better choice for institutions. However, even in an institutional context, public blockchains still have a lot of value, and, in fact, this value lies to a substantial degree in the philosophical virtues that advocates of public blockchains have been promoting all along, among the chief of which are freedom, neutrality and openness.” 

In this case, you work directly with the given blockchain tools and stack. Assembly is required, so this isn’t for the faint of heart at this point, as many of the technologies are still developing and evolving. However, working directly with the blockchain provides a good degree of innovation, for example in building decentralized applications. This is where entrepreneurs are creating ambitious end-to-end, peer-to-peer applications, such as OpenBazaar (on Bitcoin), or Ujo Music (on Ethereum).
The paper outlines some critical developments and associated problems that were both currently trending and forward-thinking at the time, many of them still very much relevant today. At the time, altcoins were quickly gaining prominence and the problems associated with their volatility, security, and lack of interoperability with Bitcoin raised concerns. The paper primarily addressed 6 issues that pegged sidechains aimed to provide a solution:
Sidechains, just like any other Blockchain, need their own miners to help protect them from nefarious actors and attacks which people would like to leverage against the network. However, since wealth isn't actually created on the Sidechain there is far less incentive for miners to actually work on it and help protect it. Because of this, transaction fees are the basic reward that is offered to miners. However, these often equate to mere pennies.

In order to spend them, you have to prove you’re entitled to do so. And you do that by providing the solution to a challenge that was laid down when they were sent to you in the first place. This challenge is usually just: “prove to the world that you know the public key that corresponds to a particular Bitcoin address and are in possession of the corresponding private key”. But it can be more sophisticated than that.
@gendal I am discussing private chains with prospects, so my interest is not superficial and theoretical. I see the benefits for the organization in using the private chain as another form of internal database, with better security properties. It can also be used where a service bus product would be today, to facilitate integration, conformance, monitoring, audit. Private chain can also, via a two way peg, be connected to the main chain, achieving a form of public/private network divide that routers created for us in the early stages of the Internet development. Anything else on the benefits side that I missed?
This is justified by observing that, in our pre-sidechain world, miners always want things to be correct. In theory, the incentives of miners and investors are very strongly aligned: both are compensated most when the exchange rate is highest. And, in practice, we do not see large reorganizations (where miners can “steal”, by first depositing BTC to major exchanges, then selling that BTC for fiat (which they withdraw), and finally rewriting the last 3 or 4 days of chain history, to un-confirm the original deposits). These reorgs would devastate the exchange rate, as they would cast doubt on the entire Bitcoin experiment. The thesis of Drivechain is that sidechain-theft would also devastate the exchange rate, as it would cast doubt on the entire sidechain experiment (which would itself cast doubt on the Bitcoin experiment, given the anti-competitive power of sidechains).
Sidechain transactions using a two-way peg effectively only allow for intra-chain transactions. A transfer from Bitcoin (parent chain) to Ethereum (sidechain) would allow a user to use the functionality of Ethereum (i.e., fully expressive smart contracts), but the underlying original asset would remain precisely that, Bitcoin. So, a Bitcoin on an Ethereum sidechain technically remains a Bitcoin.

“Such a move could allow retailers to lower prices and incentivize consumers to shop at one retailer over a competitor,” Cohen noted. “This idea is not as ludicrous as it might seem. Amazon recently registered three cryptocurrency-related domain names, suggesting a potential move into the cryptocurrency space. If large companies like Amazon, Walmart or Starbucks issued digital coins that inspired public trust, blockchain-based cryptocurrencies might gain acceptance by the public and other retail giants.”

@mowliv I think a good way to think about it is by looking at our economy. The Federal Reserve prints US dollars for the US Government (the main blockchain) to boost the US economy. However, US dollars can be exported to other countries (a side chain) that could have a completely independent economy but still use a currency backed by the US government. – Olshansk May 30 '17 at 0:56
The second option will be to use sidechains. Blockstream first announced side chain in 2014 and published its whitepaper (https://blockstream.com/sidechai...). I believe in the future, bitcoin will have its desired flexibility with its sidechains. The idea of the sidechain is you can innovate and design your solution freely in the sidechains. These sidechains are independent, if they are failed or hacked, they won't damage other chains. So damage will be limited within that chain, for that reason you can be less conservative. Otherwise you would be more risk averse, if you had 42.5 billion dollar market cap like Bitcoin.
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